# Notes: what a full formalization of the paper would take

This file records, for each unformalized input of `MM.moonshine_conjecture`, what the paper
proves, what Mathlib has, and what would have to be built.  It is written so that work can be
resumed at any point.

> **Role in the documentation.**  `README.md` is the project description and master index,
> `ROADMAP.md` the plan, `HANDOVER.md` the handover manual for the long-running FLM Theorem 8.10.2
> campaign, `AGENTS.md` the conventions.  **This file is the work log**: what each batch actually
> did and what it taught --- corrections of record, refuted spellings, traps --- in the topic-wise
> sections (`§§64--127` for the campaign) and, verbatim, in **Appendix A** (the dispatcher's
> per-batch records for batches 151--229, moved out of `ROADMAP.md` in batch 230).

## 0. Environment

* Lean `leanprover/lean4:v4.33.1`; Mathlib pinned to
  `db584cd6d46c92f209a44c0f1c829460d327499d` (`master-2026-08-10`, v4.33.0).
* `.lake/packages/mathlib` is a junction to the already-built Mathlib of this workspace, so no
  Mathlib rebuild is needed.  Lake does not resolve the transitive package paths through the
  junction, so `lake build` (and `lake env lean`) fail with "object file ... does not exist"
  for `Batteries`/`Aesop`; `tools/build.ps1` works around this by passing an explicit
  `LEAN_PATH` built from every package's `.lake/build/lib/lean` directory plus the project's
  own `build/lib/lean`.  Use that driver, or replicate its `LEAN_PATH` construction.
* If the package is ever moved inside a Mathlib workspace (for example to
  `fermats-last-theorem/MonstrousMoonshine/`), plain `lake build` works with no changes,
  since Lake then resolves every package through the manifest of that workspace.

## 1. The monster group

The paper's `M` is the Fischer--Griess sporadic simple group of order
`2⁴⁶·3²⁰·5⁹·7⁶·11²·13³·17·19·23·29·31·41·47·59·71`, acting on the FLM module.

**Mathlib status.**  Nothing.  No sporadic group of any kind; the string "monster" occurs
once in Mathlib, in a bibliography comment in `Mathlib/Algebra/Vertex/HVertexOperator.lean`.

**Least-effort route.**  The Lean 3 project `formalabstracts/formalabstracts` (unmaintained
since 2019) defines it in four lines:

```lean
noncomputable def Y443 : Group := coxeter_group $ matrix_of_graph (coxeter_edges [4,4,3])
private def a  : Y443 := generated_of torso
private def b₁ : Y443 := generated_of $ arm (by to_dfin 0) (by to_dfin 0)
-- ... b₂,c₂,b₃,c₃
noncomputable def spider : Y443 := (a * b₁ * c₁ * a * b₂ * c₂ * a * b₃ * c₃)^10
noncomputable def Monster : Group :=
  mk_ob $ quotient_group.quotient $ is_subgroup.center $ Y443/⟪{spider}⟫
```

i.e. the Coxeter group of the `Y₄₄₃` diagram modulo the spider relation (which yields
`M × C₂`) and then modulo the centre.  Porting this needs
`coxeter_group`, `matrix_of_graph`, `coxeter_edges`, `coxeter_vertices.arm/torso`, and a
quotient of a finitely presented group.  Mario Carneiro called this "the right definition"
on Zulip; Johan Commelin proposed porting it to Lean 4, which has not happened.

**Is that presentation the monster?  Yes — it is a theorem** (verified 2026, this used to be the
project's largest risk):

* **Ivanov, *Y-groups via transitive extension*, J. Algebra 218 (1999) 412--435**: Conway
  conjectured (c. 1980) that the Coxeter diagram `Y555` plus the single spider relation
  `(ab₁c₁ab₂c₂ab₃c₃)¹⁰ = 1` presents `M ≀ C₂`; "**This conjecture was proved by S. P. Norton and
  the author in 1990.**"  Second treatment: Ivanov, *The Monster Group and Majorana Involutions*,
  CUP 2009.
* **Borcherds, *Problems in moonshine* (2001), Problem 5** independently calls it the "Y
  presentation" of the monster [ATLAS], "conjectured by Conway and Norton and finally proved by
  Ivanov ... roughly the presentation of a Coxeter group with one extra relation".  Borcherds
  lists the *connection between this presentation and moonshine* as an **open problem** — the
  presentation is not in doubt, the conceptual link was.
* **Bray's 1997 tables** record the 12-generator `Y₄₄₃` form we use: arms 4, 4, 3; 79 relators
  `12 + 11 + 55 + 1`; same spider relation; with the remark that `Y₅₅₃` and `Y₅₄₃` present the
  same group.  Bray also warns: "There is more than one definition of `Y_pqr` in existence" —
  our convention (arm lengths = components of `[4,4,3]`, every bond labelled `3`) is fixed
  explicitly in `Definitions/MMCoxeter.lean`.

So the `formalabstracts` attribution to a MathOverflow comment should not be propagated as the
justification.  Full quotes are in `refs/README.md`.

**What would still be needed after the port.**  `M` finite, of order
`2⁴⁶·3²⁰·5⁹·7⁶·11²·13³·17·19·23·29·31·41·47·59·71` (task S3).  Because the presentation is now a
theorem, the obligation is "does the presented group have this order" rather than "is the
presentation correct" — still very hard (the monster's smallest faithful permutation degree is
196883), and Lean 3 never proved it.  After that: the conjugacy classes, the character table of
the first seven irreducible representations (used at the end of section 3 of the paper), and the
identification `2²⁴·2·Co₁ = Aut(Λ̂)` used in sections 12--14.

**The Griess-algebra cross-check is not feasible in Mathlib.**  Defining `M` as the isometry
group of the Griess algebra (the note `魔群月光.pdf` suggests exactly this) is a genuinely
*non-circular* route — unlike `M := Aut(V^♮_1)` from the FLM module, which presupposes the
monster action, since FLM construct the module together with the action — but it needs the Leech
lattice, the Griess product on a 196884-dimensional commutative non-associative algebra, and
`Aut` of that.  Mathlib has no Leech lattice and no usable non-associative algebra API.  The cheap
sanity check that *is* available is a nontriviality/abelianisation argument (in the S2 brief).
Two conventions not to mix: the Griess algebra has dimension **196884**, while Griess constructed
the monster as automorphisms of a **196883**-dimensional space, and `196884 = 196883 + 1` is the
first moonshine decomposition.

## 2. The FLM module

**Paper.**  Frenkel--Lepowsky--Meurman, *Vertex operator algebras and the monster* (1988).
`V = ⊕_{n∈ℤ} V_n` with `dim V_n = c(n)`, `V_{−1} = 1`, `V_0 = 0`, and the monster acting and
preserving the vertex algebra structure, the conformal vector and the bilinear form.

**Mathlib status.**  `Mathlib/Algebra/Vertex/VertexOperator.lean` (98 lines) and
`HVertexOperator.lean` (161 lines), by Scott Carnahan, define `HVertexOperator` as
`V →ₗ[R] HahnModule Γ R W` and its `coeff`/`of_coeff` API.  There is **no** vertex algebra
structure (no locality, vacuum, translation axioms), no Virasoro algebra, no module theory,
no Zhu algebra.  The related Mathlib PR #25822 is open and self-described as a testbed.

**What would be needed.**  A vertex algebra structure with the three axioms listed in section
3 of the paper (in the paper's own formulation: a family of bilinear products `u_n v` with
`u_n v = 0` for `n ≫ 0`, the Borcherds identity, and a vacuum `1`), a conformal vector of
central charge 24, the graded pieces as `L₀`-eigenspaces, and the FLM construction itself.
Also needed for sections 6--8: the no-ghost theorem (Goddard--Thorn), which is nowhere in
Lean.

## 3. The replication identity (8.3)

**Paper.**  Sections 6--8:
1. build the monster Lie algebra `M = ⊕_{m,n} M_{m,n}` from `V` as the space of physical
   states of a bosonic string, with `M_{m,n} ≅ V_{mn}` for `(m,n) ≠ (0,0)`;
2. identify it as a generalized Kac--Moody algebra with root lattice `II^{1,1}`, simple roots
   `(1,n)` of multiplicity `c(n)`, and Weyl group of order 2;
3. write its denominator formula `p⁻¹ ∏ (1 − p^m q^n)^{c(mn)} = j(p) − j(q)` and use the
   product formula for `j` to compute the simple roots (Theorem 7.2);
4. compute `H(E)` for the positive-root subalgebra: `H₀ = ℝ`, `H₁ = ∑_{n∈ℤ} V_n p q^n`,
   `H₂ = p ∑_{m>0} V_m p^m`, `H_i = 0` for `i ≥ 3`;
5. substitute into `Λ(E) = H(E)` and take the trace of `g` to get

   `p⁻¹ exp(−∑_{i>0} ∑_{m>0,n∈ℤ} Tr(g^i|V_{mn}) p^{mi} q^{ni}/i) = ∑_m Tr(g|V_m)p^m − ∑_n Tr(g|V_n)q^n`  (8.3).

**Mathlib status.**  Lie algebra homology, exterior powers of graded vector spaces, and
generalized Kac--Moody algebras are all absent.  The trace identity (8.3) itself is formal
power series algebra over `ℚ⟦p,q⟧`, for which Mathlib has `PowerSeries.exp` (`PowerSeries/Exp.lean`)
and `PowerSeries.log` (`PowerSeries/Log.lean`, with `logOf`); note that **`MvPowerSeries.exp`
does not exist**, so a two-variable statement has to be set up as power series in `p` with
coefficients in `HahnSeries ℤ ℚ` (Laurent series in `q`), which does have the needed
`Algebra ℚ` instances.  That was the plan of the (removed) `MMPowerSeries.lean` draft; the
declaration to prove there is, for a finitely supported `c : ℤ → ℤ → ℤ`,

```
PowerSeries.exp _ (∑' n, C (mult m n / m) * X m ...) = 1 + (F - 1)
```

where `F = ∑_m c 1 m p^m − ∑_n c 1 n q^n` and `mult` is the Möbius-inversion expression
(10.4) of the paper.

**What a partial formalization could achieve today.**  Steps (4) and (5) are elementary once
the objects are set up: the homology computation is a statement about exterior powers of a
graded vector space with the degrees given, and (8.3) is a formal power series identity.  A
reasonable intermediate theorem is: *if* the monster Lie algebra has the stated simple roots
and homology, *then* `c` is completely replicable in the sense of `MM.IsReplicable`.  That
would remove the paper's single largest gap (the no-ghost theorem) from the conditional
statement, leaving the Lie-theoretic part.

## 4. The Conway--Norton table (section 9)

**Paper.**
* (8.3) determines the coefficients `c_g(n)` from `c_g(1), c_g(2), c_g(3), c_g(5)` and the
  coefficients of all the powers `g^j` — this is the recursion (9.1), obtained by comparing
  the coefficients of `p²` and `p⁴` of (8.3);
* Koike proved that the modular functions of Conway--Norton's table satisfy the same
  relations;
* seven monster elements of ATLAS type `2B, 3B, 5B, 7B, 9B, 13B, 15D` have a nonsingular
  character matrix `χ_i(g_j)` (determinant `2²²·3⁵·5·7·17`), so the first five coefficients
  of every `T_g` are determined; the first five coefficients of the tabulated functions are
  equal to them.

**What is formalizable here.**  The recursion (9.1) is an identity between integer sequences
and is a candidate for a real proof: it needs the formal power series expansion of (8.3) to
order `p⁴`, which is finite combinatorics.  Once `MM.IsReplicable` is connected to the
recursion, `MM.moonshine_conjecture`'s uniqueness step could be replaced by "the coefficients
agree with the tabulated function's", which is a finite verification per conjugacy class.

**What is not.**  The genus-zero computations for `Γ₀(N)` and its normalizer `Γ₀(N)⁺`
(Riemann surface theory for modular curves; Mathlib has no genus, no modular curve, no
`IsHauptmodul`), and the construction of the `Γ₀(N)⁺` Hauptmoduls as modular functions.  The
`fermats-last-theorem` repository in this workspace has roughly 200 files of modular-curve
infrastructure (`Definitions/Def_ModularCurve_*`) that would be the starting point, but it
also has no genus theory.

## 64. Batch 151, second half: `Hrel` is closed, and three ways a dispatched file lies about itself

**The result.**  `Definitions/MMFLM85HrelClose.lean` (297 lines, 9 guards) proves

    MM.FLM85HrClose.hrel (bL) (form) (base) : MM.FLM85Hr.Hrel (F := F) bL form base

with **no hypothesis on `base`** — FLM (8.5.6), the well-definedness of the (8.5.5) normal-ordered
prescription on the spanning states (8.5.4).  The route is the landed
`MM.FLM85PS.permSpanningStatement` (`ker stateOfVL ≤ permRel ⊔ labelRel`) plus one new lemma for each
summand: the exchange at an *arbitrary* position (`normalOrderedFam_swap_at`, generalising the landed
head-only `MM.FLM85Hr.normalOrderedFam_swap`) and the label additivity at an arbitrary position
(`normalOrderedFam_label_add_at`).  With it, `MM.FLM85Un.generalVertexOperator_existsUnique` produces
the general vertex operator of (8.5.5)–(8.5.6) on all of `V_L` with nothing left to assume, and
`MM.FLM810F.theorem8102_of_hrel_modelBase` loses its last input on the relation side.  Batch 122's
head-only `RelSpanningStatement` stays refuted and is not used.

### 64.1 An unproved declaration with no `sorry` in the file

The file as delivered by the dispatched agent contained a proof whose first line failed to synthesize
`Module.Free ℤ L` (the instance is not derivable from an explicit `bL : Module.Basis κ ℤ L`
parameter).  Lean's diagnostic was one line at that line number; the *consequence* was that the
declaration and everything downstream — including the theorem the file is named for — acquired the
axiom footprint

    [propext, sorryAx, Classical.choice, Quot.sound]

**`Probe/forbidden_tokens.py` reports `sorry=0` on that file and is right to**: there is no `sorry`
token anywhere in the text.  The `sorryAx` enters through the failed elaboration.  So the two checks
this repository treats as acceptance are not redundant and not ordered by strength:

* the token stripper catches a *declared* gap;
* the `#guard_msgs` axiom guards catch a *failed* proof that produced a gap.

A file that passes only the first is a file with an unproved theorem in it, and the guard is the only
thing that says so.  Both were run here; had the guards been written *after* the last green build
instead of carried from the author's intent, the file would have been committed as proved.

### 64.2 A file written against an import closure that was never rebuilt

Two imports were missing — `MMFLM85Action` (for `MM.FLM85Ac.factorFamily`) and `MMFLM85LabelLinearity`
(for `MMFLM85LL.factorFamily_add`).  The author had them in its working closure; the interrupted run
never rebuilt the file from disk.  The lesson is the ordinary one and is recorded only because it
compounded: **the dispatcher's first action on any handed-back file is a build from the file as it
stands on disk, not a read of its header.**

### 64.3 A name that resolves in a minimal probe and not in the file — cause unknown

After the two structural fixes, `MM.FLM85Ac.factorFamily` still failed as a *simple* identifier under
`open FLM85Ac` inside this file (`unknown identifier factorFamily`, with `autoImplicit` suggested),
while the same term resolved in a minimal probe with byte-identical imports, options and `open` list
(`Probe/b151_probe4.lean`, `--- OK ---`, exit 0).  Fully qualifying the two occurrences fixes it.
The cause is **not known** and is recorded as unknown rather than explained away; the practical rule
that follows is cheap and worth keeping: *in a file that references another namespace's value,
qualify it — an `open` that works in a probe is not evidence that it works in the file.*

### 64.4 What is still open, in one paragraph

`Hrel` closed the relation side of the general vertex operator; it did **not** close Theorem 8.6.1.
By `MM.FLM86TDA.residue863_iff_corollary851` the display is *equivalent* to `MM.FLM85Co.Corollary851`
at the states of (8.6.2) — a statement whose every symbol is landed and which mentions no
normal-ordered product — and by `MM.FLM86TDA.display863_of_jacobiIdentity` it follows from a Jacobi
identity plus the finiteness `hvan`.  Two things stand between the repository and it: the **concrete**
field on `V_L` (`annihExp · actVL · creationExpVL`, all landed, unlike the abstract carrier
`MM.FLM87.LatticeElement` which has no annihilation half — §63.2) and its two-point product identity;
and the **uniform-in-`(m,n)`** vanishing `hvan`, which the VOA axioms supply only per component.  Both
are stated with exact types in `Probe/briefs/B151-W3-vacuum851.md` and
`Probe/briefs/B151-W4-assembly.md`.

> **§65 supersedes the last sentence of §64.4.**  The uniform vanishing is *not* a gap: it is the axiom
> (8.10.5) at the pair `(a,b)`, one line (`MM.FLM86UV.exists_uniform_bracketTerm_vanishing`).  What the
> axioms supply only per component is the *operator-valued* `MM.FLM88.JacobiIdentity` window, which the
> display does not consume.

## 65. Batch 152: Theorem 8.6.1's display is a theorem, and the finiteness it needed was an axiom all along

### 65.1 The display, at a fixed window, from two named inputs

`MM.FLM86DVS.display863_fixed_window` (`Definitions/MMFLM86DisplayVOASplit.lean`):

    FaithfulJacobi Y u v  →  (∀ m n i, K ≤ i → bracketTerm (toVertexOp Y) u v m n i = 0)
                          →  1 ≤ K  →  Display863 form E Y u' v' a b K

with `u = stateOf u' a`, `v = stateOf v' b` the states of (8.6.2).  Neither hypothesis restates the
conclusion:

* `MM.FLM810JF.FaithfulJacobi Y u v` is **verbatim the `jacobi` field** of the landed
  `MM.FLM810.VertexOperatorAlgebra` (`MMFLM810Axioms.lean:585`, the field's type being `JacobiAxiom Y`
  = `∀ u v, FaithfulJacobi Y.coeff u v`, `:438`).  So the display is a *consequence of the axiom
  (8.10.8)* for any field that satisfies it — no construction of the field, no normal ordering, no
  two-point product identity.
* the finiteness hypothesis is the source's own parenthetical of (8.5.26).

`display863_of_voa_self` is the same at a structure's own field; `display863_of_remainingFields` states
it at the boundary of `MM.FLM810CA.theorem8102_of_remainingFields` and consumes **only** the
`JacobiAxiom` conjunct of the four `ActionRemainingFields`.

### 65.2 Why the axiom's window suffices here, when it does not for `JacobiIdentity`

`corollary851_of_jacobiIdentity` consumes `MM.FLM88.JacobiIdentity Y u v N`: **one** window for all
monomials, with `MM.FLM88.CrossWindow` (operator-valued vanishing).  The axiom gives neither — it gives
a window per monomial *and per element* — which is why earlier batches recorded a "uniform vs
per-component" gap (§61.2) and why the history of that field records `CrossWindow` as equivalent in
characteristic zero to the vanishing of every component product
(`MMFLM810JB.jacobi8108_iff_zeroProduct`).

The display does not need that.  It is the **`l = 0`** face, and at `l = 0` the two cross sums collapse:
`deltaExpCoeff i 0 = 0` for `i ≥ 1` (`MM.FLM85Ex.deltaExpCoeff`, with `binomCoeff 0 i = 0`), so

    crossBracket Y u v N 0 m n  =  u_m v_n − v_n u_m  =  commutatorFamily Y u v m n

at **every** `N ≥ 1` — the landed `MM.FLM88.crossBracket_zero_eq_commutatorFamily`, whose window
hypothesis is only `1 ≤ N`.  The cross side therefore carries no window at all, the axiom's per-element
freedom has nothing to act on, and the single remaining obligation is the *iterate* side:
`crossIterate Y u v N 0 m n = bracketSum Y u v N m n`, which stops moving once `bracketTerm` vanishes
past `N`.  That is exactly the finiteness hypothesis, and `residue863_of_faithfulJacobi` is the
three-line proof.  **This is the structural reason batch 152's route is short where five earlier batches
were long**: those batches read (8.5.13) at general `l`, where the two windows genuinely differ.

### 65.3 The uniform vanishing is (8.10.5)

FLM's parenthetical at (8.5.26), printed p. 201, is *"`ι(a)_i·ι(b) = 0` for `i` sufficiently large"* —
a statement about the **state**, not about the operator family.  That is verbatim the axiom (8.10.5)
`u_n v = 0 for n sufficiently large` (`MM.FLM810.ComponentTruncation`) at the *pair* `(u,v) = (a,b)`.
`MM.FLM86UV.exists_uniform_bracketTerm_vanishing` is the one-line consequence:

    obtain ⟨N, hN⟩ := A.componentTruncation a b
    ...  bracketTerm = binomCoeff m i • (a_i·b)_{m+n−i} = binomCoeff m i • 0 = 0

with no grading, no homogeneity, no weight, and the window `N.toNat` **independent of `m` and `n`**
because what is killed is the state `a_i·b` and the outer component is linear in it.  The earlier
framing — "the VOA axioms supply truncation *per component*, not uniformly in `(m,n)`" — is therefore
wrong for the pair-indexed form the display consumes.  The window that is genuinely per-component is the
one of the operator-valued `JacobiIdentity`; the display does not use it.

A **second, independent** route is landed for the record:
`bracketTerm_vanishing_of_gradingOn_uniform` gets the same uniform window from the grading (8.5.22) plus
the nonnegativity of (8.10.3), the window being `(S.wt a + S.wt b).toNat`.  Note that the brief's sketch
of this route — "take the maximum of the binomial window `m+1` and the state window" — is
**unnecessary**: the state window alone is already uniform in `(m,n)`, and the binomial `binomCoeff m i`
(which for `m < 0` never vanishes) never enters.

### 65.4 A landed definition that is refutable: `WeightGrading.IsHomogeneousOp`

`MM.FLM85.WeightGrading.IsHomogeneousOp S T p` is

    ∀ a, S.IsHomogeneous a → S.IsHomogeneous (T a) ∧ S.wt (T a) = S.wt a + p.

The weight clause is demanded at `a = 0` too, where it reads `S.wt 0 = S.wt 0 + p`.  Hence
`MM.FLM86UV.isHomogeneousOp_eq_zero`: **`p = 0` for every operator `T`**; `not_isHomogeneousOp`
refutes the predicate at every `p ≠ 0`; and `not_forall_isHomogeneousOp_component` refutes the `∀ n`
form used as the landed (8.5.22) hypothesis.  So every landed lemma whose hypothesis is an
`IsHomogeneousOp` is **vacuous**:

* `MM.FLM85Br.bracketTerm_vanishing_of_grading` (the lemma the (8.5.12) contraction work was built on),
  `MM.FLM85Br.component_eq_zero_of_toNat_le`, `component_eventually_zero`,
  `component_apply_eq_zero_of_weight_lt`;
* `MM.FLM85Br.weightOfComponents_of_isHomogeneousOp` and the whole Corollary 8.5.6 chain
  (`corollary856_iff_componentBracket_of_isHomogeneousOp`, `MMFLM85Corollary.lean:781`);
* the `hOp` parameters of `MMFLM88`'s grading lemmas
  (`componentBracket_of_jacobiIdentity_grading`, `corollary854_of_jacobiIdentity_grading`).

**The repair** is `MM.FLM86UV.IsHomogeneousOpOn`: the same statement with the `T a = 0` case dropped —
precisely the case where `WeightGrading.wt_eq_of_mem` has no content, since it constrains `wt` only on
*nonzero* elements of a graded piece.  `IsHomogeneousOp.isHomogeneousOpOn` records that the landed
predicate implies the repaired one, so the repaired theorems generalise their landed counterparts, and
`*_on` versions of the engine and of the uniform vanishing are landed under it.  Nothing in batch 152's
files consumes `IsHomogeneousOp`.

*The moral, and it is the third of its kind in this project: a hypothesis that has never been
instantiated is a hypothesis that has not been read.  `IsHomogeneousOp` was carried through six batches
of ledger entries as "the landed (8.5.22)" without anyone writing down its `a = 0` instance.*

### 65.5 What Theorem 8.6.1 now rests on

By §65.1 and §65.3 the display for the corrected action prescription is conditional on
`FaithfulJacobi` for that field — i.e. on (8.10.8) — and **nothing else**.  The repository already
reduces that to generators:

    MM.FLM810RR.JacobiOnStateTargets bL form Y   →   MM.FLM810RR.jacobiAxiom_of_stateTargets
                                                 →   JacobiAxiom Y
                                                 →   FaithfulJacobi Y u v

(`Definitions/MMFLM810RemainingReduction.lean:166`, `:180`).  So the chapter's "main commutator result"
is now conditional on **one landed `Prop`**, `MM.FLM810RR.JacobiOnStateTargets`, whose type is a
per-monomial, per-*spanning-state* window plus the component identity.  The two-point product identity of
the concrete field, the normal-ordering identities (8.5.11)/(8.5.12) and the `DiagOrdinary` dictionary —
five batches' worth of residual — are consumed **nowhere** in this route.

Not claimed: FLM's own proof of (8.6.3) through (8.6.11)--(8.6.27).  The display is reached through the
Jacobi axiom, which is the source's own Remark 8.8.11 (*"Theorem 8.6.1 is `Res_{z₀}` of the Jacobi
identity"*), not through the coefficient extraction of §8.6.

## 66. Batch 153, reconnaissance: the residual of Theorem 8.6.1 is not open, it is **false** — and the repair is already landed

Two reconnaissance agents (fresh, self-contained briefs; reports `Probe/B153_R1_PROVED.md`,
`Probe/B153_R2_VOA.md`), plus the dispatcher's own re-verification of every cited line.

### 66.1 The defect, in one line

`MM.FLM85Cp.heisFam a = fun m => modeVL a (−m−1)` is the (8.4.7) Heisenberg field **without its zero
mode**: at the power `m = −1` it evaluates `modeVL a 0`, and `modeVL a 0 = 0` by construction —
whereas FLM (7.1.33)/(7.1.41) has `a(0)` act on the lattice factor by the pairing.  Everything else in
this section follows from that one missing term.

### 66.2 Consequences, all landed and unconditional

* `MM.FLM810MD.l0OnWeightGenerators_forces_zero` (`Definitions/MMFLM810ModesL0.lean:370`): for *any*
  base whose field at `0` is the identity field — in particular the (8.4.17) model base of the §8.10
  assembly — `L0OnWeightGenerators` **forces `k = 0`** at every generator of weight `k`.
  `not_l0OnWeightGenerators_modelBase` (`:398`) is the instance: the predicate is **false** as soon as a
  lattice vector of nonzero even norm exists, which for the Leech lattice (and every nonzero positive
  definite even lattice) it does.
* `MM.FLM810MD.not_translationOnStateData_modelBase`
  (`Definitions/MMFLM810TranslationStateObstruction.lean:271`): `TranslationOnStateData` is false there
  for the same reason (`L(−1)ι(a) = 0` while (8.10.13) demands the derivative of a non-constant field).
* Therefore `MM.FLM810AFT.ActionRemainingFields` is **unsatisfiable** at that field, and
  `MM.FLM810RR.JacobiOnStateTargets` — batch 152's "single named residual" — is *not* a proof
  obligation one can discharge: two of its three sibling conjuncts are false at the same object.

**This retracts, for cause, the closing claim of §65.5** ("Theorem 8.6.1 is now conditional on that one
landed `Prop` and nothing else").  The `Prop` is landed; it is refuted; the correct statement is that
Theorem 8.6.1 is conditional on a field that satisfies the §8.10 axioms, and the repository's assembly
boundary does not produce one.

### 66.3 The repair, landed since batch 80

* `MM.FLM85ZM.heisFamZero` (`Definitions/MMFLM85ZeroModeField.lean:116`): (8.4.7) with
  `hZeroVL (ofLattice a)` at the power `−1`, agreeing with the landed field at every other power
  (`heisFamZero_apply_of_ne`).
* `MM.FLM85ZP.actionPrescriptionTAZero`, and **unconditionally**
  `existsUnique_actionPrescriptionTAZero` (`Definitions/MMFLM85ZeroPrescriptionSpanning.lean`): (8.5.1)
  with (8.5.6) for the corrected prescription.  The relation-spanning input is literally the *same
  proposition* as the landed one — `PermSpanningStatement` has no prescription argument, and `stateOfVL`
  reads only strictly negative modes — so no re-proof was needed and no zero-mode hypothesis enters.
* `MM.FLM810ZMU.Yzero` + `Yzero_spec` + `Yzero_unique`
  (`Definitions/MMFLM810ZeroModeL0Uncond.lean:82`, `:87`, `:107`): the unique concrete `Y` of the
  corrected (8.5.5), with no hypothesis.  **This is the field of the moonshine-module construction.**
* `MM.FLM810ZMW.L0OnWeightGenerators_zero` (`Definitions/MMFLM810L0General.lean:301`): (8.10.12) at the
  vacuum generators for that field, where the landed prescription gave `0`.

`MM.FLM810CA.theorem8102_of_remainingFields` still consumes `actionPrescriptionTA`, and no assembly
line for `actionPrescriptionTAZero` exists.  Batch 153's task P2 builds it.

### 66.4 What is *not* a gap, and what is

* **Not a gap**: the finiteness conjuncts.  `crossWindowVectorwise_of_truncation` and
  `iterateWindowVectorwise_of_truncation` (`MMFLM810JacobiFaithful.lean:186`, `:213`) need only
  `ComponentTruncation`, which is a theorem for the concrete field
  (`MMFLM85ActionTruncationTA.componentTruncation_of_actionPrescriptionTA:63`).
* **Not a gap, but a named open `Prop`**: the display *is* the `l = 0` face of the Jacobi identity
  (`MM.FLM86TDA.display863_iff_corollary851`), and that face is *equivalent* to
  `MM.FLM85Co.Corollary851` (`MMFLM88Jacobi.lean:588`, `corollary851_iff_jacobiIdentity_zero_face`).
* **A gap**: every `l ≠ 0` face.  `Definitions/MMFLM88Jacobi.lean:189` records that the objects FLM
  uses — the localisation `F[z₁,z₂]_S` and the maps `ι_{i₁⋯iₙ}` of (8.10.36)--(8.10.42) — do not exist
  in the repository.
* **A dead route**: the weight/grading route through `IsHomogeneousOp` (unsatisfiable, batch 152), and
  the vendored `VertexAlg` route (a class field with no construction upstream; its carrier is not `VL`;
  its one concrete field is refuted).

### 66.5 What the batch does about it

Two proof tasks on disjoint files, dispatched together:

* **P1** `Definitions/MMFLM86ExpProduct.lean` — the concrete exponential two-point product on `V_L`,
  `Y(ι(a),z₁)Y(ι(b),z₂) = ε(a,b)·:Y(ι(a),z₁)Y(ι(b),z₂):·(z₁−z₂)^{(a,b)}`, and its commutator corollary.
  This is FLM's own engine for Theorem 8.6.1, independent of the §8.10 rationality layer.
* **P2** `Definitions/MMFLM810CorrectedZeroAssembly.lean` — the corrected four-field package and the
  `theorem8102` boundary at `Yzero`, reusing the landed generator reductions verbatim.

The meta-lesson, and it is the fourth of its kind here: **a residual carried across batches as "one
landed `Prop`" had never been instantiated either.**  Batch 152's `IsHomogeneousOp` lesson (a
hypothesis that has never been instantiated has not been read) applies verbatim to the *conclusion*
side: a target that has never been attempted at the object it names has not been read either.  Both
refutations that killed it were already in the repository, in files whose headers say so.

### 66.6 The empty-oscillator case of Theorem 8.6.1 is **one named `Prop`**, and the two mis-shaping defects are now distinct

Measurement agent M1 (`Probe/B153_M1_EXPAPI.md`) mapped the concrete exponential layer.  Three results
change the plan:

1. **The concrete exponentials exist and their contractions are landed.**  `MM.FLMLatSt.creationExpVL`
   (`Definitions/MMFLMLatticeState.lean:199`) is the creation half `E^-`, `MM.FLM84CM.annihExp`
   (`MMFLM84Completion.lean:682`) the annihilation half `E^+`; the five lemmas that move a Heisenberg
   mode or the zero mode past `E^-` are all landed (`modeVL_comm_creationExpVL` at `:375` and `:436`,
   `MMFLM84TwoVarProduct.lean:600`, `hZeroVL_comm_creationExpVL` at `:639`, `hZeroVL_contraction` at
   `:659`, `creationExpVL_mul_actVL` at `:333`).  **No normal-ordered product formula for two creation
   exponentials exists**; the only product statement is the commutation `creationExpVL_comm`
   (`MMFLM84TwoVarProduct.lean:624`).  That is exactly the gap batch 153's task P1 was aimed at, and
   the flat form of it (`creationExpVL a i * creationExpVL b j = creationExpVL b j * creationExpVL a i`)
   is now landed there.
2. **The object mismatch is named.**  `actionPrescriptionTA` folds factors built from
   `heisFam a m = modeVL a (−m−1)` (`MMFLM85Compat.lean:213`), the (8.4.36) display uses
   `modeFam Y a m = Y (iotaVL a) (−m−1)` (`MMFLM8512Done.lean:403`), and the two are identified only by
   `MMFLM8512H.ModeFamHeisShift` (`MMFLM8512Heis.lean:221`) — a `Prop` with **no producer anywhere**.
   So the construction carries *two* independent mis-shaping defects: the missing zero mode (§66.1) and
   this missing dictionary.  Any future plan that assumes the prescription's factors are the field's
   modes is wrong by this `Prop`.
3. **The reduction that matters.**  `theorem842Concrete_iff_corollary851` (`MMFLM8512Done.lean:805`)
   with `theorem842Concrete_iff_apply` (`:793`) makes `Theorem842Concrete Y E a b (form a b) K`
   *equivalent* to `Corollary851 Y (iotaVL a) (iotaVL b) K`; `stateOf_one_eq_iotaVL` is `rfl`, so this is
   `Display863Unshifted form E Y 1 1 a b K` (`MMFLM86TheoremDisplay.lean:163`), the empty-oscillator
   case of (8.6.3)'s second sentence.  **Theorem 8.6.1's `(1,1)` case is therefore a single named
   `Prop`.**  It does **not** lift to general `u', v''`: `theorem861Source_of_coeffFamily_and_spanning`
   (`MMFLM86SpineGap.lean:632`) is proved but needs spanning families with their own coefficient
   conditions, the pure lattice generators do not span `V_L = M(1) ⊗ F{L}`, and the spanning input
   (Remark 8.3.9) is not landed.

Priced remainder, in FLM's own steps: (a) (8.5.12) at `k ≥ 1`, i.e. (8.5.5) at the Fock states
`E_k·ι(a+b)`, plus `ModeFamHeisShift`; (b) the (8.4.35) commutator identity with its contraction
factor; (c) the `z_1 = z_0 + z_2` shifted resummation (`MMFLM86MixedProduct.jointNO:345` lands the
object, `DiagOrdinary` its identification with the iterate, unproved); (d) the Fock half of (8.5.5).

## 67. Batch 153, acceptance: the corrected boundary is landed — and the "repaired" field is a **third** defect

Both proof tasks of §66.5 are accepted, after the dispatcher's own verification: each file was
rebuilt **from a deleted `.olean`** (`tools/build.sh` → `--- OK ---`, exit 0), passes
`Probe/forbidden_tokens.py` at `sorry=0 admit=0 axiom=0 native_decide=0`, and every in-file
`#guard_msgs in #print axioms` guard printed `[propext, Classical.choice, Quot.sound]`; an independent
probe over the headline declarations confirmed the footprints and the exact type of
`MM.FLM810CZA.theorem8102_of_generatorFieldsZero`.

* `Definitions/MMFLM86ExpProduct.lean` (349 lines, `MM.FLM86EP`): `creationExpVL_mul_comm`; the
  concrete two-point product `Yzero_iotaVL_mul_Yzero_iotaVL`; its state form
  `Yzero_iotaVL_apply_iotaVL` ((8.5.11) at `p = 0`); and the commutator family
  `commutatorFamily_Yzero_iotaVL = E.eps a b • X − E.eps b a • X`, `…_eq_c = ((E.c a b − 1) * E.eps b a) • X`.
* `Definitions/MMFLM810CorrectedZeroAssembly.lean` (299 lines, `MM.FLM810CZA`): `componentTruncationZero`,
  `vacuumOperatorZero`, `creationRegularZero`, `creationPropertyZero`,
  `actionRemainingFieldsZero_of_generators`, `theorem8102_of_remainingFieldsZero`,
  `theorem8102_of_generatorFieldsZero` — the corrected Section 8.10 assembly line at `Yzero`.

### 67.1 A wrong-prescription statement with a prescription-free proof

`MM.FLM85ATT.componentTruncation_action_modelBase` / `componentTruncation_of_actionPrescriptionTA`, and
`MM.FLM810AFT.vacuumOperator_action_modelBase` / `creationRegular_action_modelBase` /
`creationProperty_action_modelBase`, all carry a compatibility hypothesis **for
`actionPrescriptionTA`**, although their proofs consume only the underlying `TruncatedField` datum and
its factor list.  So they do not transfer to `Yzero`, and `MMFLM810AFT`'s docstring — which calls them
the fields "of the corrected object" — overstates what the statements say.  (8.10.5) had to be ported
(~40 lines).  The bridges that are generic in the factor list
(`MM.FLM85TAV.actionFold_vacuumNonnegative`, `…_eq_cutOff`, `…_zero`) transferred with no work, and the
four generator reductions transferred verbatim.

### 67.2 `Yzero` still fails (8.10.13), so the zero mode was not the only defect

`MM.FLM810TR.not_translationOnStateData_modelBase`
(`Definitions/MMFLM810TranslationStateObstruction.lean:271`): `TranslationOnStateData` is **false** for
`vertexOpOfField (Yzero … (modelBaseField E weight))` at the conformal vector whenever
`(form a b : F) ≠ 0` and `actVL E a (iotaVL b) ≠ 0`.  The uniform statement is
`translationOnStateData_modelBase_forces_zero`, whose conclusion is
`(form a b : F) • actVL E a (iotaVL b) = 0`.  So `theorem8102_of_generatorFieldsZero` at that base is
**vacuous**, and the obstruction is the *vacuum-supported base field*, not the missing `a(0)`.

This is the correction of record for §66: the construction carries **three** stacked defects —

1. the missing (7.1.33)/(7.1.41) zero mode in `heisFam` (repaired: `heisFamZero` →
   `actionPrescriptionTAZero` → `Yzero`, all unconditional);
2. the missing grading shift `z^{a(0)}` in the base field.  P1 measured the honest assembled (8.4.17)
   field `MM.FLM84LF.YFam` (`Definitions/MMFLM84LatticeField.lean:720`, the three-factor
   `E^-(z)·(az^∂)·E^+(z)`) and proved `Yzero_iotaVL_apply_iotaVL_eq_YFam`: the model base **is**
   `YFam` at a pairing-shifted index.  `Yzero` is creation-half only and has neither `z^{a(0)}` nor
   `E^+`, which is why no contraction and hence no scalar `(z₁−z₂)^{(a,b)}` appears in its two-point
   product;
3. the missing dictionary `MM.FLM8512H.ModeFamHeisShift` (`MMFLM8512Heis.lean:221`) between the
   prescription's factors `heisFam a m = modeVL a (−m−1)` (`MMFLM85Compat.lean:213`) and the field's
   modes `modeFam Y a m = Y (iotaVL a) (−m−1)` (`MMFLM8512Done.lean:403`).

**The object the next tasks must be stated about is therefore `YFam`, not `Yzero` and not
`modelBaseField`.**  P1's `## What this file does not prove` block names the landed next steps:
`MM.FLM84CT.fullPlus_contraction`, `MM.FLM84RWC.annihExp_mul_creationExpVL_eq_sum_of_subset`
(= `MM.FLM84TV.halfPair2_eq_window_of_subset` with `MM.FLM84RW.ratioCoeff`/`ratioWindow`),
`MM.FLM84LF.YFam_iotaVL_eq_exp`, `MM.FLM85Co.Corollary851`, `MM.FLM86MXP.Display8625One`.

### 67.3 Corrections to the batch's own briefs

* `Yzero` is at `MMFLM810ZeroModeL0Uncond.lean:81`, `Yzero_unique` at `:100` (the briefs said `:82`
  and `:107`).
* The two batch-79 refutations are `MM.FLM810MD.exists_not_l0OnWeightGenerators` (identity base) and
  `MM.FLM810MD.exists_not_translationOnStateData` (counterexample base `translationTestBase`), in
  `Definitions/MMFLM810ModesL0Trans.lean` — **not** at the (8.4.17) model base.  The model-base
  refutation is the separate `MM.FLM810TR.not_translationOnStateData_modelBase` of §67.2.
* The corrected `hL0` of the assembly boundary is already a theorem
  (`MM.FLM810ZMW.L0OnWeightGenerators_zero`, `MMFLM810L0General.lean:301`), so the boundary needs only
  `hjac`/`htrans`/`hvir`.
* Namespaces: `creationExpVL`/`creationExpVL_comm_actVL` are `MM.FLMLatSt.*`
  (`Definitions/MMFLMLatticeState.lean:333`, `:341`); `modeVL_comm_creationExpVL` and
  `hZeroVL_comm_creationExpVL` are `MM.FLM84CT.*`.  `E.eps` is the section 2-cocycle (7.1.12) and
  `E.c` the commutator cocycle (7.1.8), with `E.eps_mul_comm : E.eps a b = E.c a b * E.eps b a`
  (7.1.26) — the product carries `eps`, the commutator's scalar is `(E.c a b − 1) * E.eps b a`.
* API trap, recorded for the next agent: `rw [← sub_smul]` / `simp only [sub_smul]` **fail** on
  `Module.End F V` (the Algebra/Module `smul` instance diamond) even though both sides print
  identically; the fix is `LinearMap.ext` plus `LinearMap.sub_apply`/`smul_apply` with `sub_smul`
  pointwise on `V_L`.

## 68. Batch 156: the `ι` maps exist, and the commutator's real dividing line is the cocycle, not the sign

### 68.1 FLM (8.10.37)–(8.10.42) for `n = 2`, and why the carrier had to be built

`Definitions/MMFLMIotaVar.lean` (`MM.FLMIotaVar`, 852 lines).  FLM's target
`F[[z₁,z₁^{-1},z₂,z₂^{-1}]]` is **not a ring in FLM's use**, and the reason is a landed theorem:
`no_ringHom_monomial` shows that no ring homomorphism into the product model can send `z₁ − z₂` to a
unit, although `z₁ − z₂ ∈ S` by (8.10.36) and the localisation `F[z₁,z₂]_S` inverts it.  So the honest
object is an `F`-linear map into the product space; but each **ordering** does live in a ring, because
expanding first in negative powers of `z_{i_1}` makes the outer exponents well-founded.  The carrier is

    IotaVar F := HahnSeries (OrderDual ℤ) (HahnSeries ℤ F)      -- a field

with two variable assignments, and both incarnations are landed: the ring homs
`iota12Ring`/`iota21Ring : F[z₁,z₂]_S →+* IotaVar F`, and the `F`-linear
`iota12`/`iota21 : F[z₁,z₂]_S → AllSeriesN F 2` that Prop. 8.10.3–8.10.5 consume.

**Injectivity is proved at all three levels**, not cited: the polynomial embeddings (including
`polyToOnePoly_injective`, which the landed `MMFLMLocalization` lacked), the ring homs out of the
localisation (`IsLocalization.lift_injective_iff` plus `IsLocalization.injective` with the landed
`S_le_nonZeroDivisors`), and the readings.  The source says "it is clear that the maps `ι_{i₁⋯i_n}` are
injective"; given that `MM.FLM81.KerThetaStatement` exists because such statements can fail, this is
worth having as a theorem.

**The bound on what the `l ≠ 0` faces can say.**  The delta series is supported on the whole
antidiagonal — unbounded above in the outer exponent — so it is **not an element of the carrier**.  In
the carrier the two readings of `(z₁−z₂)^{-1}` differ by a sign: `iota12_invShift = expand12`,
`iota21_invShift = expand21`, `iota12_sub_iota21_invShift = deltaRatio`.  This is the `n = 2` form of
FLM's Remark 8.10.1, and it is the same phenomenon batch 155 met on the product: the two expansion
orders are not related by a constant, and the difference lives only in the target.

### 68.2 The commutator at the lattice generators: the cocycle is the dividing line

`Definitions/MMFLM86CommIota.lean` (`MM.FLM86CI`, 421 lines).

* `commutator_YFam_eq_twoVarField_sub_swap2` writes the commutator through
  `twoVarField a b − swap2 (twoVarField b a)` **for every pairing** — no sign hypothesis.
* `commutator_YFam_eq_smul_mul_of_nonneg`: at `0 ≤ ⟨a,b⟩` it is the **constant**
  `1 − E.c b a · (−1)^⟨a,b⟩` times the product; **no delta term** in this carrier.
* `evenCocycle_c_mul_neg_one_zpow`: for the source's own even-lattice cocycle
  (`MM.FLMLattice.EvenCocycle`, from the landed `C.c_eq_signPow`) that constant is `1`.  Hence
  `commutator_YFam_eq_zero_of_even`: `[YFam a m, YFam b n] = 0` for `0 ≤ ⟨a,b⟩` at the honest fields and
  the intended cocycle, **unconditionally**.  *This is the compatibility `c(a,b) = (−1)^⟨a,b⟩` that the
  landed `CentralExtension` does not assume and `EvenCocycle` supplies — and it is exactly the
  hypothesis under which Theorem 8.6.1's (8.6.1) is stated.*
* `not_YFamOnGenerators_and_corollary851_rankOne`: the pair `(hY, h851)` is **refuted** at the rank-one
  datum `a = b = 1`, where the constant is `2` for the trivial cocycle and the product is nonzero on the
  vacuum.  So `MM.FLM86II.commutatorFamily_lattice_eq_zero_of_nonneg` is not "the `0 ≤ ⟨a,b⟩` case of
  (8.4.41)" in landed generality: its hypothesis pair is empty at that datum.
* `commutator_rankOne_negTwo_ne_zero`: the commutator is genuinely nonzero at `⟨a,b⟩ = −2`, which is why
  the missing range `⟨a,b⟩ ≤ −2` is not a defect of the formalization: the finite-support carrier cannot
  express `contrFam N` for `N < 0`, and no scalar makes the exchange law hold there
  (`MM.FLM86TS.not_swap2_twoVarField_rankOne`).  The only route to that range is the residue sum of
  (8.5.13) itself.

### 68.3 Theorem 8.10.2, field by field, after this batch

`L0` (8.10.12) is **proved** (`MM.FLM810ZMW.LOf_conformalVectorZero_on_weightPiece`,
`L0OnWeightGenerators_zero`), so it is not an open field.  `Virasoro` (8.10.9) is reduced to two named
`Prop`s (`MM.FLM87ZBC.virasoroOnStateData_of_named`; the honest form is
`MM.FLM87SA.FreeBosonSugawaraBracketZero`, FLM (8.7.9), whose header records it is **not refuted**).
`translation` (8.10.13) is false at the model base and is being tested at the repaired base.  `Jacobi`
(8.10.8) is the only field with no finite reduction, and its `l ≠ 0` faces are now unblocked by §68.1
while its `l = 0` face at negative pairing is blocked by §68.2.

## 69. Campaign state at the end of the batch-157--164 run: what Theorem 8.10.2 now rests on

This section is the **handover**: it lists, for each of the four inputs of
`MM.FLM810TA2.theorem8102_of_generatorFieldsAligned` (Theorem 8.10.2 at the book's field), exactly what
is proved, what is named, and what has no route in the repository.  Everything named here is landed,
verified by the dispatcher by a rebuild from a deleted `.olean` plus token and axiom-guard checks, and
wired into `FinalCheck.lean`.

### 69.1 The alignment (the owner's directive), all four steps landed

1. **The base field**: `MM.FLM85BA.baseAligned` / `genFam E a m := YFam E a (−m−1)`, with
   `baseAligned_zero` (8.5.3) and the **sharp** truncation `genFam_lower_truncation_iotaVL`
   (`m < ⟨a,c⟩ ⇒ 0`), which is the `z^{−(a,a)/2}` of (8.4.17).  The old `modelBase`/`heisFam` are
   defective (no zero mode, no `a z^∂`, no `E^+`) and are **refuted on the sibling axioms**
   (`MM.FLM810MD.not_l0OnWeightGenerators_modelBase`,
   `MM.FLM810TR.not_translationOnStateData_modelBase`).
2. **The canonical field**: `MM.FLM85CF.canonicalField := heisFamZero`, with the defect itself a theorem
   (`splitPlus_defect_at_zero`) and a migration block plus pointer lemmas (the `deprecated` attribute is
   **impossible** cross-module in this pin).
3. **The `^+`-side counterparts**: `MM.FLM85SP`, 17 declarations, with a consumption table.
4. **The assembly**: `MM.FLM810TA2.theorem8102_of_generatorFieldsAligned(_l0)`.

**Why this mattered, in one line each**: with the old field, (8.4.23) is **false** at `p = −1`
(`MM.FLM84DY`), and the "tail Leibniz" shape that two landed files asserted is **false**
(`MM.FLM84TLP.not_tailLeibnizResidual`) — two theorems that would have been "proved" and then built on.

### 69.2 `hL0` (8.10.12) — DISCHARGED

`MM.FLM810TA2.l0OnWeightGenerators_aligned`, from `MM.FLM810ZMW.L0OnWeightGenerators_zero` (which is
base-generic given `(base 0).fam = 1`) plus `baseAligned_zero`.  Needs only `hnd` and symmetry of `form`.

### 69.3 `hvir` (8.10.9) — discharged **modulo a hypothesis that is false on even lattices**

* The bracket FLM (8.7.9) at `c = dim 𝔥` **was already a theorem** (batch 117,
  `MM.FLM87ZBF.freeBosonSugawaraBracketZero_of_orth`), from `hOrth : OrthonormalBasis bL form`.
* `MM.FLM87SBC.not_orthonormalBasis_of_even` proves `hOrth` **false** whenever `2 ∣ (a,a)` — i.e. on the
  Leech lattice, because FLM (8.7.3) asks for an orthonormal basis of the **`F`-space `𝔥 = L ⊗ F`**
  (scaling by `1/√(b_i,b_i)` allowed) while the repository demands it of the **`ℤ`-basis `bL`**.
* Hence `theorem8102_of_generatorFieldsAligned_dimH` has **no Virasoro hypothesis** but does carry
  `hOrth`.  The residual is exactly `MM.FLM87SBC.SugawaraBracketAtDimH` (no `hOrth`).
* `MM.FLM87GS` **refutes the only shortcut**: the Gram twist is not a transport
  (`form_eq_of_gramTwist`: conjugation preserves the CCR coefficient `m⟨a,b⟩`, so the form cannot change;
  `not_gramTwist_rankOne_even`; and `2c² = 1` has no rational solution, so "extend `F`" *is* (8.7.3)).
* **What is needed**: a new **two-contraction** Sugawara computation, the dual pair `(b_i, u^i)` of
  (8.7.2).  Landed starting points are cited in `MM.FLM87GS` §5 — `MM.FLM810CV.dualCoeff`,
  `MM.FLM810ZM.dualHBasis_sum`, `MM.FLM810ZMT.dualCoeff_sum_smul_hBasis`, the unconditional
  decomposition `MM.FLM87ZDG.sugawaraModeZero_apply_eq_regular_add_zeroCorr` with
  `MM.FLM87SG.zeroCorrPart`, and the Gram-weighted `[Z,Z]` bracket `MM.FLM87ZBK.zeroPart_comm`.  **The
  missing piece is the bracket of the regular inverse-Gram part.**  Smallest open case:
  `MM.FLM87GS.RankOneEvenSugawaraBracket` (`rankOneEvenSugawaraBracket_iff` is `Iff.rfl`).

### 69.4 `htrans` (8.10.13) — the induction step and the state half proved, three pieces named

* **Proved**: the corrected-fold Leibniz rule (`MM.FLM810TG.actionFold_deriv_field`,
  `actionFold_deriv_dFold`, with the derivative field `dField`); the **state half**
  `MM.FLM810TG.LMinusOneCreationMode` (`MM.FLM810SHC.lMinusOneCreationMode`, at arbitrary `b` and
  arbitrary vector, no symmetry hypothesis); and the **creation intertwining at arbitrary `w`**
  (`MM.FLM810CI.Yzero_modeVL_eq_actionFamily_nat`), whose enabling tool is `Yzero_trunc` — (8.5.6) at an
  arbitrary vector.
* **Named and open**: (i) the `dFold` assembly of the induction step needs `dFactor (factor n b) =
  n • factor (n+1) b` as bundled `Factor`s (there is no scalar action on `Factor` and no `actionFamily`
  congruence in `Factor.bound`) and the derivation form of the state half at `sugawaraModeZero`;
  (ii) the base case, landed as `MM.FLM810CI.BaseResidual`, **equivalent** to the base-field identity
  `actionFamily (baseFactor a) (base a) = dField (base a)` — i.e. to `MM.FLM84LT.generalStateDeriv`,
  which is the one genuinely open analytic input on this front.
* The reduction is recorded as `MM.FLM810SHC.translationOnStateData_iff_translationResidual`.

### 69.5 `hjac` (8.10.8) — open, with its faces mapped

* The display (Theorem 8.6.1) is a theorem **given** the axiom (batch 152).
* The `l = 0` face is **equivalent** to `MM.FLM85Co.Corollary851`; at the lattice generators the
  commutator vanishes at the intended even cocycle **unconditionally**
  (`MM.FLM86CI.commutator_YFam_eq_zero_of_even`, from `evenCocycle_c_mul_neg_one_zpow`), and the missing
  range is exactly `⟨a,b⟩ ≤ −2`, where the finite-support carrier cannot express the contraction series
  (`commutator_rankOne_negTwo_ne_zero` shows the commutator is genuinely nonzero there).
* The `l ≠ 0` faces have their prerequisite: the several-variable `ι` maps for `n = 2`
  (`MM.FLMIotaVar`), with injectivity proved at three levels and the carrier `IotaVar F :=
  HahnSeries (OrderDual ℤ) (HahnSeries ℤ F)`.
* `MM.FLM810RR.JacobiOnStateTargets` has **no producer** for any field, and the field it used to be
  stated about (`modelBase`/`heisFam`) is refuted on the sibling axioms — so the Jacobi work must be
  done at the aligned field.

### 69.6 Update after batches 163--165: the two fronts have MERGED into one computation

The single most useful thing to know before resuming: **the Virasoro field and the translation field now
depend on the same open input.**

* `MM.FLM810SHC.lMinusOneCreationMode` — the state half of translation, which batch 162 *thought* it had
  closed — carries `hOrth : OrthonormalBasis` (`Definitions/MMFLM810StateHalfClose.lean:226`--`:228`,
  verified by the dispatcher), and `MM.FLM87SBC.not_orthonormalBasis_of_even` shows that hypothesis is
  **false on every even lattice**, the Leech lattice included.
* Batch 165's Y2 then proved `sugawaraDerivation_iff_lMinusOneCreationMode`: the derivation form needed by
  the `dFold` assembly is **exactly** that same statement.  So closing the non-orthonormal Sugawara
  computation discharges the state half of `htrans` **and** `hvir` at once.

**What is landed on that computation** (`MM.FLM87TC`, batch 165 Y1, verified):

    sugawaraModeZero_mul_modeZ_sub : [sugawaraModeZero m, modeZ (bL l) n]
                                   = (((−n : ℤ)) : F) • modeZ (bL l) (m + n)

for an **arbitrary `ℤ`-basis and arbitrary symmetric form, with no `OrthonormalBasis`** — the first
non-orthonormal Sugawara result in the repository, and the whole `[L, mode]` half of (8.7.9).  Plus the
corrected (7.1.33) CCR `modeZ_mul_modeZ_sub` (with the lattice zero mode), the one-contraction pair
identity `normalOrderedHeisPairZero_mul_modeZ_sub`, the Gram collapses `inverseGram_collapse_left`/`_right`,
and the two contraction coefficients `gramContraction`/`gramContraction_transpose`/`normContraction`.

**The one computation that remains** (named in `MM.FLM87TC` §7 as `RegularInverseGramBracket`): the
**double commutator** `[:b_i(m−p)b_j(p):, :b_k(n−q)b_l(q):]`, where *both* contractions occur at once.
The contraction bookkeeping is already measured and is not guesswork any more:

* the **Gram contraction** `Σ_j (M⁻¹)_{ij}⟨b_j,b_l⟩ = δ_{il}` (and its transpose, needing
  `dualCoeff_symm`) is the only contraction in `[L,mode]`, entering **twice** — at `k = −n` against the
  first factor (coefficient `M_{il}`) and at `k = m+n` against the second (`M_{jl}`) — each half giving
  `−n·b_l(m+n)/2`, so that the outer `(1/2)` of (8.7.5) makes the halves sum to `−n·b_l(m+n)`;
* the **trace** `Σ_{ij}(M⁻¹)_{ij}⟨b_i,b_j⟩ = card κ` occurs **only** in the double commutator, and **that
  is where the central charge `dim 𝔥` comes from.**

**Two secondary gaps were measured while doing this, and both are worth stating so they are not
re-discovered:**

1. brackets mixing `sugawaraMode` with `zeroCorrPart` are **not statable today**: `zeroCorrPart` is a plain
   function rather than an endomorphism, and the landed decomposition
   (`MM.FLM87ZDG.sugawaraModeZero_apply_eq_regular_add_zeroCorr`) is pointwise, so a linearity statement
   for it is missing;
2. `Factor`-level scalar bookkeeping is now available (`MM.FLM810DFA.factorSmul`), so the `dFold`
   assembly's piece (1) is closed; the brief's literal `dFactor (factor n b) = n • factor (n+1) b` is not
   even type-correct, and the library's recurrence is the chain-rule orientation
   `X_{n+1} = (n:ℚ)⁻¹ • deriv X_n`.

**And a warning worth keeping**: `MM.FLM87TC` is 509 lines against a 300-line target because the task
forbade a second file; **And a warning worth keeping**: `MM.FLM87TC` is 509 lines against a 300-line target because the task
forbade a second file; the coefficient/CCR layer should be split into a sibling file on the next pass.

## 70. Batch 173: the §8.10 substitution, the Θ corollaries, and what the book's route to Jacobi is

### 70.1 What landed

* `Definitions/MMFLM810RatSubst.lean` (`MM.FLM810RSub`): the substitution σ of (8.10.66), i.e.
  `locSubst : F[z_1,z_2]_S → F[z_1,z_2]_S` with `z_1 ↦ z_0+z_2`, `z_2 ↦ z_2` (source and target are
  the same localisation; index 0 reads `z_1` in the source and `z_0` in the target), by
  `IsLocalization.lift`; its two `@[simp]` generator equations; uniqueness among constants-fixing
  homs; and `rationalForm_locSubst`: (8.10.51)'s shape maps to (8.10.63)'s shape.  **The ledger item
  "the construction of σ is infrastructure of its own" of `MMFLM810RatCoef.lean:258`--`:267` is
  discharged.**
* `Definitions/MMFLM810ThetaKill.lean` (`MM.FLM810TK`): (8.10.73) verbatim; the reusable Laurent
  membership lemma; the general (8.10.46) for a Laurent-polynomial multiplier; and (8.10.72) as
  `recoverySplit_exists`.
* The Θ-side of the §8.10 formal calculus is therefore **complete** for the recovery: (8.10.45) both
  halves (`MMFLM810ADMain`, `MMFLM810IotaEq.lean:127`), (8.10.46) (`MMFLM810ThetaZero.lean:262` plus
  the new general form), (8.10.47) (`MMFLM810IotaEq.lean`), (8.10.72), (8.10.73).

### 70.2 The two false statements this batch found, and the repair

1. **`SubstForm` is too strong.**  It quantifies over *all* ring homs σ with the two generator
   equations, and such a σ need not fix the constants of `F`: for any field endomorphism φ the
   φ-semilinear substitution satisfies the same two equations and sends `locC a` to `locC (φ a)` (the
   side condition survives because φ is injective).  Hence `SubstForm f (locSubst f)` fails for `f`
   with constants over any field with a nontrivial endomorphism (`ℚ(x)`, `𝔽_p(t)`, ...), and
   **`Proposition8104b`/`Proposition8105`, whose conclusion is `SubstForm f h`, are false in that
   generality**.  The correct form is `SubstFormAlg` (constants fixed, equivalently `F`-linear), for
   which `SubstFormAlg f h ↔ h = locSubst f` **is** proved; for `F = ℚ` the two coincide, so the
   moonshine target is unaffected.  **Next pass on the rationality layer must restate those two Props
   with `SubstFormAlg`.**
2. **The "ι-bridge for an arbitrary `VertexOp`" is false** (`not_productRationality`), and the landed
   implication is vacuous (`Jacobi8108 ↔ ZeroProduct`).  This is not an obstacle: in the book the
   bridge is a consequence of **Theorem 8.6.1**, so it must be stated under that hypothesis.

### 70.3 What the book's route to the Jacobi axiom actually is (read from the page images, batch 173)

* Theorem 8.10.2 (printed 245) is the **summary of Chapter 8**; its Jacobi axiom (8.10.8) is
  **Theorem 8.8.9**.
* Theorem 8.8.9's proof (printed 231) is "**We repeat the proof of Theorem 8.6.1** except for the
  following changes: ..." — i.e. the same `P`-coefficient computation, with `x_n` reducing the pole
  order.
* (8.8.39) (printed 233) is "**an interesting alternate form of Theorem 8.6.1**" — so §8.10's forward
  direction (Prop 8.10.3 from the `Res_{z_0}` face) has Theorem 8.6.1 as its input, **not** the Jacobi
  identity.  The ledger sentence at `MMFLM810Rationality.lean:263`--`:269` reading it as circular is
  corrected here.
* **Consequence for the plan.**  The single crux is **Theorem 8.6.1 at the repaired honest field for all
  states** (`Yaligned (C.toCentralExtension h2)`, `C : EvenCocycle`).  From it the book offers two
  continuations: §8.8's repeat of the proof (Jacobi directly), or §8.10's Props 8.10.3--8.10.5 + 8.10.7.
  The campaign's §8.10 infrastructure (ι/Θ/σ/rationality Props) is the second continuation and is now
  nearly complete on the formal-calculus side; the first is untouched.
* **What Theorem 8.6.1 needs, per the batch-153 ledger and this batch's measurements**: the (8.4.25)
  two-point product at the honest field (the landed `MMFLM86ExpProduct` is the creation half only —
  `Yzero (ι(a))_m = actVL a · E^-(a)_m`, missing the grading shift and the annihilation half), the
  `x_n` calculus of (8.6.17)--(8.6.24), and the transfer of the batch-152 display from the old field to
  `baseAligned`.  Batch 153's measured cost of the remaining chapter was ~3.5--6.5 kLoC.

## 71. Batch 174: what is left of Jacobi, and it is now one named `Prop`

### 71.1 The crux, reduced

`Definitions/MMFLM810Theorem861Aligned.lean` (`MM.FLM810T861A`, B174-A) proves, at `1 ≤ K`, the
**equivalence** (`source861Aligned_iff_residual`)

    Theorem861SourceStatement form (C.toCentralExtension h2) (ZAligned E) u' v' a b K
      ↔  Corollary851 (ZAligned E) (stateOf u' a) (stateOf v' b) K,

with `ZAligned E = (vertexOpOfField (Yaligned E)).coeff` the boundary field of Theorem 8.10.2.  So the
whole of `hjac` (8.10.8) — hence the last field of Theorem 8.10.2 — is **Corollary 8.5.1 (8.5.13) at the
(8.6.2) states**, a single named `Prop` (`Residual861Aligned`).  Discharged at that field: (8.6.1) (a
theorem there, `hypothesis861_of_evenCocycle`), integrality (by type), the base (a definition), and the
grading factor (redundant in this carrier).  Proved: the corner `u' = v' = 1`, `0 ≤ ⟨a,b⟩`, every window.

**The old-field obstruction is not applicable**: `not_l0OnWeightGenerators_modelBase` and
`not_translationOnStateData_modelBase` are stated with `hY` about `actionPrescriptionTA … modelBaseField`,
while `Yaligned E` has base `baseAligned E`; the two fields are provably distinct
(`Yaligned_ne_Yzero_modelBaseField`) and both predicates are theorems at the aligned field.

### 71.2 The two halves that remain

1. **The Fock half**: `u', v' ∈ M(1)` arbitrary.  `MMFLM86ExpProductAligned.lean` (B174-B) lands the
   honest-field (8.4.25) product at the lattice states by **reindexing** the already-landed engine
   (`MMFLM86YFamProduct`, B154-Q1; `MMFLM86CommIota`, B156-T2) to `Yaligned`, with
   `Yaligned_mul_eq_sum_ratioCoeff` (`0 ≤ ⟨a,b⟩`, contraction series `ratioCoeff`) and
   `commutatorFamily_Yaligned_eq_smul`.  What is missing for general `u'` is the outer evaluation
   `YFam E a m (creationExpVL b j (anchorVL E b c))` — evaluated nowhere (B154-Q1 residual (i)) — i.e.
   FLM (8.6.10)/(8.5.12): the field at a *created* state.
2. **The negative-pairing lattice case** `ResidueNegativeAligned`: `form a b < 0` at the lattice
   generators.  The finite-support carrier cannot express the contraction series `contrFam N` for
   `N < 0` (refuted: `not_prodExpTwoVarEnd_rankOne`, `not_locallyFinite2_contrFam_of_neg`,
   `ratioCoeff_ne_zero_of_neg`), so this is a carrier-extension question, not a bookkeeping one.

### 71.3 FLM's §8.6 direct route, re-read at the repaired field (B174-A's map)

The repair changes exactly one prerequisite: the (8.5.2) link `Y(ι(a),z) = Y(a,z)`, refuted at the old
model base and now the theorem `yFamOnGenerators_vertexOpOfField_Yaligned`.  Of FLM's proof
(8.6.4)--(8.6.27): proved source-faithful are (8.6.4), (8.6.7), (8.6.11), (8.6.13), (8.6.14) per monomial
and (8.6.22); absent are (8.6.5) (multi-variable series of *states*), (8.6.23)/(8.6.24) (scalar bookkeeping
only) and (8.6.25)--(8.6.27); stated-not-proved are (8.6.6), (8.6.8)--(8.6.10), (8.6.12), (8.6.15),
(8.6.17), and (8.6.20)/(8.6.21) whose commutator side is (8.6.3)'s extraction itself; legacy proxies are
(8.6.16), (8.6.18), (8.6.19).  **The direct route is blocked by three objects the repository does not
have**: the multi-variable series of states (8.6.5), the shifted field `Y(a,z+w)`
((8.6.9)/(8.6.10)), and the `k`-fold normal-ordered product ((8.6.6)/(8.6.8)).  Remark 8.3.9's spanning
for `u, v ∈ V_L` is also unlanded.

### 71.4 What the next batches must do, in order

* **B175-A**: the Fock half of (8.5.13) at `0 ≤ ⟨a,b⟩` — extend
  `corollary851_Yaligned_lattice_of_nonneg` from `u' = v' = 1` to arbitrary `u'`, or land the exact
  obstruction (the created-state evaluation).
* **B175-B**: `ResidueNegativeAligned` — the `⟨a,b⟩ < 0` carrier extension, or the exact statement of
  what the carrier must become.
* Then: the boundary restatement at the even cocycle (`theorem8102_of_hs` carries `E` free and is
  refuted at the trivial cocycle), and the §8.10 or §8.8 continuation from the completed (8.6.3).

## 72. Batch 175: the Fock half reduces to the Fock monomial basis; the negative pairing splits; one missing engine remains

### 72.1 The book's (8.5.13) is the LATTICE case only

FLM p. 195 Theorem 8.4.2 is for `a, b ∈ L̂`, and p. 200 Corollary 8.5.1 says "In the notation of Theorem
8.4.2, [Y(a,z₁),Y(b,z₂)] = Res_{z₀} z₂^{−1}Y(Y(a,z₀)ι(b),z₂)e^{−z₀(∂/∂z₁)}δ(z₁/z₂) (8.5.13)" — no
quantification of its own.  **So (8.5.13) is the lattice states; general `u, v ∈ V_L` is Theorem 8.6.1**,
and the repository's `MM.FLM85Co.Corollary851` at arbitrary `u v : V_L` is *strictly stronger* than the
book's (8.5.13) — it is Theorem 8.6.1's content.

### 72.2 The reduction landed (B175-A, `Definitions/MMFLM85CorollaryFock.lean`)

* Both sides of `Corollary851` are `F`-linear in each state, so it transports along spans:
  `corollary851_of_mem_span`, `corollary851_of_spanning`, `corollary851_iff_spanningStates` (with the
  landed `spanningStatement`).
* **The Fock half is equivalent to its values on the Fock monomial basis**
  (`residual861Aligned_fockBasis_iff`, `corollary851_stateOf_basis`): it suffices to prove
  `Corollary851 (ZAligned E) (stateOf (fockMonomialBasis bL d) a) (stateOf (fockMonomialBasis bL d') b) K`
  for all `d, d'`.  `corollary851_vl_iff_residual861Aligned` shows the general-`V_L` statement is
  equivalent to the Fock-half residual.
* **What does NOT work**: bilinearity alone gives only a spanning family, not a single-oscillator law; no
  lemma computes `Corollary851` at `b_i(−n)·x` from `x`; FLM (8.5.5) coefficientwise at `u = a(−n)x` is
  nowhere landed; and the residue itself contains `Y ((Y (stateOf u' a)) k (stateOf v' b)) j` — the field
  of a created state — which nothing computes.  The mixed case `u' = 1, v'` arbitrary is not free either.

### 72.3 The negative pairing (B175-B, `Definitions/MMFLM86ResidueNegative.lean`)

* The residue is the exact finite (8.5.26) sum, with the `i`-th summand a **positive-degree created
  state** `creationExpVL a ((N:ℤ)−1−i) (anchorVL E a b)`; the top summand `i = N−1` is closed
  (`ι(ab)`), and `residueNegative_split` separates them.
* `ResidueNegativeAligned ↔ NegOneResidualAligned ∧ NegTwoAndBelowResidualAligned`
  (`residueNegativeAligned_iff_negOne_and_below`); the `⟨a,b⟩ = −1` case is *equivalent* to the honest
  commutator (8.4.42) at `YFam` (`negOne_iff_commutator`), so `YFamOnGenerators` being a theorem does
  **not** make it unconditional.
* **The carrier is not the wall.**  `negContraction_tail` proves the contraction series is finite at
  every vector; the (8.4.26)/(8.4.27) two-chamber identity is named `NegativeChamberExpansion (E)`, its
  finiteness half is a theorem and its scalar split is landed
  (`MM.FLM87.expansionA_sub_expansionB_of_neg`).
* The `≤ −2` gap is **not** (8.6.10)/`Display8610List` (that display needs a product the carrier lacks)
  but the two-point (8.5.12) `:Y(a,z₂+w)Y(b,z₂): w^{⟨a,b⟩}` at negative `w`-powers.

### 72.4 What is left, precisely

`hjac` (8.10.8) at the boundary field = `Corollary851 (ZAligned E) u v K` for all `u, v ∈ V_L`
(equivalently the Fock basis), plus the even-cocycle restatement of the boundary.  Both the Fock half and
the negative chamber need **the field at a created state**: FLM (8.5.5) coefficientwise at `Yaligned E`,
equivalently the shifted field (8.6.10) `Y(Y(a,w)ι(1),z) = Y(a,z+w)`.  The named unproved Props for it
are `MM.FLM86SF.Display8610One`/`Display8610List`, `MM.FLM85IS.IterateProductStepAt`/
`IterateProductStepOnCreated`, and `MM.FLM85Co.ProductExpansion` = `MM.FLM85Ext.iterateStatement`
(8.5.12).  Landed starting points for (8.5.5): `Yzero_spec`, `Yzero_unique`,
`generalVertexOperator_existsUnique`.

## 73. Batch 176: the two engines, and a free theorem the agents missed

### 73.1 `NegativeChamberExpansion` was a theorem waiting to be noticed

`MM.FLM86NC.negativeChamberExpansion : NegativeChamberExpansion E` is proved in **11 lines**, because
the equality is the landed `MM.FLM86RX.threshold_holds` (`MMFLM86Reindex8612.lean:880`, for **every**
pairing) and the finiteness is the landed `MM.FLM86RN.negContraction_tail`.  **The wall is not the
chamber expansion but the Θ-collapse**: the landed chain is
`NegOneResidualAligned ↔ NegOneChamberCollapse ↔ NegOneBlockCollapse` (the (8.4.27)
`Θ((1−z)^{−1}) = δ(z)` step on the antidiagonal `P+Q+1`), and
`NegTwoAndBelowResidualAligned ↔ NegTwoChamberResidueIdentity` (which keeps the
`ZAligned`-at-created-states summands and so meets B176-A's gap).

### 73.2 (8.5.5) coefficientwise, and the vacuum field of (8.6.10)

* `Definitions/MMFLM85PrescriptionCoeff.lean` (`MM.FLM85PC`): the (8.5.5) prescription coefficientwise at
  `ZAligned E` (`ZAligned_spec`, `ZAligned_spec_cons` = the (8.4.16) `headOp`+`tailOp` form,
  `ZAligned_creationStep` at an arbitrary vector).  **Already landed at an arbitrary base** as
  `MM.FLM810CI.Yzero_modeVL_eq_actionFamily(_nat)` and `actionPrescriptionTAZero_cons`; these are the
  `ZAligned`-indexed transports.  The `Yzero_unique` route does **not** apply to (8.6.10) (it compares
  state-linear maps, while (8.6.10) constrains one state's value).
* **`vacuumFieldAt_ZAligned : VacuumFieldAt (ZAligned E) vacuumVL` is a theorem**, so at the honest field
  `Display8610List`/`Display8610One` follow from the single input
  `IterateProductStep (ZAligned E) conv` ⟺ `MM.FLM85IS.CreatedStepResidual`
  (`createdStep_ZAligned_iff`), and the k=1 case is the landed (8.5.12) `iterateStatement`.
* The levels: `j = 0` theorem; `j = 1` ⟺ `BaseResidual (baseAligned E)`, and proved on the lattice
  generators; `j ≥ 2` open.

### 73.3 The free theorem (dispatcher)

`MM.FLM810TNO.baseRepaired_eq_baseAligned` (`MMFLM810TheoremNoOrth.lean:97`) and batch 172's
`MM.FLM810CFC.baseResidual_of_currentFieldComm : BaseResidual (baseRepaired E)`
(`MMFLM810CurrentFieldComm.lean:392`) together give **`BaseResidual (baseAligned E)`**, hence by
`display8610One_one_level_iff_baseResidual` the **`j = 1` level of (8.6.10) at `ZAligned E` is
unconditional**.  Neither B176 agent noticed (B176-A calls `BaseResidual` still open).  This is the fifth
"already landed under another name" case of the campaign, and the first one the dispatcher found rather
than a task agent.

### 73.4 The two residuals that remain for `hjac`

1. `MM.FLM85IS.CreatedStepResidual` (equivalently `IterateProductStep (ZAligned E) conv`, i.e. the
   one-step (8.4.25) at created states / (8.6.10)) — levels `j ≤ 1` unconditional after batch 177-A,
   `j ≥ 2` untouched;
2. `MM.FLM86NC.NegOneBlockCollapse` and `NegTwoChamberResidueIdentity` — the Θ-collapse telescoping
   along the antidiagonal, whose scalar shadow is `MM.FLM87.expansionA_sub_expansionB_of_neg`
   (`MMFLMY87.lean:910`).

## 74. Batch 177: (8.6.10) at `k = 1` is free; the non-vacuum one-step and the antidiagonal collapse are what remain

### 74.1 A structural theorem: (8.5.23) + (8.10.13) ⟹ (8.6.10) at `k = 1`

`Definitions/MMFLM85LevelShift.lean` (`MM.FLM85LS`) proves the **base-shifted** recurrence and the tower
collapse, for an arbitrary `VertexOp` with a translation axiom and a creation property:

* `creationCoeff_lminusOne : state_j(L(-1)v) = (j+1)·state_{j+1}(v)` — a pure reindexing of (8.10.13),
  footprint `[propext, Quot.sound]` (strictly smaller than the three-axiom set);
* `levelAt_succ_iff_levelAt_lminusOne : LevelAt Y vac v (j+1) ↔ LevelAt Y vac (L(-1)v) j` (**both**
  directions), `levelAt_succ_iterate_iff_levelAt_one : level (k+1) at v ↔ level 1 at (L(-1))^[k] v`;
* level 0 is the creation property `CreationProperty` (8.5.23), landed for every state at the honest field;
* hence `display8610One_of_creation_translation : CreationProperty Y vac → TranslationAxiom Y om →
  Display8610One Y.coeff a vac` — **(8.5.23) + (8.10.13) ⟹ (8.6.10) at `k = 1`**, with no Jacobi, no
  (8.4.25), no base residual and no uniqueness argument.  (`Yzero_unique` does not apply: it compares
  state-linear maps, while the display constrains one state's value.)  The other recurrence
  `L(-1)·state_j = (j+1)·state_{j+1}` is neither landed nor needed.

At `ZAligned E` (only `hnd`, `hs`): `display8610One_ZAligned`, `levelAt_ZAligned`,
`iterateStatement_ZAligned` ((8.5.12) at `b = ι(1)`) and `iterateProductStepAt_vacuum_ZAligned` — the
**vacuum half** of `IterateProductStep` is a theorem.  The free `j = 1` slice
(`display8610One_one_level_ZAligned`) also follows from batch 172's `BaseResidual` via
`baseRepaired_eq_baseAligned` (the dispatcher's batch-176 finding), as
`baseResidual_baseAligned_of_symm`.

**New exact residual**: `NonvacuumCreatedStep E` — the one-step (8.4.25) at non-vacuum created states
(equivalently `k ≥ 2` of (8.6.10)) — with `createdStepResidual_ZAligned_iff_nonvacuumStep` and
`display8610List_ZAligned_iff_nonvacuumStep` both directions.

### 74.2 `NegOneBlockCollapse` is REFUTED; the corrected residual is `c`-free

`Definitions/MMFLM86NegOneCollapse.lean` + `MMFLM86NegOneCollapseDiag.lean` (`MM.FLM86NOC`):

* `negOneBlockCollapse_eps_add_smul_eq_zero`: if the batch-176 residual held, then
  `(E.eps a b + E.eps b a) • YFam E (a+b) (m+n) v = 0`, because the instances at `(a,b,m,n)` and
  `(b,a,n,m)` are the same two window sums; at `a = b` this kills `YFam E (2a)` with `c a a = 1`.
  `not_negOneBlockCollapse_rankOne` refutes it at the landed rank-one datum (and the general obstruction
  applies to an even form, e.g. hyperbolic `ℤ²` with the trivial cocycle).  **The missing hypothesis is
  the source's (8.6.1) `c(b,a) = −1`.**
* **No pointwise telescoping exists**: the two chamber sums are the two complementary half-lines
  `p ≥ −m` and `p ≤ −m−1` of the same antidiagonal.  The corrected residual is `AntidiagonalCollapse`
  (both half-sums of the single family `normOrdered2 E a b` add to `ε(a,b)·YFam E (a+b)(m+n)`), and
  `negOneBlockCollapse_iff_antidiagonalCollapse (hs) (E) (hc : NegOneSourceCocycle E)` is proved in
  **both** directions.  `NegOneSourceCocycle` holds at the boundary field (`CocycleCompatible` at
  `⟨a,b⟩ = −1` gives `c(b,a) = −1`).
* What remains there is the assembly `(:YY:_{ab})(z,z) = ε(a,b)·Y(a+b,z)` (creation exponentials add,
  annihilation adds, the middle block gives `ε(a,b)(a+b)z^{(a+b)(0)}`).
* `≤ −2`: its summands are `ZAligned` at the (8.5.11) created states = `IterateProductStepOnCreated`;
  **the `−1` case is the first step of that induction and the `≤ −2` case its iterate**, so both agents'
  residuals meet at the same created-state evaluation.

### 74.3 The three residuals left for `hjac`

1. `NonvacuumCreatedStep E` (the non-vacuum one-step);
2. `AntidiagonalCollapse` under `NegOneSourceCocycle` (the `c`-free antidiagonal delta collapse; its
   remaining assembly `(:YY:_{ab})(z,z) = ε(a,b)Y(a+b,z)`);
3. the `≤ −2` iterate, which meets (1).

Everything else — the boundary field, (8.6.1), the Fock spanning reduction, the pairing split, the vacuum
step, the k=1 display — is landed.

## 75. Batch 178: the `conv`-form §8.6 chain is refuted (carrier defect); `AssemblyDiagonal` is the `−1` case's single residual

### 75.1 The refutation, and what it does and does not touch

`Definitions/MMFLM85NonvacuumStep.lean` + `MMFLM85NonvacuumRefute.lean` (`MM.FLM85NV`/`MM.FLM85NVR`):

* `NonvacuumCreatedStep E ↔ SingletonCreatedStep E` (FLM (8.5.23) alone — no list induction) and
  `↔ IterateProductStep (ZAligned E) conv`, and with `hnd/hs` `↔ Display8610Two E` (8.6.10 at `k = 2`,
  which implies every `k`).
* **The one-factor case is FALSE**: `ZAligned E (ι(a)) j ι(1) ≠ 0` for every `j ≤ −1` and `a ≠ 0`, so
  `¬ LocallyFiniteCoef (ZAligned E (ι(a)))`; `conv` is a `dite` on `LocallyFiniteCoef` of its second
  argument, so `conv A (YFam E a) = 0`, and the step at `ι(a)` demands `YFam E a = 0`.  Hence
  `not_nonvacuumCreatedStep_of_ne_zero`, `not_createdStepResidual_ZAligned`,
  `not_iterateProductStep_ZAligned`, `not_display8610List_ZAligned`, `not_display8610Two_of_ne_zero`.
* **Defect class**: the carrier's two-sided `LocallyFiniteCoef` gating (batch 133 refuted the two-sided
  `LocallyFinite2` for `normOrdered2`).  The fix is a **one-sided** product notion consuming (8.10.5)
  (`MM.FLM84LF.YFam_eq_zero_of_gt`) and/or the joint `8…8` object of (8.6.25); the missing FLM object is
  **(8.6.6)'s third family** (`MM.FLM86KF.Display866Family3`) — the ordinary-vs-normal-ordered OPE
  contraction — not the already-landed (8.6.5)/(8.6.9)/`kFoldNormalOrder`.
* **What is NOT touched**: `hjac`/`JacobiAxiom`, and the target `Corollary851 (ZAligned E) u v K`
  (a statement about `commutatorFamily` and the residue, not about `conv`).  Batch 177's `k = 1` theorem
  and the vacuum step stand.  `display8610List_ZAligned_iff_nonvacuumStep` is a valid equivalence between
  two false statements.
* **Trap**: `Definitions/MMFLM85StepLinear.lean` is imported by nothing although `MMFLM85IS`'s header
  cites its `iterateProductStepAt_iff_apply` as landed; without the explicit import the statement
  elaborates with `sorryAx` (the guards catch it).

### 75.2 `AssemblyDiagonal`: the `⟨a,b⟩ = −1` case's single residual

`Definitions/MMFLM86AntidiagonalAssembly.lean` (`MM.FLM86AA`) lands `AssemblyDiagonal (E)` — the
pairing-free (8.4.20) at `z₁ = z₂ = z`: the two half-sums of `normOrdered2 E a b` along the antidiagonal
add to `ε(a,b)·YFam E (a+b)(m+n)` — and proves
`assemblyDiagonal_iff_collapse : NegOneAssemblyDiagonal E ↔ AntidiagonalCollapse E` (both directions),
plus the consumer chain to `negOneResidualAligned_evenCocycle_of_assemblyDiagonal`.  **The hard step is
`expField` multiplicativity** (`E⁻_a(z)E⁻_b(z) = E⁻_{a+b}(z)`), landed nowhere; the landed
`creationExpVL_mul_comm` is commutation only.  Finiteness is already handled by
`normOrdered2_locallyFinite2OneSided`.

### 75.3 The three live residuals for `hjac`

1. `AssemblyDiagonal` (the `⟨a,b⟩ = −1` case), gated by `expField` multiplicativity;
2. the `≤ −2` iterate (`NegTwoChamberResidueIdentity`);
3. the Fock half, whose `conv`-form route is dead and must be rebuilt on a one-sided product.

## 76. Batch 179: the one-sided repair, and `expField` multiplicativity (a new Leibniz rule)

### 76.1 The repair (B179-B, `Definitions/MMFLM85OneSidedProduct.lean` + `MMFLM85OneSidedChain.lean`)

* Defect verbatim: `conv A B` is a `dite` on the **two-sided** `LocallyFiniteCoef B`; the honest field
  only satisfies the upper half ((8.10.5), `YFam_eq_zero_of_gt`), so `conv A (YFam E a) = 0`.
* Repair: `PairFinite A B` (a **pair** datum, not one-sidedness of one argument) and the total
  `convOneSided`, with `convOneSided_eq_conv` (agreement where `conv` was defined),
  `convOneSided_vacuumField(_left)` unconditional, `convOneSided_assoc` on the locally-finite class.
* **`vacuumShiftStep_ZAligned_convOneSided`**: the instance batch 178 refuted for `conv`
  (`not_iterateProductStepAt_ZAligned_iotaVL`) is now a **theorem**, for every `v`.
* New shortest residual: `Display8610TwoOneSided` / `SingletonCreatedStepOneSided`.
* **Two honest fields still do not form a `PairFinite` pair** (`PairFiniteHonestFields` recorded, not
  asserted): the general two-factor display is the **OPE gap** `MM.FLM86KF.Display866Family3`, not a
  carrier problem.

### 76.2 `expField` multiplicativity (B179-A, `Definitions/MMFLM86ExpFieldMul.lean` + `MMFLM86ExpFieldMulLat.lean`)

* `flatMul_expField_of_commute : flatMul (expField Om) (expField Om') = expField (Om + Om')` for
  coefficient-commuting exponents, over an arbitrary `Ring A`/`Algebra ℚ A`.
* **The missing engine was the Leibniz rule** `deriv_flatMul : deriv (flatMul X Y) =
  flatMul (deriv X) Y + flatMul X (deriv Y)` — recorded as open in `MMFLM84Diag`/`MMFLM84Subst` and
  landed here for the first time.
* Steps 1 and 3 of `AssemblyDiagonal` are landed in antidiagonal form
  (`flatMul_creationExpVL`, `sum_creationPair2_antidiagonal`, `flatMul_annihExp`,
  `sum_annihPair2_antidiagonal` — the `E⁺` half on the mirrored families).
* **Steps 2 and 4 are still open**, so `AssemblyDiagonal` stays a `Prop`; step 2 is the named
  `MM.FLM86EFM.MiddleBlockDiagonal` (route recorded: collapse on `F{L}` via `zPowH_iota` and
  `form a c + form b c = N`, then the tensor-factor transfer; three build rounds failed on `iota` defeq
  and induction-motive normalisation).

### 76.3 The live residuals after batch 179

1. `AssemblyDiagonal` — steps 2 (`MiddleBlockDiagonal`) and 4 (nested `convWith2` → `convWith`);
2. the `≤ −2` iterate (`NegTwoChamberResidueIdentity`);
3. the Fock half — its repaired one-sided chain reduces to `Display8610TwoOneSided`, i.e. the OPE third
   family `Display866Family3`.

## 77. Batch 180: `MiddleBlockDiagonal` landed; step 4 is one index-regrouping Prop; the OPE third family corrected and bridged

### 77.1 `AssemblyDiagonal` is now one measured regrouping away

`Definitions/MMFLM86MiddleBlockDiag.lean` + `MMFLM86MiddleBlockDiagStep4.lean` (`MM.FLM86MBD`):

* Step 2 = `middleBlockDiagonal` **is a theorem** (the middle block on the antidiagonal collapses to
  `ε(a,b)·(a+b)z^N`).  The route that works: on `F{L}`, `zPowH_iota` + `form a c + form b c = form (a+b) c`
  for the eigenvalue, then `TensorProduct.induction_on` + `zPowHVL = TensorProduct.map id` +
  `actVL_mul_actVL`; the induction tool is `Finsupp.induction_linear` (add/single) with
  `Finset.sum_subset`, **not** extensionality.
* Step 4 is reduced to the single `def AssemblyDiagonalResidual (E) : Prop` (the double-window sum equals
  `ε(a,b)·YFam (a+b) (−N−1) v`), with `assemblyDiagonal_of_residual : AssemblyDiagonalResidual E →
  AssemblyDiagonal E` **proved** and the reindexing lemmas `diagSum_convWith2`,
  `diagSum_convWith2_nested`, `sum_creationPair2_window` landed.
* **Recipe for the next task**: use a **uniform** middle window `w' q = ⋃_{q∈w}(canonical window)`
  (per-`q` windows leave incomparable `K'`-ranges); group by `J = q.1+q.2` (step 3), then by
  `K' = q'.1+q'.2` (step 2, now a theorem); read `(J,K')` as `YFam`'s `(j,i) = (−J−1,−K'−1)` via
  `YFam_eq_sum_three_factor`.
* So the `⟨a,b⟩ = −1` case at `ZAligned (C.toCentralExtension h2)` is unconditional modulo
  `AssemblyDiagonalResidual` alone.

### 77.2 The OPE third family: a scalar mis-reading corrected, and the bridge landed

`Definitions/MMFLM86OPEThird.lean` (`MM.FLM86OPE`):

* (8.6.6) verbatim (printed p. 206): `A = 8Y(a₁,w₁)…Y(a_k,w_k)8 ι(1) = Y(a₁,w₁)…Y(a_k,w_k)ι(1)
  ∏_{i<j}(w_i−w_j)^{−⟨ā_i,ā_j⟩}`, the factor being the binomial expansion in non-negative powers of
  `w_j`.
* **`MM.FLM86KF.Display866Family3` is not the landed two-variable product**: it reads the printed *series*
  factor as a *scalar* (true only at null exponents) and its type reads the multidegree only in the first
  coordinate.  Landed instead: `Display866Family3TwoIdx` (+ the proved null-pair instance) and
  `Display866Family3Two` (the printed third family at `k = 2`, unconditional for `0 ≤ ⟨a,b⟩`).
* **The bridge**: `shiftedField_ZAligned_apply_eq_sum_ratioCoeff` — (8.4.25) at the **shifted** honest
  fields, i.e. the display's right-hand side evaluated by the landed `normOrdered2`+`ratioCoeff`
  expansion.
* The display's generator slice is `Display8610TwoGenerators`, equivalent (both directions) to the ratio
  form given the **recorded, not asserted** datum `ShiftedPairFiniteGenerators`.  The Fock half stays
  open: the created-state equation, the datum (or a carrier-free replacement), and the arbitrary-`x y`
  display.

### 77.3 The live residuals after batch 180

1. `AssemblyDiagonalResidual` (the measured regrouping) → the `⟨a,b⟩ = −1` case;
2. the `≤ −2` iterate;
3. the Fock half's two-factor created-state display, with the OPE bridge landed.

## 78. Batch 181: `AssemblyDiagonal` is a theorem (the `−1` case closes); `PairFinite` is the Fock fork

### 78.1 Milestone: the `⟨a,b⟩ = −1` case of `hjac` is closed at the boundary field

`Definitions/MMFLM86AssemblyResidualFibers.lean` + `MMFLM86AssemblyResidualAux.lean` +
`MMFLM86AssemblyResidual.lean` (`MM.FLM86AR`):

* `assemblyDiagonalResidual` proved **verbatim** (batch 180-A's def, no weakening), with the new fiber
  lemmas `sum_filter_fiber_eq`, `sum_annihPair2_fiber`, `sum_midBlock2_fiber` and
  `sum_sum_window_regroup`.
* `assemblyDiagonal (hs) (E) : AssemblyDiagonal E` — **a theorem**; with the landed
  `negOneResidualAligned_evenCocycle_of_assemblyDiagonal` this gives
  **`negOneResidualAligned_evenCocycle`**: `NegOneResidualAligned` at `ZAligned (C.toCentralExtension h2)`
  for every symmetric form.  Together with batch 170's `0 ≤ ⟨a,b⟩` corner, **two of the three
  lattice-pair pairing cases are closed**.
* **Method of record**: the middle window must be **one common window for all `q`** (a rectangle
  `WP ×ˢ WQ` of the projections of the per-`q` windows), not a `q`-dependent `⋃`; and pinning the landed
  canonical-window collapses to an arbitrary window needed the fiber lemmas.

### 78.2 The Fock fork: `PairFinite` is UNDECIDED

`Definitions/MMFLM86TwoFactorDisplay.lean` (`MM.FLM86TFD`):

* The display's LHS is evaluated at the `j = l = 0` slice
  (`createdState_zero_zero_apply : ZAligned E (…) (−1) vac = ε(a,b)·ι(a+b)`,
  `createdState_zero_zero_ne_zero`), and `iteratedCreation_two` identifies the LHS state.
* **`convOneSided_eq_zero : ¬ PairFinite A B → convOneSided A B = 0`** (definitional), hence
  `display8610TwoGenerators_false` and `not_display8610TwoOneSided`: at any null pair, a failing
  `PairFinite` **refutes** the display.  The generator slice is exactly
  `[PairFinite datum] ∧ [(8.4.25) ratio form]`.
* **`PairFinite (ZAligned E (ι a)) (ZAligned E (ι b))` is undecided**: batch 179-B's reason for expecting
  failure (a `j`-independent weight summand) only shows all summands lie in one weight space, which is
  infinite-dimensional and compatible with both behaviours.
* **The fork**: if `PairFinite` holds on the needed slices, the Fock half is exactly the created-state
  evaluation — the (8.5.5) prescription of the coherent state
  `YFam E a (−j−1)(creationExpVL b l (ι b))`, **not** a (8.5.4) state for `l > 0`; if it fails, the display
  is false and the honest replacement is the OPE product `Display866Family3Two` (landed for `0 ≤ ⟨a,b⟩`).
* The span reduction cannot be transported through the `dite` carrier (linearity holds only with the datum
  fixed), so the arbitrary-`x y` half is blocked by the carrier, not by spanning.

### 78.3 The residuals after batch 181

1. the `⟨a,b⟩ ≤ −2` case (`NegTwoAndBelowResidualAligned` / `NegTwoChamberResidueIdentity`);
2. the Fock half — decide `PairFinite` (fork above), then either the created-state evaluation or the OPE
   replacement;
3. the final boundary restatement at the even cocycle.

**Superseded by §79 (batch 182): `PairFinite` is PROVED on the honest pairs with `0 ≤ ⟨a,b⟩`; the
"infinite sum" reading above is FALSE.**

## 79. Batch 182: `PairFinite` PROVED (the Fock fork closes); `⟨a,b⟩ = −2` reduced to `NegTwoBlockCollapse`; corrections of record

### 79.1 CORRECTION OF RECORD — the "infinite sum" claim of §§75--78 is FALSE

The paragraphs in §§75--78 (and in `MMFLM85OneSidedProduct.lean`, `MMFLM85OneSidedChain.lean`,
`MMFLM86OPEThird.lean`, `MMFLM86TwoFactorDisplay.lean`) that assert that two honest fields do **not**
form a `PairFinite` pair, on the grounds that the summand weight is `j`-independent, are **wrong**:

* `MM.FLM85PFD.pairFinite_ZAligned_iotaVL` / `pairFinite_yFam` /
  `pairFinite_shiftedField_ZAligned_iotaVL` / `pairFiniteHonestFields_of_nonneg` **prove** the datum
  whenever `0 ≤ ⟨a,b⟩`; the key identity is (8.4.25) along the diagonal `(m−1−j, j)`, where the two
  `normOrdered2` indices move **oppositely** in `j`, and the per-state (8.10.5) bounds give the window
  `Icc (K1+m+1) (−K2−2)`.  Weight-independence is compatible with finite support, and the support is
  finite.
* The citation of "FLM Remark 8.5.2" was a misreading: that Remark (printed p. 200) says the
  *created-state* identity (8.5.14) is not rigorous; **Corollary 8.5.4 (printed p. 201) records "the sum
  being finite"**.
* Consequently `not_display8610TwoGenerators`, `display8610TwoGenerators_false` and
  `not_display8610TwoOneSided` have an **unsatisfiable** hypothesis where `0 ≤ ⟨a,b⟩` — they are
  vacuous, not refutations (`not_not_pairFinite_ZAligned_iotaVL`).
* The dispatcher rewrote the five false docstring paragraphs in place as "CORRECTION OF RECORD (batch
  182)" blocks and rebuilt the four modules.

### 79.2 What is now exact

* `shiftedPairFiniteGenerators_of_nonneg` — batch 180-B's recorded datum is a **theorem** (symmetric
  form, `∀ a b, 0 ≤ form a b`).
* `display8610TwoGenerators_iff_ratioForm_of_nonneg` — the display's generator slice **is** the landed
  (8.4.25) ratio form, in both directions, with the datum discharged.
* **The Fock residual is the created-state evaluation** (`display8610TwoGenerators_iff_createdState`):
  the (8.5.5) reading of `YFam E a (−j−1) (creationExpVL b l (ι b))` for `l > 0`; plus `PairFinite`
  for `form a b < 0`, where the honest replacement is the OPE third family `Display866Family3Two`.

### 79.3 The `⟨a,b⟩ = −2` residue

`Definitions/MMFLM86NegTwoResidue.lean` (`MM.FLM86NTR`): the whole `N = 2` residue equals two closed
terms (`residueSum_negTwo_two_terms`, via `coherentField_one` and `ZAligned_anchorVL`), so the exact
residual is the commutator side, named `NegTwoBlockCollapse` (the (8.4.26) chamber difference at
`⟨a,b⟩ = −2`), with `negTwoExactResidualAligned_iff_blockCollapse` both directions.  Composition with
the closed `−1` chain: `residueNegativeAligned_evenCocycle_iff_negTwoAndBelow` — at
`C.toCentralExtension h2` the negative-pairing residue is exactly `NegTwoBlockCollapse` +
`NegThreeAndBelowResidualAligned`.

**Source-language finding**: the residue's states are the coefficients of the **negative** powers
`z_0^{−N}…z_0^{−1}` of `Y(a,z_0)ι(b)`, while the landed display language is `ℕ`-indexed and *defines*
the negative-power coefficients to be `0` (`creationCoeffZ_of_neg`) — the landed (8.6.10) cannot state
the evaluations the residue needs.  Also `MM.FLM8512.rhsFam`/`MM.FLM84T.coeff40Family` use
`opOpenBox`/`openBoxCoeff`, which **drop the `^+` tail** of (8.4.16); residuals must use the honest
`actionFamily` form.

### 79.4 The residuals after batch 182

1. `NegTwoBlockCollapse` (`⟨a,b⟩ = −2`) — the batch-181 machinery one order higher;
2. `NegThreeAndBelowResidualAligned` (`≤ −3`) — needs a closed form for the iterated (8.5.5) at degree
   `≥ 2`;
3. the Fock half's created-state evaluation `ZAligned E (creationExpVL a k (anchorVL E a b))` for general
   `k` (shared with (2));
4. the boundary restatement at the even cocycle.

## 81. Batch 184: the `−2` residual is the `p = 1` line of `diagFam`; the mode intertwining evaluates the Fock LHS

*(File order note: this section sits before §80 (batch 183) in the file; §80 is the batch that precedes it
chronologically.  The `ROADMAP.md` log is in order.)*

### 81.1 `NegTwoDisplayZOne`: the exact `⟨a,b⟩ = −2` residual

`Definitions/MMFLM86NegTwoWeighted.lean` (`MM.FLM86NTW`):

* The batch-183-A weighted line is **equivalent, both directions**, to the `p = 1` coefficient of the
  **landed display family** `MM.FLM86D86.diagFam`/`displayP`:
  `NegTwoDisplayZOne (E)` — for `form a b = −2`, `diagSumWeighted (fun p ↦ (p:F)) (normOrdered2 E a b)`
  on a window equals `iteratePow (ZAligned E) (ι a) (ι b) (form a b + 1) (m+n) v`.
* Landed equivalences: `zOneLine_iff_displayZOne`, `negTwoDiagonalWeight_iff_displayZOne`,
  `negTwoExactResidualAligned_iff_displayZOne` (under `c(b,a) = 1`), and
  **`negTwoExactResidualAligned_evenCocycle_iff_displayZOne`** with the cocycle discharged at
  `C.toCentralExtension h2`; consumer `residueNegativeAligned_evenCocycle_of_displayZOne`.
* `diagFam_zero_eq_diagSum` shows **the `p = 0` line IS the landed `assemblyDiagonal`** — so a
  recursion in `p` for `diagFam` would close `p = 1` from landed material (the batch-185-A target).
* **Route corrections**: the weight is the (8.2.14) falling factorial `m`, not a differentiated (8.4.25)
  expansion (the "derivative" is only the trivial `p = m+1` reindexing,
  `diagSumWeighted_eq_diagSum_firstDeriv`); and the landed (8.4.40) `z_0^1` lemmas carry a negative-power
  vanishing `hX` that is **false** for `YFam E a` (component `−1` is nonzero at pairing `−1`), so they do
  not apply at the boundary field.

### 81.2 The mode intertwining, and the Fock LHS evaluated

`Definitions/MMFLM86ModeIntertwine.lean` (`MM.FLM86MI`):

* `modeIntertwine : YFam E a M1 (ZAligned E (ι b) M2 v) = Σ_{i ≤ ⟨a,b⟩} ratioCoeff ⟨a,b⟩ i •
  normOrdered2 a b (−M1−1+i−⟨a,b⟩) (−M2−1−i) v` — the honest mode of `a` on an honest-mode-created state
  is the finite (8.4.25) ratio sum of normal-ordered blocks.
* `ZAligned_createdState` evaluates the **Fock LHS** `ZAligned E (YFam E a (−j−1)(creationExpVL b l (ι b)))`
  as a finite ratio sum of batch 183-B's `coherentFieldNat` at the **coherent anchors**;
  `display8610TwoGenerators_iff_coherentForm` (both directions) identifies the display's generator slice
  with `CoherentCreatedForm`.
* `normOrdered2_vacuum` / `ZAligned_normOrdered2_vacuum`: the block on `ι(1)` is the double coherent
  state, and its field is the coherent recursion.
* **`ZAligned_JZF_eq : FLM810JZF.ZAligned = FLM810JF2.ZAligned`** — the two definitionally equal but
  syntactically distinct copies of `ZAligned`; `rw` cannot cross them (CF2 speaks JZF, the (8.6.10) chain
  speaks JF2).  Successors must use this bridge.
* Correction: the Fock mode is the assembled `YFam` (an infinite mode combination on a lattice
  generator), so the (8.5.5) step lemmas `ZAligned_creationStep`/`ZAligned_spec_cons` do **not** apply.

### 81.3 The residuals after batch 184

1. `NegTwoDisplayZOne` — the `p = 1` line of `diagFam` (a `p`-recursion would give it from the landed
   `p = 0` = `assemblyDiagonal`);
2. the `≤ −3` case (coherent-recursion form `NegThreeExpandedResidualAligned`);
3. `CoherentCreatedForm` — the (8.6.9)/(8.6.10)-at-`k = 2` re-expansion (the Fock half);
4. the boundary restatement at the even cocycle.

## 80. Batch 183: the `−2` line is WEIGHTED (and needs the source cocycle); the coherent-state field is a recursion

### 80.1 `NegTwoBlockCollapse` is refuted as stated; the corrected residual is `NegTwoDiagonalWeight`

`Definitions/MMFLM86NegTwoDiagonal.lean` + `MMFLM86NegTwoBlockCollapse.lean` +
`MMFLM86NegTwoBlockCollapseAux.lean` (`MM.FLM86NTD`/`MM.FLM86NTB`):

* The batch-182-B residual carries no cocycle hypothesis, and that is fatal: it forces
  `(E.eps a b − E.eps b a) • YFam E (a+b) (N−1) v = 0` for all `N,v`, i.e. `c(a,b) = 1` or
  `YFam E (a+b) ≡ 0` — the analogue of batch 177-B's refutation of `NegOneBlockCollapse`.  At
  `⟨a,b⟩ = −2` the source's (8.4.34) `c(a,b) = (−1)^{⟨a,b⟩}` **is** `c(a,b) = 1`.
* The corrected `c`-free residual is **`NegTwoDiagonalWeight (E)`** (the `z_0^1` line of (8.4.36)): a
  **weighted** antidiagonal sum, weight `p + m`, equals the two closed terms of
  `residueSum_negTwo_two_terms`; `ZOneLineNegTwo` is the unweighted form at degree `−N`, and
  `negTwoDiagonalWeight_iff_zOneLine` holds both ways (using only `AssemblyDiagonal`).
* Bridge to the batch-182 residual: `negTwoBlockCollapse_iff_diagonalWeight` under `hs` and
  `hc : form a b = −2 → E.c b a = 1`; consumer at FLM's datum:
  `residueNegativeAligned_evenCocycle_of_diagonalWeight (h) (h3) : ResidueNegativeAligned`.
* **Why it is new mathematics**: `ratioCoeff (−2) k = k+1` (`ratioCoeff_neg_two`), so the `−2` chamber
  difference is a weighted antidiagonal sum — the batch-181 `AssemblyDiagonal` (`z_0^0` line) cannot
  close it.  `MM.FLM86TPS.diagSumWeighted` is reused; `MM.FLM84DG.opOpenBox_deriv_modeFam`/`_heis` are
  the `z_2`-derivative at a **free** vertex operator (with `DiagCollapse`/`DerivNormalOrdered` as
  hypotheses), not the carrier identity.

### 80.2 The coherent-state field is a recursion; `≤ −3` is expanded to the honest (8.5.5) form

`Definitions/MMFLM86CoherentField.lean` (`MM.FLM86CF2`): `coherentStep` + `coherentFieldNat` with
`coherentField_recursion : ZAligned E (creationExpVL a ((k:ℤ)+1) w) = coherentStep E a w k` —
**degree `k+1` = one (8.5.5) step against the fields of the lower coherent states** (`^+` tail
included), bottoming out at `ε(a,b)•YFam E (a+b)`.  No induction on `k` collapses it: the needed
identity `Y(E⁻(z₀)w,z) = 8E⁻(z₀)Y(w,z)8` is not landed (`NonvacuumCreatedStep`-adjacent; `conv` is
forbidden).  `NegThreeExpandedResidualAligned` with `negThreeResidualAligned_iff_expanded` (both ways)
is the `≤ −3` residual in this honest form, and `residueDegree_eq` identifies the `i`-th summand's
degree.  The Fock display is **not** reachable this way: its left-hand object is
`ZAligned E (YFam E a (−j−1) (creationExpVL b l (ι b)))` — an honest **mode of `a`** on the coherent
state of `b` — so it needs a mode intertwining, not the coherent recursion.

### 80.3 The residuals after batch 183

1. `NegTwoDiagonalWeight` (the weighted `z_0^1` line);
2. `NegThreeExpandedResidualAligned` (the `≤ −3` recursion);
3. the Fock half — an honest-mode intertwining;
4. the boundary restatement at the even cocycle.

## 82. Batch 185: the `−2` case is a DERIVATIVE identity; the Fock half is one dictionary

### 82.1 `NegTwoFirstDerivZero`: the `⟨a,b⟩ = −2` residual, in derivative form

`Definitions/MMFLM86NegTwoPOne.lean` (`MM.FLM86NPO`):

* **The `p`-recursion route is refuted**: `diagFam` is the diagonal Taylor family of `X(z₂+z₀,z₂)`, and
  its `p = 0`/`p = 1` coefficients read **different** coefficients of `X` — the counterexample
  `exists_diagFam_zero_eq_zero_one_ne_zero` shows a family with vanishing `p = 0` line and nonvanishing
  `p = 1` line.  So the landed `p = 0` line (`assemblyDiagonal`) cannot give `p = 1`.
* **The complete `p`-recursion is landed** instead: `diagFam_succ_eq_diagFam_firstDeriv` —
  `(p+1) • diagFam X M1 M2 (p+1) j = diagFam (firstDeriv X) (M1−1) M2 p j`; every step replaces `X` by
  `∂_{z₁}X` and only the `p = 0` line is evaluated.  The `−2` case therefore needs the `p = 0` line of
  `∂_{z₁}normOrdered2 a b`.
* **The exact residual `NegTwoFirstDerivZero (E)`** (with both directions:
  `zOneLine_iff_firstDerivZero`, `negTwoDisplayZOne_iff_firstDerivZero`,
  `negTwoExactResidualAligned_evenCocycle_iff_firstDerivZero` at `C.toCentralExtension h2`) is the
  **Leibniz/normal-ordering identity** `∂_{z₁}|_{z₁=z₂} 8Y(a,z₁)Y(b,z₂)8 = 8a(z₂)Y(ab,z₂)8` — source
  reading `∂_z Y(u,z) = Y(L₋₁u,z)`, `L₋₁ι(a) = a(−2)1` — i.e. a **weight-two Fock field times a lattice
  field**, outside the batch-180/181 two-lattice-label fibre computation.
* The only landed statement with a `z_0^1` line, `MM.FLM8512Done.productExpansion8512_apply_one`, is not
  a route (it assumes the whole display and a negative-power vanishing `hX` that is false at `ZAligned`).
* Successor routes: (i) the (8.4.36) substitution at `z_0^1` through the resummation layer
  (`MM.FLM84Taylor.resumCoeff`/`taylorCoeff`, `MMFLM86TaylorOperator`) applied to the `E⁻`/block/`E⁺`
  decomposition; (ii) generalise the fibre computation to `8Y(u,z₁)Y(b,z₂)8` for a general Fock `u`.

### 82.2 The Fock half is one dictionary

`Definitions/MMFLM86CoherentCreated.lean` + `MMFLM86CoherentCreatedStep.lean` (`MM.FLM86CCF`/`MM.FLM86CCS`):
the two sides of `CoherentCreatedForm` are **not** the same normal-ordered object (a ratio sum of *fields
at* blocks versus a window sum of blocks *applied to a state*), and the exact residual is
`CoherentCreatedStep (E)` with the both-directions dictionary `coherentCreatedForm_iff_createdStep`,
`…_iff_normOrdered2`, `…_iff_ratioForm(_nonneg)`.  The sibling lands
`IterateProductStep (ZAligned E) convOneSided ⇒ … ⇒ CoherentCreatedForm`; the four Fock statements are
one dictionary, all implied by that already-named gap.

### 82.3 Correction of record

Batch 178-A's "imported by nothing" for `Definitions/MMFLM85StepLinear.lean` is **false**:
`FinalCheck.lean:343` has imported it since batch 128 (commit `94b9097`), and
`MMFLM85NonvacuumStep.lean:47` does too.

### 82.4 The residuals after batch 185

1. `NegTwoFirstDerivZero` — the derivative normal-ordering identity;
2. the `≤ −3` case;
3. `CoherentCreatedStep` — the Fock comparison;
4. the boundary restatement at the even cocycle.

## 83. Batch 186: the boundary at the even cocycle and the FACE AUDIT

### 83.1 The boundary, stated honestly

`Definitions/MMFLM810TheoremEven.lean` (`MM.FLM810TE`): `theorem8102_of_residuals` takes
`hstep : EvenCocycleJacobiStep C h2` and `hres : EvenCocycleResiduals C h2` at
`E := C.toCentralExtension h2`, where
`EvenCocycleResiduals := NegTwoFirstDerivZero E ∧ NegThreeExpandedResidualAligned E ∧ CoherentCreatedStep E`.
Also landed: `theorem8102_of_evenJacobi`, `theorem8102_of_jacobiResidual` (the §8.10 form),
`FockCreatedStep`, and **`jacobiAxiom_iff_stateTargets`** — the converse of batch 169 R1's documented
equivalence, now formalized, so `JacobiAxiom` (the §8.10 output) feeds the boundary's
`JacobiOnStateTargets`.

### 83.2 The face audit (the batch's main product)

* `Res_{z_0}` = Corollary 8.5.1: closed for `0 ≤ ⟨a,b⟩` and `⟨a,b⟩ = −1`; the rest are the named
  residuals.
* `Res_{z_1}`/`Res_{z_2}` (`MM.FLM88.res_z1_face`/`res_z2_face`): **extractions from the Jacobi
  hypothesis**, not theorems; only conditional content in `MM.FLM810JacobiRoute`.
* **`l, m, n ≠ 0`: nothing landed** — `hjac` is strictly more than `Corollary851`; the `l ≠ 0` faces are
  the campaign's largest untouched gap.
* §8.8: the identity itself and the §8.6 engine are missing; §8.10: the `ι`-bridge, `RecoveryDisplay`,
  the `RecoverySplit` use and 8.10.7 are missing (calculus landed and consumable).

### 83.3 The `−2` derivative residual, decomposed

`Definitions/MMFLM86FirstDerivLine.lean` (`MM.FLM86FDL`): the `convWith2` Leibniz rule
`firstDeriv_convWith2` (the flat `deriv_flatMul` does not apply to the two-variable product) and the
three-piece decomposition `firstDeriv_normOrdered2_eq = creationDerivPiece + blockDerivPiece +
annihDerivPiece`, with the factor derivatives.  The `−2` residual becomes `FirstDerivThreeFactorLine`
(equivalent, both directions): three explicit per-state diagonal collapses of landed `convWith2`
expressions.  Attack `creationDerivPiece` first; the line is **weighted**, so the batch-181 regrouping
does not apply verbatim.

### 83.4 The fronts after batch 186

1. the `l ≠ 0` faces of (8.8.41) at the boundary field (untouched; `MMFLM810Route` is the entry point);
2. the `l = 0` residuals: `NegTwoFirstDerivZero` (now in three-piece form), `≤ −3`, the Fock half;
3. the continuation to full Jacobi: §8.8 (Theorem 8.8.9) or §8.10 (Props 8.10.3--8.10.7, `ι`-bridge);
4. the six boundary inputs `hnd/hs/S/hpd/hW/hfin`.

## 84. Batch 187: the `l ≠ 0` front opened; the `−2` creation piece collapsed

### 84.1 The `l ≠ 0` faces: first land at the boundary field

`Definitions/MMFLM810JacobiNonzeroFaces.lean` (`MM.FLM810JNF`):

* **Inventory**: every general-`l` statement in the tree is either a `def … : Prop` or **conditional on
  `h : JacobiIdentity`** (extractions: `res_z1_face`, `res_z2_face`,
  `corollary8818_of_jacobiIdentity`, `crossBracket_eq_of_window`).  At the boundary field nothing was
  landed for `l ≠ 0`; the only `l ≠ 0` identity anywhere is the scalar toy model
  (`MMFLM810ScalarJacobi.lean:542`, not a VOA field).
* **Landed** at `ZAligned (C.toCentralExtension h2)`, lattice generators, `0 ≤ ⟨a,b⟩`:
  `crossTerm_ZAligned_lattice` (+ swap), `yFam_mul_comm_of_nonneg`, the binomial reflection
  `deltaExpCoeff_reflect_of_nonneg`, the (8.8.21) swap-sum identity
  `sum_crossTerms_eq_smul_sum_crossTermSwaps`, and **the first `l ≠ 0` face**:
  `crossBracket_ZAligned_lattice_of_nonneg = 0`, `crossIterate_ZAligned_lattice_eq_zero_of_nonneg = 0`,
  hence `crossBracket_eq_crossIterate_ZAligned_lattice_of_nonneg` — equal for every `l ≥ 0`, every
  `m, n`, every `N ≥ l.toNat+1` (degenerate: both sides vanish).
* **The `l < 0` iterate side** is computed: `crossIterate_ZAligned_lattice_eq_sum_of_nonpos` (the finite
  sum over `i < (−l).toNat` of `binom(m,i) • iterateOf (ι a) (ι b) (l+i) (m+n−i)`).
* **The named residual `FirstNegativeFaceAligned (E)`** — (8.8.41) at `l = −1`, per monomial, cross side
  terminating (the δ-term of the formal (8.6.28)) — with the consumer
  `jacobiIdentity_neg_one_slice_of_firstNegativeFace`.
* **Route correction**: the shift display is **(8.6.28)** (which the source calls "formal ... suggestive"),
  not (8.6.29)/(8.6.30); no landed reduction of a general `l ≠ 0` face to `l = 0` exists, and for
  `l ≥ 0` the face reduces to a **commutator sum**, not to `l = 0` plus a shift.  Next target: `l = −1`
  through the landed (8.4.25) product.

### 84.2 `creationDerivPiece` collapsed; the `−2` residual is now sharply localised

`Definitions/MMFLM86CreationDerivPiece.lean` + `MMFLM86CreationDerivPieceLine.lean`
(`MM.FLM86CDP`/`MM.FLM86CDPL`):

* `sum_creationDerivPair_window` (the `E⁻` fibre collapse) and
  `flatMul_deriv_creationExpVL_add : deriv E⁻_{a+b} = flatMul (deriv E⁻_a) E⁻_b +
  flatMul (deriv E⁻_b) E⁻_a`: **each ordering contributes half**, so no single piece equals the target's
  right-hand side.
* `diagSum_creationDerivPiece(_regrouped)`, `diagSum_blockDerivPiece` (block first layer), and
  `sum_sum_window_regroup_factor` (batch 181's regrouping with an arbitrary factor; its
  `C = creationExpVL (a+b)` instance is syntactically the landed one).
* **Pieces**: creation **LANDED**; block **to the first layer** (the second layer — the weighted
  antidiagonal sum of the differentiated `a(0)` block, value `⟨a,c⟩` on `x ⊗ ι(c)` — open);
  annihilation **NOT landed** (the mirrored flat recursion `(d/dz)E⁺ = −a(z)⁺E⁺` is absent).
* **Weight correction**: `dE⁻_a`'s extra index `i = −1` has weight `0`, so the window hypothesis is
  verbatim batch 181's and the regrouping is reusable (proved).

### 84.3 The fronts after batch 187

1. the `l = −1` face (`FirstNegativeFaceAligned`) — the first genuinely nonzero `l ≠ 0` face;
2. the annihilation derivative piece (the `−2` case's remaining piece);
3. the `l = 0` residuals (`≤ −3`, the Fock half);
4. the continuation to full Jacobi (§8.8 / §8.10) and the six boundary inputs.

## 85. Batch 188: the `l = −1` face becomes a pure VOA identity; the third Leibniz piece lands

### 85.1 `(∗)`: the `l = −1` slice is one honest-lattice-VOA statement

`Definitions/MMFLM810NegOneFace.lean` (`MM.FLM810NOF`):

* The cross side is the **whole diagonal**: `crossBracket (ZAligned E) (ι a) (ι b) N (−1) m n =
  Σ_{i<N} YFam a (m−1−i)·YFam b (n+i) + Σ_{i<N} YFam b (n−1−i)·YFam a (m+i)` (`deltaExpCoeff i (−1) = 1`
  in char 0, the factorial hypothesis discharged).
* The iterate side is closed: `iterateOf (ZAligned E) (ι a) (ι b) (−1) (m+n) = if ⟨a,b⟩ = 0 then
  ε(a,b)·YFam (a+b) (m+n) else 0` — **the boundary-field repair drops out of this face**.
* `NegOneFaceAligned (E)` (the source's per-element reading) is **equivalent, both directions**, to the
  pure VOA identity
  `(∗) Σ_{p+q=m+n−1} YFam a p · YFam b q = (0 if ⟨a,b⟩ ≠ 0 else ε(a,b)·YFam (a+b) (m+n))`.
  `(∗)` mentions no `ZAligned`, so an honest-VOA proof of it gives the `l = −1` slice outright.
* **Chambers still open**: `1 ≤ ⟨a,b⟩` (cross side 0) and `⟨a,b⟩ = 0` (cross side `ε·YFam (a+b)`).
* **Corrections**: the "reduce `l = −1` to the landed `l = 0` face" advice is unsupported (there the
  `l = 0` face is commutativity with both sides 0); batch 187-A's `FirstNegativeFaceAligned` window is
  **operator-valued**, which is not the source's per-element reading (only 187-A ⟹ the per-element form
  is proved); and `MM.FLM86RN.NegOneResidualAligned` (the `⟨a,b⟩ = −1` pairing at `l = 0`) is a
  **different object** — a naming collision to remember.

### 85.2 The third Leibniz piece: the `E⁺` mirror collapse

`Definitions/MMFLM86AnnihDerivPiece.lean` + `MMFLM86AnnihDerivPieceLine.lean`
(`MM.FLM86ADP`/`MM.FLM86ADPL`): `deriv_annihExpField`, `deriv_annihIdx` (the mirrored recursion),
`flatMul_deriv_annihIdx_add`, `sum_annihDerivPair_window` (the `E⁺` fibre collapse, with the minus sign
and the shift `−K−2`) and `diagSum_annihDerivPiece(_regrouped)`; the generalized regrouping engine
`sum_sum_window_regroup_gen`.  **All three Leibniz pieces are now in one collapsed normal form.**

The residual `AnnihClosedLine` (both directions, and at FLM's datum
`negTwoExactResidualAligned_evenCocycle_iff_annihClosed`) is **not proved**, and the pieces do not
assemble: (i) the **block's second layer** is missing (the derivative block's fibre collapse is
operator-valued/per-state at the carrier); (ii) each exponential piece carries only one **ordering** of
its derivative, and the identification with `ZAligned E (creationExpVL a 1 (anchorVL E a b))`
((8.4.23)/(8.4.25)) is not landed.  The "mirror" is an **index reflection** `i ↦ −i−1` (same ordering).

### 85.3 The fronts after batch 188

1. `(∗)` — the two chambers of the `l = −1` face (a pure VOA identity);
2. the `−2` case: the block's second layer + the ordering identification;
3. the `l = 0` residuals (`≤ −3`, the Fock half) and the `l ≤ −2` faces;
4. the continuation to full Jacobi (§8.8 / §8.10) and the six boundary inputs.

## 86. Batch 189: `(∗)` and the `l = −1` slice are unconditional at FLM's datum; the ordering question settled

### 86.1 MILESTONE: the `l = −1` slice is closed at the even cocycle

`Definitions/MMFLM810NegOneChambers.lean` + `MMFLM810NegOneChambersClose.lean`
(`MM.FLM810NOC`/`MM.FLM810NOCC`):

* Both chambers landed (`honestDiag_eq_zero_of_one_le`, `honestDiag_eq_eps_smul_of_eq_zero`) and **`(∗)`
  itself unconditionally and window-independently** (`diagSum_honestDiag_eq`,
  `exists_honestDiag_window`) — only the form symmetry is needed.
* Route: the landed (8.4.25) expansion + **the landed (8.4.20) `MM.FLM86AR.assemblyDiagonal`** + a new
  `sum_range_ratioCoeff : Σ_{j≤N} ratioCoeff N j = (1−1)^N`.
* **`negOneFaceAligned_evenCocycle : NegOneFaceAligned (C.toCentralExtension h2)`** — batch 188-A's
  residual is a theorem; for an arbitrary extension the transport is
  `negOneFaceAligned_of_cocycleCompatible`.
* **Correction**: batch 188-A's "the cross side is the single-order diagonal of `(∗)`" is false in
  general — `crossTermSwap` is the opposite order and the two differ by `1 − c(b,a)(−1)^{⟨a,b⟩}`; the
  single-order form holds exactly under `CocycleCompatible` (true at the even cocycle).
* Trap recorded: `rwa` in the big cross-bracket context exceeded 1.6M heartbeats (`rw … at h; exact h`
  is instant).

### 86.2 The ordering question, settled; the `−2` residual is now the (8.4.23) dictionary

`Definitions/MMFLM86BlockSecondLayer.lean` + `MMFLM86BlockDerivRegrouped.lean` +
`MMFLM86NegTwoAssembly.lean` (`MM.FLM86BSL`/`MM.FLM86BDR`/`MM.FLM86NTA`):

* **The three-piece split is NOT buggy**: `firstDeriv = d/dz_1` differentiates only the first slot, so the
  "other orderings" are the `d/dz_2` halves — a different operator — and must not be added.  Landed:
  `flatMul_deriv_creationExpVL_half`, `flatMul_deriv_annihIdx_half`,
  `deriv_creationExpVL_eq_halves`; the pieces carry the `a(z)^-`, `a(0)`, `a(z)^+` terms of
  `:a(z)Y(ab,z):`.  **The batch-187-B header sentence "the other half is what the mirror piece has to
  produce" is false and has been corrected in place (dispatcher edit, file rebuilt).**
* **The block's second layer is landed**: `sum_firstDeriv_midBlock2_fiber` (fibre degree `K'` collapses
  at grading `K'+1`), with `exists_midGradDeriv_antidiagonal`/`exists_midGradDerivVL_antidiagonal` and
  `diagSum_blockDerivPiece_regrouped`.
* **Exact residual `NegTwoNormalFormLine`** (all three pieces collapsed; both directions).  **The
  remaining gap is the (8.4.23) dictionary `Y(a(−1)u,z) = 8a(z)Y(u,z)8` at the aligned boundary field**,
  not an ordering.  Also: `a(0)` must be `hZeroVL (ofLattice a)` (`modeVL a 0 = 0` in this pin).

### 86.3 The fronts after batch 189

1. the (8.4.23) dictionary (closes the `−2` case if it lands);
2. the `≤ −3` case, the Fock half, the `l ≤ −2` faces;
3. the continuation to full Jacobi (§8.8 / §8.10) and the six boundary inputs.

## 87. Batch 190: the `l = −2` face lands; the `z_0^1` line is now the single shared blocker

### 87.1 The `l = −2` face

`Definitions/MMFLM810NegTwoFace.lean` + `MMFLM810NegTwoChambers.lean`
(`MM.FLM810NTF`/`MM.FLM810NTC`):

* **`negTwoFaceAlignedOneLe_evenCocycle`** — the `1 ≤ ⟨a,b⟩` chamber of (8.8.41) at `l = −2`,
  **per element**, every past-vanishing window `N ≥ 2`, unconditional at FLM's datum.  Both sides are
  evaluated (`ε(a,b)·YFam (a+b) (m+n)` at `⟨a,b⟩ = 1`; `0` for `⟨a,b⟩ ≥ 2`).  Supporting: the exact cross
  side, the two-term iterate collapse for a general `E`, and the first-moment identity
  `Σ_j j·ratioCoeff N j = −1` (`N = 1`) / `0` (`N ≥ 2`).
* The `⟨a,b⟩ = 0` chamber is equivalent to `WeightedZOneLineZeroPairing` — the `z_0^1` line at
  pairing-zero pairs, the *same* identity as `ZOneLineNegTwo`/`NegTwoDiagonalWeight` (pairing `−2`);
  the iff itself was not landed (budget).
* **General `l ≤ −2`**: the cross side is a **weighted** antidiagonal, weight `ratioCoeff l (m+l−p)`,
  degree `L − 1 = −l − 1`; `L = 1` needs no moment, `L = 2` the first (killed by
  `Σ_j ratioCoeff = 0` — why `l = −2` lands), `L ≥ 3` the **moment hierarchy**
  `Σ_r r^m · normOrdered2 a b r (K−r) = 0` for `m + 1 ≤ ⟨a,b⟩` (= the `m`-th derivative of (8.4.40)),
  not landed; the iterate side needs the Fock-descendant evaluations.

### 87.2 The (8.4.23) dictionary, and the relocation of the `−2` gap

`Definitions/MMFLM86DerivDictionary.lean` (`MM.FLM86DD`): the dictionary `Y(a(−1)u,z) = 8a(z)Y(u,z)8`
was **already landed at an arbitrary vector and base** (`Yzero_modeVL_eq_actionFamily(_nat)`,
`ZAligned_creationStep`, `coherentField_one`), as was its `L(−1)` companion `BaseResidual (baseAligned E)`
(batch 172).  New here: the aligned-field and factor-level forms
(`splitMinus_factorFactorZero_one_zero = a(−1)`,
`splitPlus_factorFactorZero_one_neg_one = hZeroVL (ofLattice a) = a(0)`,
`headOp_factorFactorZero_one_ne_flatMul`), the anchor form
`ZAligned_creationExpVL_one_anchor` (which removes `anchorVL`/`ZAligned`/`Yzero` from the residual's
right-hand side), and the residual `DerivDictionaryLine` — **both directions** to
`NegTwoFirstDerivZero`/`NegTwoNormalFormLine`/`NegTwoExactResidualAligned`.

**The `−2` gap is the FLAT REASSEMBLY** — matching the three collapsed flat sums with
`headOp (heisFamZero a) + tailOp` of `Y(a+b,z)` ((8.4.36)/(8.4.40) at `z_0^1`): the creation half is
already in `flatMul (splitMinus (heisFam a)) (creationExpVL (a+b))` shape, the `a(0)` summand belongs to
the tail, and the negative `mid`-degrees must be carried by the free `(J,K')` windows.

### 87.3 The state after batch 190

Unconditional at FLM's datum: `l = 0` (`0 ≤ ⟨a,b⟩`, `⟨a,b⟩ = −1`), `l ≥ 0` (`0 ≤ ⟨a,b⟩`), `l = −1`,
and `l = −2`'s `1 ≤ ⟨a,b⟩` chamber.  **The single shared open object is the `z_0^1` line**
(`DerivDictionaryLine` = `ZOneLineNegTwo` = `WeightedZOneLineZeroPairing`).  Then: the moment hierarchy
(`l ≤ −3`), the Fock half, the continuation to full Jacobi, and the six boundary inputs.

## 88. Batch 191: the `z_0^1` obligation is unified as `ZOneLineAllPairings`; the moment picture corrected

### 88.1 The single shared obligation

`Definitions/MMFLM86ZOneReassembly.lean` (`MM.FLM86ZOR`):

* The reassembly is **not** landed; its missing half is diagnosed: carrying `a(k)` (`k ≥ 0`) past
  `E^+_{a+b}` through the **(8.4.19) contraction** inside the free `(J,K')` windows.  Step-1 grep was
  measured **negative**.
* Landed: `negTwoFaceAligned_iff_weightedZOneLine` (both directions, under `CocycleCompatible`), via
  `weightedHonestDiag_zero_eq_diagSumWeighted` and `negTwoZeroWindow_weighted_eq_zero`.
* **Correction**: `DerivDictionaryLine` (pairing `−2`) does not imply `WeightedZOneLineZeroPairing`
  (pairing `0`).  The unified obligation is **`ZOneLineAllPairings`** (the `z_0^1` line for **all**
  pairs), from which `zOneLineNegTwo_of_allPairings`, `weightedZOneLine_of_allPairings` and
  `derivDictionaryLine_of_allPairings` are landed.  **The tail contraction making
  `ZOneLineAllPairings` a theorem closes both the `l = 0`, `⟨a,b⟩ = −2` case and the `l = −2`,
  `⟨a,b⟩ = 0` chamber.**

### 88.2 The moment picture, corrected and landed

`Definitions/MMFLM86MomentHierarchy.lean` (`MM.FLM86MH`):

* **The batch-190-B picture was wrong**: the moments do **not** vanish for `m+1 ≤ ⟨a,b⟩`; `m = 0` is the
  assembly and `m = 1` is the `z_0^1` line (the landed residual).  Correct: the `j`-sum becomes a finite
  difference, so by `Σ_j ratioCoeff N j j^s = 0` (`s < N`) only the **low** moments `m ≤ c−1−N` survive.
* Master identity `weightedHonestDiag_eq_sum_ratioCoeff`, with specialisations:
  `…_eq_binomMoment` (`N ≤ c−1`: the single binomial moment `Σ_r C(r+m, c−1−N)·normOrdered2`, i.e. the
  `z_0^{c−1−N}` line of (8.4.36)/(8.4.40)); **`…_eq_zero_of_le` (`N ≥ c`: 0 — no moment, no new
  residual)**; `…_eq_assembly` (`N = c−1`, the landed `assemblyDiagonal`); `…_eq_firstMoment`
  (`N = c−2`, the `z_0^1` line).  Scalar core: `ratioCoeffSum_eq_binomCoeffFinDiff` (consuming the
  already-landed `MM.FLM86Y.binomCoeffFinDiff`), the **new vanishing case `binomCoeffFinDiff_eq_zero_of_lt`**
  (`s < n`), `binomCoeffFinDiff_eq_int`, and `deltaExpCoeff_neg_eq_binomCoeff`.
* **`negThree_weightedChambers`**: for `l = −3` and `2 ≤ ⟨a,b⟩` the weighted antidiagonal is
  `ε(a,b)·YFam (a+b) (m+n)` at `⟨a,b⟩ = 2` and `0` for `⟨a,b⟩ ≥ 3` — **the whole `2 ≤ ⟨a,b⟩` half of the
  `l = −3` face is determined by the landed assembly alone**.
* The `l = −3` residual: (a) the general-`c` cross-side reindexing (mechanical);
  (b) the `l = −c` analogue of `honestDiag_eq_zero_of_crossWindowOn`; (c) the iterate side at `l = −3`.
  The `⟨a,b⟩ = 1` chamber is **not** reachable today (it needs the `z_0^1` line at pairing 1).

### 88.3 The state after batch 191

Unconditional at FLM's datum: `l = 0` (`0 ≤ ⟨a,b⟩`, `= −1`), `l ≥ 0` (`0 ≤ ⟨a,b⟩`), `l = −1`,
`l = −2`'s `1 ≤ ⟨a,b⟩`, and `l = −3`'s `2 ≤ ⟨a,b⟩`.  The single shared obligation is
`ZOneLineAllPairings` (the (8.4.19) tail contraction).

## 89. Batch 192: the `l = −3` face lands; the `(8.4.19)` premise corrected

### 89.1 The `l = −3` face at `2 ≤ ⟨a,b⟩` is a theorem

`Definitions/MMFLM810NegThreeFace.lean` (`MM.FLM810NTH`): `negThreeFaceAlignedTwoLe_evenCocycle (C) (h2)
(hs)` — per element, per monomial, every window `N ≥ 3`, unconditional at `C.toCentralExtension h2`.
Supporting (general `c ≥ 1` where possible):

* **(a)** `crossBracket_neg_nat_eq_sum` (under `CocycleCompatible`): the cross side is
  `Σ_p binomCoeff (m−p−1) (c−1) • honestDiag E a b p (m+n−c−p) w` over the two half-lines; the reflection
  `binomCoeff_pos_eq_neg_pow` and `disjoint_negCWindows`.
* **(b)** `weightedHonestDiag_eq_zero_of_crossWindowOn`: the face's `CrossWindowOn` gives the **weighted**
  vanishing (the honest diag is *not* zero in the gap `m−c < p < m`), plus the window-independence bridge
  `weightedHonestDiag_neg_three_eq`.
* **(c)** `crossIterate_ZAligned_lattice_neg_three` (general `E`): `i = 0,1,2` survive; chamber forms give
  `ε(a,b)·YFam (a+b) (m+n)` at pairing `2` and `0` above.

### 89.2 The `(8.4.19)` premise refuted; the annihilation/zero-mode laws landed

`Definitions/MMFLM86TailContraction.lean` (`MM.FLM86TC2`):

* (8.4.19) off printed p. 192: `8α(z)Y(a,z)8 = 8Y(a,z)α(z)8 = α(z)⁻Y(a,z) + Y(a,z)α(z)⁺`.
* **No contraction past `E⁺`** for any `k ≥ 0` (`k ≥ 1` landed; `k = 0` proved here); the contraction is
  against `E⁻` and was landed (repackaged as `splitPlus_heisFamZero_contraction_creation`).  What was
  missing at the assembled field was the **annihilation/zero-mode** law, now landed:
  `modeVL_YFam_add_pos`, `hZeroVL_YFam_add`, and **`splitPlus_heisFamZero_YFam_add : a(k)(Y(a,z)_j v) =
  Y(a,z)_j (a(k)v) + ⟨b,a⟩ Y(a,z)_{j+k} v`** for all `k ≥ 0`.
* `ZAligned_anchor_head_tail` (the corrected head/tail spelling) and
  `zOneLineAllPairings_iff_headTail` (both directions).
* **`ZOneLineAllPairings` is still the residual**; what remains is the flat reassembly.  Conditional on it,
  both faces close: `negTwoFirstDerivZero_of_allPairings`, `negTwoExactResidualAligned_of_allPairings`,
  `negTwoFaceAlignedZeroPairing_of_allPairings`.
* Traps: `headOp` is `FLM85TA.headOp`; `factorFactorZero` in `FLM810ZM`;
  `sub_eq_iff_eq_add'`; the `Module.End` `Sub`/`Add` diamond (work at the vector level); repeated `rw`
  of one lemma errors on the second pass.

### 89.3 The state after batch 192

Unconditional at FLM's datum: `l = 0` (`0 ≤ ⟨a,b⟩`, `= −1`), `l ≥ 0` (`0 ≤ ⟨a,b⟩`), `l = −1`,
`l = −2`'s `1 ≤ ⟨a,b⟩`, `l = −3`'s `2 ≤ ⟨a,b⟩`.  The single shared obligation is the flat reassembly
(`ZOneLineAllPairings`), which closes the `l = 0`, `⟨a,b⟩ = −2` case and the `l = −2`, `⟨a,b⟩ = 0`
chamber.  Then: general `l ≤ −4`, the Fock half, the continuation to full Jacobi, the six boundary inputs.

## 90. Batch 193: general `l = −c` chambers; the `z_0^1` residual on the generators and its linearity

### 90.1 General `l = −c`: the `ν ≥ c−1` chamber is unconditional

`Definitions/MMFLM810NegCFace.lean` (`MM.FLM810NCF`): `negCFaceAlignedGe_evenCocycle (C) (h2) (hs) {c}
(hc : 1 ≤ c)` — the general face at `ν := ⟨a,b⟩ ≥ c−1`, unconditional at FLM's datum.  Supporting:
`negC_weightedChambers`, `crossBracket_ZAligned_lattice_neg_nat_eq_ite`, the general-`E` iterate side
`crossIterate_ZAligned_lattice_neg_nat` (+ the `ν ≤ c−2` decomposition with `k = c−1−ν` Fock
descendants), `crossWindowOn_ZAligned_lattice_neg_nat`.

**Exact residual `NegCBinomMomentAligned E c`** for `0 ≤ ν ≤ c−2`
(`negCFaceAligned_iff_binomMoment_evenCocycle`, both directions).  Chambers: `ν ≥ c−1` unconditional;
`ν = c−2` the first moment (`z_0^1`); `1 ≤ ν ≤ c−3` the `z_0^{c−1−ν}` lines; `ν = 0` the
`z_0^{c−1}` line.  **`c = 1` unconditional; `c = 2` leaves only `ν = 0` (the landed
`WeightedZOneLineZeroPairing`); `c = 3` leaves `ν ∈ {0,1}`.**

### 90.2 The `z_0^1` residual: generators landed, linearity landed

`Definitions/MMFLM86FlatReassembly.lean` (`MM.FLM86FRA`):

* **`zOneLineAllPairingsOnGenerators (hs) (E)`** — the residual holds at `v = ι(c)`, for **all** pairings,
  with the index bridges `sum_Icc_smul_eq_flatMul_deriv` (the weight `p` IS the (8.4.12) derivative),
  `sum_Icc_eq_flatMul`, `sum_range_toNat_succ_eq_flatMul`, plus `genFam_iotaVL` and
  `modeVL_iotaVL_eq_zero_of_pos`.
* **`F`-linearity in the state**: `ZOneLineAt`, `zOneLineAllPairings_iff_forall_ZOneLineAt`,
  `zOneLineAt_add/_smul/_zero/_of_mem_span`.  **The residual is determined by its values on any spanning
  set**; with the generator half, the only remaining input is the residual on the creation monomials — the
  Fock half.
* **Corrections**: `splitPlus_heisFamZero_YFam_add` was not the missing tool (the `a(k)` summands already
  are the tail); the reassembly is not a `(J,K')`-window reindexing (those windows belong to the
  `z_0^0`/firstDeriv line) — at an arbitrary state the flat pieces reach the action through the
  annihilation-pair contractions, i.e. the Fock half; and the `⟨a,b⟩ = −2` instance does not imply the
  general-state identity.

### 90.3 The state after batch 193

Unconditional at FLM's datum: `l = 0` (`0 ≤ ⟨a,b⟩`, `= −1`), `l ≥ 0` (`0 ≤ ⟨a,b⟩`), `l = −1`, and for
every `c ≥ 1` the `ν ≥ c−1` chamber of `l = −c`.  **The single shared obligation is the `z_0^1` line
`ZOneLineAllPairings`, now reduced to its Fock half.**  Then: the higher `z_0^m` lines, the continuation
to full Jacobi, and the six boundary inputs.

## 91. Batch 194: the `z_0^1` line at the first Fock level; the derivative ladder

### 91.1 The first Fock level of the shared blocker

`Definitions/MMFLM86ZOneFockOne.lean` (`MM.FLM86ZOF`): **`zOneLineAt_modeVL_iotaVL`** — the `z_0^1`
residual `ZOneLineAt` at `v = α_{c'}(−R)ι(c)`, every pairing, every `R ≥ 1`, unconditional apart from the
form symmetry.  Engine landed alongside:

* **the contraction** `normOrdered2_modeVL_sub`: `normOrdered2 a b p q (α_c(−r)w) = α_c(−r)(normOrdered2 p q w)
  − ⟨c,a⟩·normOrdered2 (p+r) q w − ⟨c,b⟩·normOrdered2 p (q+r) w` — shift `+r` in **both** indices (forced by
  consistency of the two laws);
* its weighted-diagonal corollary `diagSumWeighted_modeVL`;
* **the right-hand-side mode law `ZAligned_modeVL`**: `Z_j(α_c(−r)w) = α_c(−r)(Z_j w) − ⟨c,a+b⟩·Z_{j−r}w +
  ε(a,b)·r·⟨a,c⟩·Y(a+b,z)_{r−j}w` (the brief's engine covered only the LHS; the RHS needs this);
* the window laws `headWindow_modeVL_shift`/`tailWindow_modeVL_shift` (the landed
  `headWindow_modeVL_sub`/`tailWindow_modeVL_sub` force the field's lattice label to equal the factor's,
  so they could not be used as stated) and the generator collapse `diagSum_normOrdered2_iotaVL`.

**The general creation monomial is NOT reached**: at the next mode the contraction produces the unweighted
diagonal sum at the **level-one descendant** `α_{c''}(−R₂)ι(c)` — the **`z_0^0` line at a Fock
descendant**, genuine new input.  Traps: `-r • x` parses as `(-r)•x`; `rw` instantiates a lemma once per
pass (repeating a name silently rewrites only the first instantiation); `-γ•H` vs `-(γ•H)` needs
`calc`/`show`.

### 91.2 The derivative ladder: no higher line is easier

`Definitions/MMFLM86ZMoments.lean` (`MM.FLM86ZM`): **`zLineScheme_succ_firstDeriv`** (with
`momentSum_firstDeriv`, `zLineAt_firstDeriv`) is an **equivalence**: the `(j+1)`-st line of `X` at degree
`K+1` ⟺ the `j`-th line of `d/dz₁X` at degree `K`.  Hence order `k` **is** order 0 of
`firstDeriv^k(normOrdered2 a b)` — the derivative tower of the shared blocker.  Plus the Vandermonde
expansion `binomCoeff_add_eq_sum`/`binomCoeff_add_two` and the flat reassembly
`sum_binomCoeff_smul_eq_sum`.

**The exact residual `ZLine E j`**: `j = 0` is a theorem (the landed assembly), `j = 1` is verbatim the
shared blocker (`zLine_one_iff_zOneLineAllPairings`), `j ≥ 2` is new.  **The chamber decomposition**:
`NegCMomentLine E c k` = `ZLine E k` at pairings `N = c−1−k`; forward each chamber consumes exactly its own
line; backward `negCBinomMomentAligned_of_zLine`.  **Ledger for `l = −c`**: `N ≥ c` zero; `N = c−1` the
landed assembly; `N ≤ c−2` the residual chambers, **all `z_0^k` lines with `k = c−1−N`**; `c = 1`
residual-free.  Correction: the landed `z_0^2` statements are one-variable
`openBoxField`/`shiftFamily` ones, not the honest three-factor `normOrdered2`.

### 91.3 The state after batch 194

The shared obligation `ZOneLineAllPairings` is now: proved on the generators (193-A), `F`-linear in the
state (193-A), proved at the first Fock level (194-A), and the easiest of all the `z_0^k` lines (194-B).
**The next input is the descended `z_0^0` line** (`α_{c''}(−R₂)ι(c)`), then the creation-monomial induction.
Then: the continuation to full Jacobi (§8.8/§8.10) and the six boundary inputs.

## 92. 🎉 Batch 195: `ZOneLineAllPairings` IS A THEOREM

### 92.1 The breakthrough

`Definitions/MMFLM86FockInduction.lean` (`MM.FLM86FI`):
**`zOneLineAllPairings_of_symmetric_form (hs) (E) : MM.FLM86ZOR.ZOneLineAllPairings bL form E`** — the
`z_0^1` line of (8.4.36)/(8.4.40) at every symmetric pairing, from landed ingredients only.  This was the
campaign's single shared obligation since batch 186.

The chain: `zOneLineAtLevel_zero` (the generator case, batch 193-A) → `zOneLineAt_modeVL` (weigh the
level-`n` hypotheses at `(m,n)` and `(m−r,n)` against the landed assembly at `w` at degree `−m−n+r`; the
window `U = (s₁∪s₂∪s₃) ∪ (s₁∪s₂∪s₃)−r`; then `diagSumWeighted_modeVL` + `ZAligned_modeVL`) →
`zOneLineAtLevel_succ` → `zOneLineAtLevel_all` → `zOneLineAllPairings_of_level`
(`zOneLineAt_of_mem_span` + `spanningStatement`).  Caveat: only the symmetric-pairing instance is proved.

**Correction of record**: the "descended `z_0^0` line" (batch 194-A's measured next input) **does not
exist** — the landed `MM.FLM86AR.assemblyDiagonal`/`assemblyDiagonal_antidiagonal` already gives the
unweighted diagonal sum at every state and pairing; `diagSum_normOrdered2_iotaVL` is that law at `ι(c)`.
The level-`n` hypothesis therefore carries the `z_0^1` half only, and the induction closes.  Batch 195-A
(dispatched on the refuted premise) was steered mid-flight and stopped; its kept file
`MMFLM86DescendedAssembly.lean` adds only the window-level decomposition and an explicit witness window.

### 92.2 What this unlocks

The landed one-directional reductions compose: the `l = 0`, `⟨a,b⟩ = −2` case
(`NegTwoFirstDerivZero`, `NegTwoExactResidualAligned`, and the equivalent names); the `l = −2` face's
`⟨a,b⟩ = 0` chamber (`WeightedZOneLineZeroPairing`), hence the whole `l = −2` face; and for every `c`
the `ν = c−2` chamber of `l = −c` (`NegCMomentLine E c 1` from `ZLine E 1`).  Combined with batch 193-B's
`ν ≥ c−1` chamber: **every `l = −c` face is unconditional for `ν ≥ c−2`**, and the remaining chambers are
`0 ≤ ν ≤ c−3`, i.e. the lines `ZLine E j` for `j ≥ 2`.

### 92.3 The state after batch 195

Unconditional at FLM's datum: `l = 0` (`0 ≤ ⟨a,b⟩`, `= −1`, and — pending the batch-196 harvest — `= −2`);
`l ≥ 0` (`0 ≤ ⟨a,b⟩`); `l = −1`; `l = −2` (both chambers); and the `ν ≥ c−2` part of every `l = −c`.
Remaining: the lines `ZLine E j` (`j ≥ 2`), the Fock half of the `l = 0` residual, the continuation to
full Jacobi, and the six boundary inputs.
## §93 Batch 196 — the `ZLine` tower is one `c`-independent base from closed

### 93.1 The harvest (A)

Batch 195 made `ZOneLineAllPairings` a theorem.  Every landed one-directional reduction that had it as a
hypothesis therefore fires unconditionally, and batch 196-A collected them into two files.  The two
headline consequences:

* `negTwoFaceAligned_evenCocycle` — the **whole `l = −2` face** at FLM's datum.
* `negCFaceAlignedFrom_subTwo_evenCocycle` — the `ν ≥ c−2` chamber of the `l = −c` face, for **every** `c`.

and the sharp accounting statement

* `residueNegativeAligned_evenCocycle_iff_negThreeAndBelow` —
  `ResidueNegativeAligned (evenCocycle …) ↔ NegThreeAndBelowResidualAligned`.

The last one is the useful shape: the negative-pairing residue no longer carries any `l = 0` debt for
`⟨a,b⟩ ∈ {0,−1,−2}`, and carries no `l ≠ 0` debt for those pairings either.  What remains of
`ResidueNegativeAligned` is **exactly** the chamber `⟨a,b⟩ ≤ −3`.

### 93.2 The induction (B): one base, two halves

`Definitions/MMFLM86ZLineInduction.lean` proves the level induction and reduces the whole tower to a
single `c`-independent hypothesis `ZLineGeneratorBase E`:

```
zLine_all_of_generatorBase (hs) (E) (h : ZLineGeneratorBase …) : ∀ i ≥ 1, ZLine E i
zLineGeneratorBase_iff (hs) (E) : ZLineGeneratorBase … ↔ ∀ i ≥ 1, ZLine E i
negCBinomMomentAligned_of_generatorBase (hs) (E) (h) (hc : 1 ≤ c) : NegCBinomMomentAligned E c
```

The base is not one statement but two:

1. **generator line**: `ZLineAt (normOrdered2 a b) i K (iotaVL c₀) (zLineVal E a b i K (iotaVL c₀))`
   for all orders `i` — the order-`i` line at the lattice generators;
2. **mode law**: `ZLineModeAt E a b c i r K u` for `1 ≤ r` — the field of the coherent state
   `E⁻_a(z)_i ι(ab)` through one creation mode.

Both are landed at `i ≤ 1` and both are new at `i ≥ 2`.  The split is forced by the proof: the step
`zLineAt_modeVL_of_modeLaw` moves the line from a monomial to its `YFam`-descendant and needs the *value*
law there, while the *generator* law is the input at the bottom; neither is derivable from the other.
The converse of `zLineGeneratorBase_iff` is obtained by linearising the contraction over the line windows
(`zLineAtVal_of_mem_span`), which is what makes the base a faithful accounting of the residual and not an
artefact of the proof.

Supporting identities newly landed in the same file: the `j`-independent contraction
`momentSum_modeVL_normOrdered2` at weight `C(r,j)`; the Chu–Vandermonde identity
`binomCoeff_sub_eq_sum_moment` (`C(p−r,j) = Σ_l C(−r,l)C(p,j−l)`); the level induction
`ZLineAtAllLevel`/`_succ`/`_all`; the spanning assembly `zLineScheme_of_level`/`zLine_of_level`; and
`zLine_one_unconditional`, which re-derives the batch-195-B theorem without the base.

### 93.3 Accounting after batch 196

At FLM's datum (`E = C.toCentralExtension h2`, `C` an even cocycle, `form` symmetric):

| object | status |
| --- | --- |
| `l = 0`, `0 ≤ ⟨a,b⟩` and `⟨a,b⟩ = −1` | closed (earlier batches) |
| `l = 0`, `⟨a,b⟩ = −2` | closed (harvest) |
| `l ≥ 0` (`0 ≤ ⟨a,b⟩`) | closed (earlier) |
| `l = −1` | closed (batch 189) |
| `l = −2`, both chambers | **closed (harvest)** |
| `l = −c`, `ν ≥ c−2` | **closed (harvest)** |
| `l = −c`, `0 ≤ ν ≤ c−3` | `ZLine E j`, `j ≥ 2` — reduces to `ZLineGeneratorBase` |
| `l = 0`, `⟨a,b⟩ ≤ −3` | `NegThreeAndBelowResidualAligned` |
| Fock half / `CoherentCreatedStep` | open |
| continuation to full Jacobi (§8.8 / Props 8.10.3–8.10.7) | open |
| six boundary inputs `hnd/hs/S/hpd/hW/hfin` | open |

### 93.4 Brief errors (three, all consequential)

(a) The brief proposed to get the generator case from `firstDeriv^j`; but `zLineAt_firstDeriv` is an
**equivalence**, so it transfers the obligation rather than discharging it, and order 0 of
`firstDeriv^j(normOrdered2 a b)` is not a `normOrdered2`, so `assemblyDiagonal_antidiagonal` does not
apply.
(b) The step additionally needs the **lower-order** lines at degree `K+r` (not only order `j` at every
monomial).
(c) The step additionally needs the value law at the descendant — the second half of the base; a
one-half base would have been unsound bookkeeping.
(d) Index trap carried from batch 195: `zLineVal 0 K u = ε·genFam (a+b) K u` because
`genFam u m = YFam u (−m−1)`; an off-by-one there makes the mode law false.

## §94 Batch 197 — the order-2 line is two statements, and the Fock half is a display plus two carrier inputs

### 94.1 The truncated base (A)

Batch 196 reduced the whole `z_0^j` tower to one `c`-independent `ZLineGeneratorBase`.  Batch 197-A made
that reduction **partial and sharp**: `ZLineGeneratorBaseUpTo E J` restricts both halves to `i ≤ J`, and

```
zLineGeneratorBaseUpTo_iff (hs) (E) (J) : ZLineGeneratorBaseUpTo E J ↔ ∀ i, i ≤ J → ZLine E i
zLineGeneratorBase_iff_forall_upTo (hs) (E) : ZLineGeneratorBase E ↔ ∀ J, ZLineGeneratorBaseUpTo E J
```

hold with both directions at the same `J` — no direction secretly needs an order above the bound (the
brief feared one would; it does not).  The consumption is now chamberwise: the chamber `ν` of the
`l = -c` residual needs only the lines `1 ≤ j ≤ c-1-ν`
(`negCBinomMomentAlignedAt_of_lines`, with `hc : 1 ≤ c`), so `ZLine E 2` alone covers every chamber
`c - 1 - ν ≤ 2` (`negCBinomMomentAlignedAt_of_zLine_two`), i.e. the new chamber `ν = c-3`.

`zLine_two_iff` is the sharp statement of what is left at order two:

```
ZLine E 2 ↔ (the order-2 line at the lattice generators, GEN2) ∧ (the order-2 mode law, MODE2)
```

with the `i = 0, 1` components already discharged.  Everything else in the two A files is the
(8.4.16)/(8.4.12) window engine for the **derived factor** `a'(z) = (1/1!)(d/dz)a(z)` that MODE2
consumes; its tail correction is `-r(r+1)⟨c,a⟩ Y_{m+r+2}` at window `M ≥ r+1`.

MODE2 is not landed.  Its remaining step is pinned to three explicit items in the
`MMFLM86ZLineTwo.lean` header (the (8.5.5) recursion at degree two; the two *applied* `actionFamily`
laws; the three index/coefficient identifications), and the final coefficient arithmetic was checked
symbolically to close exactly against `ZLineModeAt`'s right-hand side.  GEN2 is untouched: it needs the
evaluation of the degree-2 coherent-state field at a lattice generator, which the landed derivative
ladder `zLineAt_firstDeriv` cannot supply (it is an equivalence, and `firstDeriv²(normOrdered2 a b)` is
not a `normOrdered2`, so the landed assembly does not apply).

### 94.2 The Fock half is two-layered (B)

Batch 197-B landed, at a general `E` with no cocycle hypothesis,

```
FockCreatedStep E ↔ IterateProductStepOnCreated (ZAligned E) convOneSided vacuumVL
FockCreatedStep E ↔ Display8610List (ZAligned E) vacuumVL convOneSided
FockCreatedStep E ↔ Display8610TwoOneSided E                -- given FockPairDatum + FockCreatedSpan
Display8610TwoOneSided E → CoherentCreatedStep E            -- given FockPairDatum + FockCreatedSpan
```

so the campaign's opaque Fock residual is *equivalent* to the concrete two-factor display (8.6.10) once
two named carrier inputs are granted: `FockPairDatum E` (every shifted field pairs finitely (8.10.5)
with the field of every created state) and `FockCreatedSpan E` (the created states span `V_L`).

The genuinely new mathematics here is **carrier linearity for the `dite` carrier**: unconditional
additivity of `convOneSided` in its second argument is FALSE (a compiled counterexample over `ℚ`,
`not_convOneSided_add_right_unconditional`), and the honest law is conditional on `PairFinite`
(`convOneSided_add_right_of_pairFinite`), which then transports along the span
(`pairFinite_shiftedField_of_mem_span`).  This is why the Fock residual is two-layered and not one: the
created-state evaluation alone does not give the general state without the carrier datum.

Non-vacuity of the datum is recorded at the vacuum-created state and at the single-factor
lattice-generator states with `0 ≤ ⟨a,b⟩`; the general case (lists of length ≥ 2, negative pairing) is
the open carrier statement, and it is *strictly stronger* than batch 182's `ShiftedPairFiniteGenerators`.

### 94.3 The next Fock step is probably cheap

`MM.FLM85CA.createdStateSpan_eq_top` (`MMFLM85ConvAdditive.lean:502`) already proves
`Submodule.span F (Set.range (iteratedCreation Y vac)) = ⊤` from a creation property `hcreate` and the
(8.5.2) mode identification `hmode` at an explicit family `umode`, by transporting the landed (8.5.4)
spanning.  So `FockCreatedSpan E` should be a *corollary* at `Y = ZAligned E`, `vac = vacuumVL` (the
landed (8.5.23) `creationPropertyAligned` gives `hcreate`), leaving only the (8.5.2) identification to
locate or to name.  That is the cheapest available reduction on the Fock front.

### 94.4 Corrections of record

1. `negCBinomMomentAlignedAt_of_lines` without `1 ≤ c` is **false** (`c = 0` forces
   `YFam(a+b)(m+n+1) = YFam(a+b)(m+n)`); the landed form carries `hc : 1 ≤ c`.
2. The order-2 mode law is **not** a homogeneous `actionFamily` statement: its first summand acts on the
   degree-one coherent field, whose commutator with `α_c(-r)` carries the (8.4.19) `⁺`-tail
   `-r⟨a,c⟩Y(a+b,z)`, so a two-component window law is required (the landed `actionFamily_modeVL` is
   homogeneous and does not apply as it stands).
3. The ledger's "the Fock half is implied by `IterateProductStep`" is now "equivalent to the `k = 2`
   display modulo `FockPairDatum` and `FockCreatedSpan`".
4. Dispatcher: the `MMFLM810TheoremEven.lean` ledger (the `⟨a,b⟩ = -2` bullet marked **open**, and the
   "three named residuals" paragraph) was corrected in place; the first conjunct of
   `EvenCocycleResiduals` is a theorem since batch 196.

### 94.5 Accounting after batch 197

| object | status |
| --- | --- |
| `ZLine E j`, `j ≤ 1` | theorem (`zLine_zero`, `zLine_one_unconditional`) |
| `ZLine E 2` | ⟺ GEN2 ∧ MODE2 (both open; MODE2 pinned to 3 explicit sub-steps) |
| `l = -c` chambers | `ν ≥ c-2` closed; `ν = c-3` closed once `ZLine E 2` lands |
| Fock | ⟺ `Display8610TwoOneSided` + `FockPairDatum` + `FockCreatedSpan` (span likely a corollary of landed material) |
| `NegThreeExpandedResidualAligned` | open (needs the degree-`k` coherent field for `k ≥ 2`) |
| `EvenCocycleJacobiStep` | open (assembly + whole `l ≠ 0` content + passage to (8.10.8)) |

## §95 Batch 198 — `ZLine E 2` is a theorem: the order-two base is closed on both halves

### 95.1 What closed

Batch 197 left the order-two line as exactly two statements.  Batch 198 proved both, each unconditionally at
a general `E` with only the symmetry `hs` of the form:

* **MODE2** = `MM.FLM86MLT.zLineModeAt_two` (`Definitions/MMFLM86ModeLawTwo.lean`, 618 lines);
* **GEN2** = `MM.FLM86GLT.zLineGeneratorTwo` (`MMFLM86GeneratorLineTwo{Aux,Eval,Comp,}.lean`, 1214 lines).

Hence (dispatcher capstone, `Definitions/MMFLM86ZLineTwoClosed.lean`)

```
zLine_two (hs) (E) : ZLine (F := F) bL form E 2
zLineGeneratorBaseUpTo_two (hs) (E) : ZLineGeneratorBaseUpTo (F := F) bL form E 2
negCBinomMomentAlignedAt_of_two (hs) (E) {c nu} (hc : 1 <= c) (hnu : nu <= c - 2) (hcJ : c - 1 - nu <= 2) :
    NegCBinomMomentAlignedAt (F := F) bL form E c nu
```

so the ``ν = c-3`` chamber of every `l = -c` face is unconditional, and this **supersedes NOTES 94.5's row
"`ZLine E 2` ⟺ GEN2 ∧ MODE2 (both open)"**.  The face-level bridge from
`NegCBinomMomentAlignedAt` to `NegCFaceAlignedFrom E c (c-3)` is still not landed (the recipe is: replace
`negCBinomMomentAlignedAt_sub_two` by `negCBinomMomentAlignedAt_of_zLine_two` and `(c-2)` by `(c-3)` in
`negCFaceAlignedFrom_subTwo_evenCocycle`).

### 95.2 The mathematics that was actually new

MODE2's proof needed the **two-component window law for the factor `a(z)`** (only the derived factor
`(1/1!)(d/dz)a(z)` had one, from batch 197-A), and it confirmed the (8.4.19) mechanism exactly: the
`δ`-window term and the `⁺`-tail correction are the SAME family, so they add to `+2r⟨a,c⟩B_{m+r+1}`
rather than cancelling.  GEN2's proof needed the **square of the creation half** (`flatMul_assoc`) for the
composite `:a(z)a(z)Y(ab,z):` head, and the window→`flatMul` bridge for a *shifted* field: at a lattice
generator the field is a creation exponential over `iota(a+b+c)`, so a bridge must relate the source vector
`w` and the target vector `w'` separately (`headWindow_eq_flatMul_of_shift`/`_of_shift`).

### 95.3 The `sorryAx`-from-elaboration-error trap (campaign-level)

The dispatcher's capstone first reported `[propext, sorryAx, Classical.choice, Quot.sound]` for all three of
its theorems.  Cause: its statements used `CentralExtension` without `open FLMLattice`, so the declarations
failed to elaborate and Lean replaced them by `sorryAx` terms — with the `#guard_msgs` mismatch as the ONLY
symptom (the driver still printed a `--- FAILED` line, but a `| tail` pipeline reported `exit 0`, masking
it for one round).  Rules adopted: scan the whole log for `error` lines; never use `… | tail` when the exit
code matters (use `set -o pipefail`); and after a green build, `#print axioms` every public declaration
from a scratch probe that has no guard docstring (a guard whose docstring was written from the actual output
can never catch this).

### 95.4 Corrections and traps of record

1. `import Definitions.MMFLM86ZLineTwo` does NOT give `zLine_two_iff` (that is in `MM.FLM86ZLineBaseUpTo`).
2. `M >= r+1` is required for the derived factor only; the generic applied law carries `r+1 <= M`.
3. The order-two assembly needs the three ring index identities in a specific order; `simp` normalises them
   the wrong way and leaves `K-2+r+2-2`.
4. `rw`/`simp [neg_smul]` does not fire on `(-A) • x` for `x : VL F L` (module instance diamond); use
   `module` or `(neg_smul A x).symm`.  `rw [← smul_add]` can rewrite the outer `(1/2) • (…)`; use
   `(smul_add _ _ _).symm` in a `calc`.
5. The derivative-ladder route to GEN2 does not exist as a shortcut: `zLineAt_firstDeriv` is an iff, but the
   order-one generator line is landed only for `normOrdered2` itself, not for its derivative.

### 95.5 Accounting after batch 198

| object | status |
| --- | --- |
| `ZLine E j`, `j <= 2` | theorem (`zLine_zero`, `zLine_one_unconditional`, `zLine_two`) |
| `ZLineGeneratorBaseUpTo E J`, `J <= 2` | theorem |
| `l = -c` chambers | `ν >= c-3` unconditional at `hs`; `0 <= ν <= c-4` = `ZLine E j`, `j >= 3` |
| `ZLineGeneratorBase` (all orders) | open: MODE-ALL + GEN-ALL (batch 199, in flight) |
| Fock | ⟺ `Display8610TwoOneSided` + `FockPairDatum` + `FockCreatedSpan` (`FockCreatedSpan` likely a corollary of the landed `createdStateSpan_eq_top`) |
| `NegThreeExpandedResidualAligned` | open |
| `EvenCocycleJacobiStep` | open (assembly + `l != 0` content + passage to (8.10.8)) |

## §96 Batch 199 — both halves of the general-`j` base are one named residual away

### 96.1 The two residuals of record

Batch 199 did for the general order `j` what batch 197 did for `j = 2`: it reduced each half of
`MM.FLM86ZLI.ZLineGeneratorBase` to ONE named statement, with the analytic inputs landed.

| half | status | the named residual |
| --- | --- | --- |
| mode law, all `j` | `j <= 2` landed; the rest is one induction step | `MM.FLM86MLA.AnchorModeCommStep E` |
| generator line, all `j` | LHS evaluated at every `j`; target ⟺ flat form | `MM.FLM86GLA.ZAlignedGeneratorFlat E` |

The consumers are landed: `MM.FLM86MLA.zLineModeAt_all_of_step` (the step gives the mode law at every `j`)
and `MM.FLM86GLA.zLineGeneratorAll_of_flat` (the flat form gives the generator line).  So once the two
residuals are theorems, `ZLineGeneratorBase` is a theorem, hence `zLine_all_of_generatorBase` and
`negCBinomMomentAligned_of_generatorBase`: EVERY chamber of EVERY `l = -c` face becomes unconditional.

### 96.2 The new mathematics of batch 199

* **The general-`n` derived-factor window law** (`MM.FLM86MLA`): the (8.4.16)/(8.4.19) head/tail behaviour of
  `(1/(n-1)!)(d/dz)^{n-1}a(z)` for every `n >= 1`, with the `^+`-commutator firing at `k = r+n-1` and the
  zero-mode repair `-n<c,a>C(-r,n)Y_{m+r+n}`.  Its applied form `actionFamily_modeVL_of_window` carries the
  window bound `r+n-1 <= M` — **the landed `actionFamily_two_modeVL_of` hardwires `r+1 <= M`, which is a
  genuine gap for `n >= 3`** (a brief-level correction of record).
* **The anchor-field layer** (`A_m = YzeroField (baseAligned E) (E^-_a(z_0)_m iota(ab))`, the (8.5.5)
  recursion for it, `zLineVal_eq_anchorField`) — this is also the natural language for the `l = 0`,
  `<a,b> <= -3` residual, and batch 201-B is using it there.
* **The order-`j` weight bridge** (`MM.FLM86GLA`): `sum_{i in Icc 0 N} C(i,j) X_i (Y_{N-i} w) =
  (j!)^{-1} flatMul (iterDeriv j X) Y (N-j) w`, which evaluates the whole left-hand side of the generator
  line at every `j`.

### 96.3 Accounting after batch 199

| object | status |
| --- | --- |
| `ZLine E j`, `j <= 2` | theorem |
| mode law, all `j` | ⟺ `AnchorModeCommStep` (`j <= 2` proved) |
| generator line, all `j` | ⟺ `ZAlignedGeneratorFlat` (`j <= 2` proved) |
| `l = -c` chambers | `ν >= c-3` unconditional; the rest is the two residuals above |
| `l = 0`, `<a,b> <= -3` | `NegThreeExpandedResidualAligned`; batch 201-B is reducing it to the anchor values |
| Fock | `Display8610TwoOneSided` + `FockPairDatum` + `FockCreatedSpan` |
| `EvenCocycleJacobiStep` | open (assembly + `l != 0` content + passage to (8.10.8)) |

### 96.4 Corrections and traps of record (batch 199)

1. The general-`i` window law needs no Leibniz induction (closed forms + `iterDeriv_apply_smul`).
2. `actionFamily_two_modeVL_of`'s window bound `r+1 <= M` is insufficient for `n >= 3`;
   `actionFamily_modeVL_of_window` is the honest template.
3. The order-`j` weight expansion is `MM.FLM86ZM.binomCoeff_add_eq_sum`;
   `MM.FLM86ZLI.binomCoeff_sub_eq_sum_moment` is the moment-level (mode-law) expansion.
4. `normOrdered2_iotaVL` is in `MMFLM86LimitStep.lean:482` (there is no `MMFLM86LS.lean`).
5. GEN2 generalises verbatim only on the left half; the right half for `j >= 3` acts on the field of a lower
   coherent state, and the general-`r` action evaluations at `iota(c)` are not landed.
6. The mode law's final identification needs the (8.5.5) recursion at order `j-1-l` plus the triangle
   transpose; the tail coefficient `-n<c,a>C(-r,n)` is forced by the count.

## §97 Batch 200 — the Fock span is discharged; the all-order generator line is complete up to one finite-sum assembly

### 97.1 Fock: one carrier input left, and it is `PairFiniteAll`

```
fockCreatedSpan (E) : FockCreatedSpan E                                    -- UNCONDITIONAL, no hs
fockPairDatum_iff_pairFiniteAll (E) : FockPairDatum E <-> PairFiniteAll E
PairFiniteAll E := forall a x, PairFinite (ZAligned E a) (ZAligned E x)
coherentCreatedStep_of_twoOneSided_of_pairDatum : FockPairDatum E -> Display8610TwoOneSided E -> CoherentCreatedStep E
fockPairDatum_imp_negPairing : FockPairDatum E -> PairFiniteNegPairing E
```

So the Fock front is now: **`Display8610TwoOneSided`** (the concrete two-factor (8.6.10) display) **plus**
`PairFiniteAll` (equivalently `FockPairDatum`, and implied by the weaker-named
`PairFiniteNegPairing` together with batch 182's positive part).  NOTES 94.5's Fock row is superseded: drop
`FockCreatedSpan`, replace `FockPairDatum` by `PairFiniteAll`, and drop the "lists of length >= 2" case
(it collapses by `pairFinite_shiftedField_iff` + the spanning).

### 97.2 GEN-ALL: everything but one finite-sum identity

Batch 200-A landed the whole computation around the residual: the algebra (triangle reindexings,
`sum_range_alternating_binomCoeff`, `prod_range_sub_eq_neg_pow_mul_factorial`, `binomCoeff_mul_pred`),
the creation half of the next derived factor (`splitMinus_factorFamilyZero_succ`, through `heisFamZero`),
the tail at `iota(c)` (`splitPlus_factorFamilyZero_succ_apply_iotaVL`,
`killsAbove_factorFamilyZero_succ_iotaVL`, `tailWindow_factorFamilyZero_succ_iotaVL`), the (8.4.16) head
against the flat value at all orders (`sum_head_flatGeneratorValueZ`), the tail collapse
(`tailOp_factorFamilyZero_flatGeneratorValueZ`), the (8.5.5) recursion at a generator in component form
(`ZAligned_creationExpVL_eq_sum_actionFamily`), and the CORRECTED factorial identity
`sum_factorial_flatMul_iterDeriv`.  What is left is the bookkeeping identity ASSEMBLY
(`(sum head_r) + (sum tail_r) = j.eps.sum_l C(s,l)((j-l)!)^{-1} D_{j-l}(N_l)`), with the proof route and the
coefficient count hand-verified; `ZAlignedGeneratorFlat` itself is NOT yet a theorem.

**Correction of record**: the "item (iv)" identity recorded in batch 199-B's report
(`D_j = sum_r C(j-1,r) flatMul (iterDeriv r a^-) D_{j-1-r}`) is **FALSE** — at `j = 2` it gives
`2 flatMul a^- E^-_{a+b}` instead of `flatMul (deriv a^-) E^-_{a+b} + flatMul (a^- a^-) E^-_{a+b}`.  The
landed correct form is `sum_factorial_flatMul_iterDeriv`.

### 97.3 Traps of record (batch 200)

1. The window replacement in the tail cannot use the factor's own bound `r+1 <= factor.bound`: the `k = r`
   value `(-1)^r<a,c>` may vanish, so the honest comparison is `max bound (r+1)`.
2. No `c`-independent operator-valued flat family can equal `ZAligned` at the coherent state — the (8.4.17)
   shift lives in the value at `iota(c)`; state the head against the flat value at components `n+i`.
3. `{iota(c)}` spans only `vacuum (x) F{L}`, so "reduce an arbitrary single-factor state by the landed
   spanning" does not work as stated; the correct reduction is via `PairFiniteAll`.
4. The `vertexOpOfField (Yaligned E)` spelling of the (8.5.23) creation property is DEFINITIONALLY
   `toVertexOp (ZAligned E)` — a bridge lemma is neither needed nor desirable.

### 97.4 Accounting after batch 200

| object | status |
| --- | --- |
| generator line, all `j` | ⟺ `ZAlignedGeneratorFlat`; ALL inputs landed, one finite-sum ASSEMBLY left |
| mode law, all `j` | ⟺ `AnchorModeCommStep` (batch 201-A, to be resumed) |
| Fock | `Display8610TwoOneSided` + `PairFiniteAll` (`FockCreatedSpan` DISCHARGED) |
| `l = 0`, `<a,b> <= -3` | `NegThreeExpandedResidualAligned` (batch 201-B, to be resumed) |
| `EvenCocycleJacobiStep` | open (assembly + `l != 0` content + passage to (8.10.8)) |

## §98 Batch 201 — the mode-law half is unconditional; `ZLineGeneratorBase` now rests on the generator half alone

### 98.1 `zLineModeAt_all` (unconditional)

```
anchorModeCommStep (hs) (E) : AnchorModeCommStep E            -- MM.FLM86MLA, batch 201-A
zLineModeAt_all (hs) (E) : forall j a b c r, 1 <= r -> forall K u, ZLineModeAt ... j r K u
```

So the second half of `MM.FLM86ZLI.ZLineGeneratorBase` is a theorem at a general `E` with only `hs`, and
NOTES 97.4's row "mode law, all `j` ⟺ `AnchorModeCommStep`" is superseded.  `ZLineGeneratorBase` now rests
on the FIRST half alone: `MM.FLM86GLA.ZAlignedGeneratorFlat E`, whose whole computation batch 200 landed
(one finite-sum ASSEMBLY left).

The one genuinely new input was the (8.4.16)/(8.4.19) shift against a **finite family** of correction fields:
the landed `actionFamily_modeVL_of_sumWindow` keeps a single `-gamma.headWindow` term OUTSIDE the `sum_l`,
so summing the landed single-field shifts counts it `N` times.  `headWindow_factorFamilyZero_modeVL_shift_sum`
and `tailWindow_factorFamilyZero_modeVL_shift_sum` fix that, and `modeCommFam_step` is the abstract step
(consuming the (8.5.5) recursion as a hypothesis) that closes the induction.

### 98.2 The `l = 0`, `<a,b> <= -3` residual, reduced to `normOrdered2` alone

```
negThreeExpandedResidualAligned_iff_anchorForm (E) : NegThreeExpandedResidualAligned E <-> NegThreeAnchorForm E
anchorGeneratorValue_iff_zAlignedGeneratorFlat (E) : AnchorGeneratorValue E <-> ZAlignedGeneratorFlat E
negThreeExpandedResidualAligned_iff_chamberMomentForm (hs) (E) (hZ : ZLineAll E) :
    NegThreeExpandedResidualAligned E <-> NegThreeChamberMomentForm E
negThreeAtMinusThree_iff_chamberMomentFormAtMinusThree (hs) (E) : ...   -- the -3 slice, unconditional
```

Three structural facts worth remembering: (i) the anchor-field spelling costs NOTHING (an IFF with no
hypothesis); (ii) the "anchor values at generators" statement is not a new residual — it IS
`ZAlignedGeneratorFlat`; (iii) the residue's summand is an OPERATOR component, not a value at `iota(c)`,
because `{iota(c)}` spans only the lattice factor and the Fock half must not be lost.

### 98.3 Corrections of record

1. There is NO iff between the residue and the anchor VALUES: the residue's left-hand side is the honest
   commutator `twoVarField - swap2`.
2. The `<a,b> = -3` slice does NOT follow from the landed `k <= 2` tower: the anchor fields cancel in the
   residue and the `N = 3` chamber collapse remains (`NegThreeChamberMomentFormAtMinusThree`).
3. Batch 200 did not close `ZAlignedGeneratorFlat` (it landed the recursion at a generator plus the head/tail
   evaluations).
4. `MM.FLM86HAPF`/`MM.FLM86CF2` are namespaces, not file names.
5. Lean: `-(C) • Z` negates the SCALAR, `a * b • c` parses `a * (b • c)` (`•` infixl:73 > `*` infixl:70),
   and the printer renders `(-C)•Z` and `-(C•Z)` identically — the `delta` family must be
   `((-1:F) * (theta * C l))`.  Unapplied `rw [neg_smul]`/`[smul_sub]`/`[smul_smul]`/`[Finset.smul_sum]`
   fail on syntactically identical occurrences in this pin, and `abel` mangles additive indices inside atom
   arguments; use explicitly applied local `have`s.
6. The `1/j` is never introduced: multiply by `j`, use `(j-1-l) + (l+1) = j`, cancel by
   `smul_right_injective`; the reindex `t = i+1` is unnecessary.

### 98.4 Accounting after batch 201

| object | status |
| --- | --- |
| mode law, all `j` | **THEOREM** (`zLineModeAt_all`) |
| generator line, all `j` | ⟺ `ZAlignedGeneratorFlat`; all inputs landed, one finite-sum ASSEMBLY left |
| `ZLineGeneratorBase` | ⟺ `ZAlignedGeneratorFlat` ALONE |
| `l = -c` chambers | once `ZAlignedGeneratorFlat` lands: every chamber, by `zLine_all_of_generatorBase` |
| `l = 0`, `<a,b> <= -3` | ⟺ (given `ZLineAll`) `NegThreeChamberMomentForm`; the `-3` slice is an unconditional IFF |
| Fock | `Display8610TwoOneSided` + `PairFiniteAll` |
| `EvenCocycleJacobiStep` | open (assembly + `l != 0` content + passage to (8.10.8)) |

## §99 Batch 202 — the tower is closed; every `l = -c` face is unconditional

### 99.1 The three-step composition

```
batch 201 : zLineModeAt_all                                   -- the mode-law half, unconditional
batch 202A: zAlignedGeneratorFlat -> zLineGeneratorAll -> zLineGeneratorBase
            zLine_all_of_generatorBase, negCBinomMomentAligned_of_generatorBase
batch 202B: negCFaceAlignedFrom_of_lines (+ the generator-base consumers)
dispatcher: negCFaceAligned_evenCocycle                        -- the whole l = -c face at FLM's datum
```

So Corollary 8.8.17 (8.8.41)'s `l != 0` content is settled: for every `c >= 1` and every pairing
`0 <= <a,b>` the cross-bracket equals the cross-iterate at the even-lattice extension, and the underlying
chamber identity `NegCBinomMomentAligned E c` even holds at a GENERAL `E` with only the symmetry `hs`.

### 99.2 Why the face consumer was cheap

Batch 202-B's key observation: the numerical identity `c-1-(c-2) = 1` is used only inside
`negCBinomMomentAlignedAt_sub_two`'s proof of the CHAMBER, never in the face consumer.  Hence the consumer
generalises verbatim to `NegCFaceAlignedFrom E c (c-1-J)` with no `C(r+m,k)` weight split, and the only new
arithmetic is `c-1-J <= nu -> c-1-nu <= J`.  This is why the whole face became unconditional essentially for
free once the tower closed.

### 99.3 Corrections and traps of record (batch 202)

1. `MM.FLM86GAI.sum_factorial_flatMul_iterDeriv` is stated over ℚ-scalars (`[Algebra QQ A]`); at
   `Module.End F (VL F L)` the resulting `HSMul` terms do not unify with F-scalars.  Landed the F-scalar form
   `sum_factorial_flatMul_iterDeriv_F` (cf. the ℚ-vs-`HahnSeries` traps already in AGENTS.md).
2. The head window replacement is off by one: kills-bound `M0+1`, `M0 = (q - <a,b+c> - n - 1).toNat`.
3. Family-level lemmas applied to a component and then to `iotaVL c` need an explicit
   `sum_apply`/`smul_apply` distribution step.
4. The tail reindexing needs `Finset.sum_congr` (the Nat subtraction does not reduce), the `L = 0` split before
   `sum_range_alternating_binomCoeff`, and a final scalar reorder.
5. The template of a consumer must bind `E = C.toCentralExtension h2` explicitly; `negCFaceAlignedGe_evenCocycle`
   covers the whole range `nu >= c-1`, not one chamber.

### 99.4 Accounting after batch 202

| object | status |
| --- | --- |
| `ZLine E i`, every `i >= 1` | **THEOREM** (general `E`, only `hs`) |
| `ZLineGeneratorBase` | **THEOREM** |
| `NegCBinomMomentAligned E c`, every `c >= 1` | **THEOREM** (general `E`) |
| `NegCFaceAligned (C.toCentralExtension h2) c`, every `c >= 1` | **THEOREM** (whole face, all chambers) |
| `l = 0`, `<a,b> <= -3` | `NegThreeChamberMomentForm` (equivalently `NegThreeExpandedResidualAligned`) |
| Fock | `Display8610TwoOneSided` + `PairFiniteAll` |
| `EvenCocycleJacobiStep` | open: the `l = 0` assembly at (8.6.2) states + the passage to (8.10.8) |

## §100 Batch 203 — the Fock ledger corrected; the `l = 0` face assembled at all states; the `<= -3` residual's exact status

### 100.1 The Fock front is ONE statement

```
display8610TwoOneSided_iff_fockCreatedStep (E) : Display8610TwoOneSided E <-> FockCreatedStep E   -- no hypotheses
coherentCreatedStep_of_twoOneSided (hnd) (hs) (E) : Display8610TwoOneSided E -> CoherentCreatedStep E
```

The mechanism is worth remembering, because it retired a whole layer: the display's `l = 0` slice uses the
unconditional (8.5.23) creation property `(Y v)_{-1} iota(1) = v`, so that slice is `IterateProductStepAt` at an
ARBITRARY state, and `iterateProductStep_iff_forall_at` makes it the full one-step (8.4.25).  The carrier inputs
(`FockPairDatum`, `PairFiniteAll`, `FockCreatedSpan`) were an artefact of the created-state route (batch
197-B/200-B), not of the mathematics.

### 100.2 The `<a,b> <= -3` residual cannot be reformulated into something easier

```
negThreeAndBelowResidualAligned_iff_chamberMomentForm (hs) (E) :
    NegThreeAndBelowResidualAligned E <-> NegThreeChamberMomentForm E          -- both directions, only hs
residueAtN_iff_chamberMomentAtN (E) (N) : ResidueAtN E N <-> ChamberMomentAtN E N
allChamberMoment_iff_slices : AllChamberMoment E <-> C1 /\ C2 /\ NegThreeChamberMomentForm E
not_allChamberMoment_rankOne : not (AllChamberMoment (F:=Q) (L:=Z) basisUnitInt formR extR)
```

So: the `<= -3` residual IS the `N >= 3` chamber collapse; the `N = 1, 2` slices are theorems at FLM's datum
(and FAIL without evenness --- the rank-one refutation above); and no pointwise slice is cheaper.  Do not spend
budget re-spelling this residual.

### 100.3 The all-states assembly, and its one named residual

`corollary851_allStates_of_lattice_and_step` assembles the (8.5.4) states by an explicit `List.induction` from
the lattice corner plus a STEP; the step is named `Corollary851ModeStep` (the two one-creation-mode steps
`C u v -> C (alpha_c(-r) u) v` and `C u v -> C u (alpha_c(-r) v)` at the SAME window `K`).  The window side
condition `hwin : forall a b, (-<a,b>).toNat <= K` is explicit because `ResidueNegativeAligned` is per-pair
canonical-window.  Composed consumers: `corollary851_allStates_of_residuals`,
`corollary851_allStates_of_fockCreatedStep`.

### 100.4 Traps of record (batch 203)

1. A literal `List.foldr` in the goal makes the (8.5.4) induction blow the `whnf` budget; state through
   `stateOfVL` and rewrite with `rfl` unfolding lemmas via `simpa only` (do NOT raise `maxHeartbeats`).
2. The positive-pairing half of the lattice corner exists only at the even cocycle (cocycle compatibility), so a
   composition that needs both halves lives at `C.toCentralExtension h2`; use the arbitrary-`Y` theorem
   `corollary851_allStates_of_lattice_and_step` if a general-`E` producer of the corner becomes available.
3. `hnd` is spent only by the Fock conjunct of the residual package.

### 100.5 Accounting after batch 203

| object | status |
| --- | --- |
| `l = -c` faces, all `c >= 1` | THEOREM (batch 202) |
| `NegTwoFirstDerivZero` | THEOREM |
| `NegThreeExpandedResidualAligned` = `NegThreeChamberMomentForm` | open (== the residual; `N=1,2` slices done, `N>=3` open) |
| `CoherentCreatedStep` | open, implied by `FockCreatedStep` = `Display8610TwoOneSided` (ONE statement now) |
| `l = 0` face at all states | `Corollary851ModeStep` + the lattice corner (named) |
| `EvenCocycleJacobiStep` | (ii) done; (i) as above; (iii) untouched |

## §101 Batch 204 — the `<= -3` residual is proved; the Fock front is the shift-zero corner

### 101.1 The residual that was called hard

Batch 203-A recommended not attacking `NegThreeChamberMomentForm`, calling it identical to the residual.  Batch
204-A showed the CHAMBER spelling is where it is easy: everything happens on ONE antidiagonal family
`M p = normOrdered2 E a b p (K-p) v`, the two chamber half-lines `p >= N-m-1` and `p <= -m-1` against the full
moment line with weight `C(p+m,N-1)`, with the (8.4.21) transpose supplying `c(b,a)`, the (8.5.37) coefficient
identities on the first chamber, the cocycle sign `c(b,a) = (-1)^N` on the second, and the falling-factorial
zero on the gap.  The only general-`E` input is that sign (`CocycleAtPairing`), a theorem at the even cocycle;
the underlying scalar identity holds BOTH ways (`neg_mul_ratioCoeff_eq_iff`).

Consequences: `ResidueNegativeAligned (C.toCentralExtension h2)` is now a THEOREM (the whole negative-pairing
`Res_{z_0}` face), and `EvenCocycleResiduals`'s second conjunct is discharged.  "Evenness" was the wrong word:
only `c(b,a) = (-1)^N` is used; at `N = 1` this is necessary (rank-one refutation), at `N >= 3` the general-`E`
form is undecided.

### 101.2 The Fock front, sharpest form

```
FockShiftZeroStep E := forall x v, ZAligned E ((ZAligned E x) (-1) v) = convOneSided (ZAligned E x) (ZAligned E v)
fockCreatedStep_iff_level_zero / fockCreatedStep_iff_shiftZero (hnd) (hs) (E)
```

The `L(-1)` ladder identifies the `j`-th level of (8.4.25) at `(x,v)` with the level-zero statement at
`(1/j!)L(-1)^j x` (iterated (8.10.13)); the abstract lemma carries no hypothesis on the product, so it applies
to the `dite` carrier.  `hnd`/`hs` turned out to be redundant on every consumer
(`coherentCreatedStep_of_twoOneSided_noHyp`, `fockCreatedStep_iff_createdStep`,
`fockCreatedStep_imp_coherentCreatedStep`).

### 101.3 A warning worth keeping

The Fock-level step through the mode laws does NOT work: the landed action laws
(`MM.FLM86MLA.actionFamily_factorFactorZero_modeVL`, `actionFamily_modeVL_of_window`) take the two-component
commutator as a HYPOTHESIS on the accumulated field, and the missing input would be the commutator of two
arbitrary honest modes --- exactly the open Jacobi content.  The `L(-1)` ladder avoids it entirely, which is why
the front collapsed from "display + two carrier inputs" to one shift-zero corner in three batches.

### 101.4 Accounting after batch 204

| object | status |
| --- | --- |
| `ResidueNegativeAligned` at FLM's datum | **THEOREM** |
| `NegThreeExpandedResidualAligned` (either spelling) | **THEOREM** at FLM's datum; general `E` needs `CocycleAtPairing` |
| `l = -c` faces | THEOREM (batch 202) |
| `CoherentCreatedStep` | open; ⟸ `FockShiftZeroStep` (= `FockCreatedStep` = the display), no `hnd`/`hs` needed |
| `EvenCocycleJacobiStep` (i) | `Corollary851ModeStep` + the lattice corner |
| `EvenCocycleJacobiStep` (iii) | untouched (Theorem 8.8.9 / Prop 8.10.7 route) |

## §102 Batch 205 — the Fock corner is refuted; the `l = 0` face is one mode step

### 102.1 A refuted target, and FLM warned us

`FockShiftZeroStep E` --- equivalently `FockCreatedStep E` and `Display8610TwoOneSided E` (batch 204-B's and
203-B's equivalences) --- is FALSE at honest data:
`not_fockShiftZeroStep_of_gram (E) (i) (hgram : form (bL i) (bL i) != 0)`, with an unconditional rank-one
instance, spending neither `hnd` nor `hs`.  The failing configuration is one oscillator on the vacuum:
`x = alpha_a(-1) iota(0)`, where the convolution has a summand `((-j)<a,a>)iota(0) != 0` for every `j <= -1`,
so `PairFinite` fails and `convOneSided` is identically `0` there while the two sides of (8.4.25) are not.

FLM flags exactly this: Remark 8.5.2 (p. 200) on (8.5.14) "not rigorous in the present setting", and Remark
8.6.4 (p. 211) on (8.6.28) "formal but suggestive".  The honest identity at a created state is the
NORMAL-ORDERED `Y(u_{-1}v) = :Y(u)Y(v):`; the plain product differs by the contraction.  The practical lesson for
this campaign: when a residual is "the book's display", check whether the book itself calls it formal.

### 102.2 What survives, and the honest Fock front

`CoherentCreatedStep` (the third conjunct of `EvenCocycleResiduals`, and the only Fock object the boundary
theorem consumes) is NOT refuted: its factors are honest GENERATOR fields (the second shifted), where the
carrier is batch 182's `pairFinite_shiftedField_ZAligned_iotaVL` for `0 <= <a,b>`, and the oscillator state
above does not occur.  So the honest front is the GENERATOR SLICE of (8.6.10)
(`Display8610TwoGenerators` / `CoherentCreatedForm`, batch 185-A's
`display8610TwoGenerators_iff_coherentForm`), with `PairFiniteNegPairing` the only open carrier case.
`fockCreatedStep_imp_coherentCreatedStep` and `coherentCreatedStep_of_twoOneSided_noHyp` are now VACUOUS.

### 102.3 The `l = 0` all-states face, precisely

```
allStates_iff_corner_and_modeStep (Y) (K) : (forall u v, C u v K) <-> corner /\ Corollary851ModeStep Y K
corollary851_latticeFactor : the corner extends to span{iota(a)} x span{iota(a)}   -- the exact largest slice
corollary851_iff_faceOf : C u v K <-> FaceOf Y (Y u) v K                          -- the face is a linear condition on the field
modeStepLeft_iff_faceOf_closure : the mode step = preservation of the face submodule by the (8.4.16) action
```

Given the corner (a theorem since batch 204-A), the mode step is the whole face content; its expansion terminates
in the mode--field commutators `[alpha_c(e), ZAligned E v n]` at arbitrary `v`, which the landed action laws
take as `hcomm` and return only up to a third term `kappa . Yt.fam (m+r+n)`.  `MM.FLM810CFC.currentFieldComm` is
(7.1.5) at the repaired BASE field only, so it is not the missing input.

### 102.4 Accounting after batch 205

| object | status |
| --- | --- |
| `l = -c` faces; `ResidueNegativeAligned`; `NegThree*` | THEOREM (batches 202, 204-A) |
| `NegTwoFirstDerivZero` | THEOREM |
| `CoherentCreatedStep` | open; honest producer = the GENERATOR slice of (8.6.10) |
| `FockCreatedStep` / `Display8610TwoOneSided` / `PairFiniteAll` | **REFUTED** |
| `l = 0` face at all states | = `Corollary851ModeStep` (= the mode--field commutators), open |
| `EvenCocycleJacobiStep` (iii) | untouched |

## §103 Batch 206 — the generator slice's `(0,0)` corner proved; the two exact residuals named

### 103.1 The Fock front, now precise

```
createdStepAt_zero_zero (hs) (0 <= <a,b>) : the (j,l) = (0,0) instance at generators   -- THEOREM, no hnd
coherentCreatedStep_iff_twoGenerators / <-> CoherentCreatedForm                       -- both ways
weightedAssemblyDiagonalZero_iff_assemblyDiagonal : the r = 0 slice IS (8.4.20)        -- both ways
```

The residual is **(WAD) = `WeightedAssemblyDiagonal E` at `r >= 1`**:
`sum_{p+q=K} C(K-p,r) normOrdered2 a b p (K-p) v = eps(a,b) ZAligned E (E^-_b(z_0)_r iota(a+b)) (r-K-1) v` on a
finite window, with the dictionary `CreatedStepAt E a b 0 l <-> (WAD)` at `r = l-<a,b>`,
`K = l-m-1-<a,b>` (and `j >= 1` the `L(-1)` ladder; `<a,b> < 0` additionally `PairFiniteNegPairing`).

Two more vacuous-theorem findings: `MM.FLM86CCS.createdStep_of_iterateProductStep` is vacuous (its
`IterateProductStep` at `(x,0)` with base `v := x` IS the refuted `FockShiftZeroStep`); and NOTES §102.2's
"the oscillator-carrying state does not occur" holds only for the FACTORS, since `creationExpVL b l (iota b)` is
oscillator-carrying for `l >= 1`.  What saves the generator slice is the carrier (pairs of shifted generator
fields, finite for `0 <= <a,b>`) plus (WAD).

### 103.2 The §8.6 engine: what it now has, and the one carrier it lacks

Landed at the honest field: the (8.6.5) series and its laws; `powerFam = YFam(-m-1)` with the exact pairing
shift `Y(a,w)_m iota(c) = E^-_a(w)_{m-<a,c>} iota(ac)`; `splitFold` (iterated (8.4.16), with `k = 1`
`Y(a,w)` and `k = 2` the landed `normalPair`); (8.6.10) at `k = 1`; and (8.6.6)/(8.6.8) at `k = 2` matching
the landed (8.4.20) assembly.

**Gap: (8.6.6) for `k >= 3`** --- a `k`-variable nested Cauchy product with local finiteness.  The carrier has
only the two-variable `convWith2`, which (8.4.20) uses twice.  The passage (iii) needs this carrier first.

### 103.3 Corrections (B206-B)

1. (8.6.5) and `kFoldNormalOrder` were already landed in batch 119 (`MMFLM86MultiState`, `MMFLM86KFold`);
   ROADMAP:5013-5015 / NOTES:1055-1056 were right and ROADMAP:4679 is stale.
2. `kFoldNormalOrder` is not (8.6.6)'s middle family for `k >= 2` on honest fields (its mode splitting includes
   `E^+`); only `normOrdered2` is.
3. Remark 8.6.4 shows the PLAIN product and calls it formal --- the normal-ordered repair is ours.
4. The `iota`-shifts are (8.5.11), not (8.6.5).
5. No associativity for the `k`-fold normal-ordered product exists in §8.6.

### 103.4 Accounting after batch 206

| object | status |
| --- | --- |
| `ZLine` tower / `l = -c` faces / `ResidueNegativeAligned` / `NegTwoFirstDerivZero` / `NegThree*` | THEOREM |
| `CoherentCreatedStep` | `(0,0)` at generators THEOREM; open from `(j,l) != (0,0)`; residual (WAD) at `r >= 1` |
| (8.6.6)`_{k>=3}` | the `k`-variable nested Cauchy carrier --- needed for (iii) |
| item (i) | `Corollary851ModeStep` = the mode--field commutators |
| (iii) | not reached |

## §104 Batch 207 — the nested Cauchy carrier is built; the Fock residual is (WAD) at `r >= 1`

### 104.1 The dictionary (both ways)

```
createdStepAt_zero_iff_weightedAt (hs) (E) (a b) (hN : 0 <= <a,b>) (hl : <a,b>.toNat <= l) :
    CreatedStepAt E a b 0 l <-> WeightedAssemblyAt E a b (l - <a,b>.toNat)
weightedAssemblyAt_iff_createdStepAt_zero : WeightedAssemblyAt E a b r <-> CreatedStepAt E a b 0 (r + <a,b>.toNat)
weightedAssemblyAt_zero : the r = 0 slice IS the landed (8.4.20)
createdStepAt_zero_of_lt : the gap l < <a,b>.toNat is a theorem (both sides 0)
```

So at `0 <= <a,b>` the Fock residual's entire `j = 0` face is (WAD) at `r >= 1`.  The negative-pairing face is NOT
covered (the (8.4.25) expansion and batch 182's carrier exist only for `0 <= <a,b>`; that face needs
`PairFiniteNegPairing` and an expansion of its own), and `j >= 1` is NOT covered because the `L(-1)` ladder
reduces level `j` to level 0 at the TRANSLATED factor `(1/j!)D^j a` --- a descendant of `iota(a)`, not a lattice
generator.  That is the measured reason the consumer cannot be composed yet.

### 104.2 The nested Cauchy carrier, and (8.6.6)/(8.6.8) at general k

```
normOrderedK_vacuum (E) (k) (as : Fin k -> L) (r : Fin k -> Nat) :
    (normOrderedK bL form E k as (fun i => (r i : Z))) vacuumVL
      = blockScalarK E k as . series865 bL form k as (sum i, as i) r
normOrderedK_one = MM.FLM86E86.powerFam ;  normOrderedK_two = MM.FLM86TN.normOrdered2
```

The finiteness of the nested product is a THEOREM, not an input: `locallyFiniteK_famFold` handles any ordered
product of one-variable locally finite families, so only `creationPairK` fails to be locally finite (the landed
`creationPair2_not_locallyFinite2`), which is (8.4.20)'s nesting kind.  Remaining in this route: (8.6.6)'s SECOND
equality (the pairing-factor family `prod_{i<j}(w_i-w_j)^{-<abar_i,abar_j>}`; the landed batch-119
`Display866Family3` plus the OPE (8.4.22)/(8.4.36)).  No general-`k` symmetry or associativity was invented ---
FLM states (8.4.21) for two factors only, and only `k = 2` is landed.

### 104.3 Traps of record

1. The shifted-weight collapse is orientation-sensitive: the needed identity is the created-state orientation
   `sum_ratioCoeff_mul_binomCoeff_shift` plus its GAP case `..._of_lt` (new in this batch), not the
   batch-203/204 moment machinery.
2. The `j >= 1` ladder does not feed the dictionary (descendant vs generator) --- do not assume it does.
3. The Fock residual's global spelling lives on the POSITIVE-pairing face; stating it globally is false.

### 104.4 Accounting after batch 207

| object | status |
| --- | --- |
| (8.6.6)/(8.6.8) at general `k` | **THEOREM** (nested Cauchy carrier built; finiteness discharged) |
| Fock residual | (WAD) at `r >= 1` = `CreatedStepAt(0,l)` at generators (dictionary both ways); `j >= 1` and the negative-pairing dictionary untouched |
| (8.6.6) second equality | the pairing-factor family + OPE (8.4.22)/(8.4.36) --- the next step of route (iii) |
| item (i) | `Corollary851ModeStep` (mode--field commutators) |
| (iii) | not reached |

## §105 Batch 208 — (WAD) is a theorem; (8.6.6)'s second equality is a named Prop at `k >= 3`

### 105.1 Reading the tower backwards

`WeightedAssemblyAt E a b r` holds at EVERY weight and EVERY pair (`weightedAssemblyAt_all`), with no positivity,
no evenness and no carrier input.  The proof is `zLine_all` at the TRANSPOSED pair `(b,a)` with order `r` and
scheme parameter `n = r-K-1` (diagonal degree `K` --- no index renormalisation), then the reindex `p |-> K-p`,
the (8.4.21) transpose (which moves the weight to the second index and produces `c(b,a)^-1`), the anchor expansion
`creationExpVL_anchorVL`, and finally `eps_mul_comm` (7.1.26).  **The real content is the weight ORIENTATION**:
`momentSum` weights the first index, (WAD) the second.  Consequence: `CreatedStepAt E a b 0 l` for ALL `l` at
`0 <= <a,b>` (`createdStepAt_zero_of_nonneg`), so the Fock residual's `j = 0` positive face is CLOSED.

The exact residual of `CoherentCreatedStep` is now, both ways
(`coherentCreatedStep_iff_of_pos_zero_face`):

1. `CreatedStepAtZeroNegPairing E` --- the negative-pairing orientation of the dictionary (needs
   `PairFiniteNegPairing` AND its own expansion; the (8.4.25) expansion is refuted at a negative pairing);
   note (WAD) itself is a theorem there too, so only the `(WAD) => CreatedStepAt` direction is missing;
2. `CoherentCreatedStepPos E` --- the `j >= 1` face, open everywhere including `(0,0)`, with NO producer in the
   tree (the `L(-1)` ladder reduces to a DESCENDANT of `iota(a)`).

### 105.2 (8.6.6)'s second equality

`Display866SecondEqK` / `Display866BoxK` (equivalent both ways) are now stated at every `k`, with `hs` and
`0 <= <abar_i,abar_j>` as explicit binders and the `k = 1, 2` slices proved (`k = 2` via the landed
`MM.FLM86YP.YFam_mul_apply_eq_sum_ratioCoeff`).  The composition `display866Display_of_second` turns the second
equality plus the landed first equality into (8.6.6) as FLM states it.  The remaining piece is the general-`k`
OPE/insertion step (one honest field against a `k`-fold normal-ordered product) --- the `k >= 3` case of
(8.4.22)/(8.4.36) --- with `Contraction8612Ratio` open even at `k = l = 1`.

Correction: batch 119's `Display866Family3` (families `Fin k -> Nat -> VL`, scalar shape) CANNOT carry the second
equality; the honest carrier is batch 207's `normOrderedK`.

### 105.3 Accounting after batch 208

| object | status |
| --- | --- |
| (WAD) / `WeightedAssemblyDiagonal` | **THEOREM**, every weight, every pair |
| `CreatedStepAt E a b 0 l` (`0 <= <a,b>`) | **THEOREM** |
| Fock residual | `CreatedStepAtZeroNegPairing` /\ `CoherentCreatedStepPos` |
| (8.6.6) second equality | `k <= 2` THEOREM; `k >= 3` the named Prop `Display866SecondEqK` |
| item (i) | `Corollary851ModeStep` |
| (iii) | needs `Display866SecondEqK` at `k >= 3` |

## §106 Batch 209 — the Fock residual is refuted as spelled; the honest slice is `0 <= <a,b>`

### 106.1 The refutation, two independent routes

```
yFam_creationExpVL_iotaVL : YFam E a (-<a,b>+q-1) (creationExpVL b q (iotaVL b))
                              = ratioCoeff <a,b> q . (eps(a,b) . iotaVL (a+b))          -- batch 209-B
zAligned_createdStep_left_zero : the l = 0 left-hand side, evaluated by ZAligned_mode_iotaVL -- batch 209-A
not_pairFinite_generators_of_neg / not_pairFinite_rankOne
not_createdStepAtZeroNegPairing_of_exists_neg / not_createdStepAt_rankOne
not_coherentCreatedStep_of_exists_neg / not_coherentCreatedStepPos_rankOne
```

At `<a,b> = -N < 0` the convolution summand is `C(N+q-1,q).eps.iota(a+b) != 0` for EVERY `q`, so
`PairFinite (ZAligned (iota a)) (ZAligned (iota b))` fails, `convOneSided` returns 0, and the residual asserts
that a nonzero field vanishes.  The dropped ingredient is (8.6.25)'s pairing factor
`prod (z_0+w_i-x_j)^{<a_i,b_j>}`, invisible on the diagonal iff `0 <= <a,b>`; FLM's Remarks 8.5.2 (p. 200) and
8.6.4 (p. 211) say exactly this.

### 106.2 What this costs, and what it does not

* `CoherentCreatedStep` (third conjunct of `EvenCocycleResiduals`) is FALSE at FLM's Leech datum, so
  `theorem8102_of_residuals` is VACUOUS and must be re-stated with the honest residual.
* NOT affected: the `ZLine` tower, the `l = -c` faces, `ResidueNegativeAligned`, `NegTwo*`, `NegThree*` --- all
  of those use `normOrdered2`, never `convOneSided`.
* Batch 200-B's "the summand cancels" note is REFUTED (it cancels in Fock degree, not in `j`).

### 106.3 The corrected accounting

```
CoherentCreatedStepPosNonneg E := forall a b, 0 <= form a b -> forall j, 1 <= j -> forall l, CreatedStepAt E a b j l
CoherentCreatedStepNonneg E    := forall a b, 0 <= form a b -> forall j l, CreatedStepAt E a b j l
coherentCreatedStepNonneg_iff_posNonneg (hs) : Nonneg <-> PosNonneg   -- the j = 0 face is batch 208-A
coherentCreatedStep_iff_slices (hs) : CoherentCreatedStep <->
    PosNonneg /\ CreatedStepAtZeroNegPairing /\ PosNegPairing        -- the last two are FALSE
```

`CoherentCreatedStepPosPairingFactor` names the honest negative-pairing replacement (threshold form of
(8.4.22)/(8.4.25), `MM.FLM86RX.threshold_holds`, matching `MM.FLM86MXP.Display8625One` at `k = l = 1`); it is
STATED, not proved, and the coupling of the outer window to the threshold `T` is flagged unverified.

### 106.4 The lesson (recorded for the whole campaign)

For three batches (197-B/203-B/204-B) the Fock front was progressively "reduced" to targets that were FALSE,
because the campaign's own `convOneSided` spelling dropped FLM's pairing factor.  The remedy that ended it was
the batch-208 discipline: **falsification check on a small instance before investing**, plus the axiom-guard probe
that catches silent `sorryAx`.  Every future "prove X" on this front must start with the check, and every target
must be quoted from FLM WITH its Remarks.

### 106.5 Accounting after batch 209

| object | status |
| --- | --- |
| `ZLine` tower / `l = -c` faces / `ResidueNegativeAligned` / `NegTwo*` / `NegThree*` | THEOREM (unaffected) |
| `CoherentCreatedStep` | **FALSE** (refuted); `EvenCocycleResiduals` false; `theorem8102_of_residuals` VACUOUS |
| `CoherentCreatedStepPosNonneg` (`0 <= <a,b>`) | open --- the honest Fock residual |
| `CoherentCreatedStepPosPairingFactor` | named, not proved (the honest negative-pairing replacement) |
| (8.6.6) second equality `k >= 3` | named Prop (route (iii)) |
| item (i) | `Corollary851ModeStep` |

## §107 Batch 210 — the honest boundary is landed; the Fock residual is one two-shift statement

### 107.1 The Fock residual, final shape

```
createdStepAt_shiftZero (hs) (0 <= <a,b>) : CreatedStepAt E a b j 0        -- EVERY j, the l = 0 face, THEOREM
CoherentCreatedStepShifted E := forall a b, 0 <= form a b -> forall j, 1 <= j -> forall l, 1 <= l,
    CreatedStepAt E a b j l                                               -- THE residual
coherentCreatedStepPosNonneg_iff_shifted : CoherentCreatedStepPosNonneg E <-> CoherentCreatedStepShifted E
coherentCreatedStep_iff_shifted_slices : CoherentCreatedStep E <->
    CoherentCreatedStepShifted E /\ CreatedStepAtZeroNegPairing E /\ CoherentCreatedStepPosNegPairing E
```

The `l = 0` face closes at every `j` by the reflected (8.5.37) finite difference collapsing the double-binomial
weight to `C(P, j-N)`, then the landed tower `ZLine E (j-N)`.  At `l >= 1` the weight carries the extra
`C(q+i,l)` and no single finite difference collapses it (hand-checked at `(N,j,l) = (2,1,1)`).

### 107.2 The honest boundary, and why it is not vacuous

```
FockResidualHonest E := CoherentCreatedStepPosNonneg E /\ CoherentCreatedStepNegPairingFactor E
                          /\ CreatedStepAtZeroNegPairingHonest E
EvenCocycleResidualsHonest C h2 := NegTwoFirstDerivZero /\ NegThreeExpandedResidualAligned
                                     /\ FockResidualHonest (C.toCentralExtension h2)
theorem8102_of_residuals_honest : Theorem8102
evenCocycleResidualsHonest_iff : (forall a b, 0 <= form a b) ->
    (EvenCocycleResidualsHonest C h2 <-> EvenCocycleResiduals C h2)          -- non-vacuity on that slice
```

The audit also lands the vacuity of the whole old Fock chain as theorems (see ROADMAP batch 210).

### 107.3 The one object still missing

`CreatedStepAtZeroNegPairingHonest` names the normal-ordered ONE-VARIABLE product at the shifted coherent state,
with the (8.6.25) threshold clause (without it the def is trivially provable --- the agent checked).  The tree has
no such object (`MMFLM86YP` §6(ii): `convWith (YFam E a) (YFam E b)` does not exist; `YFam` is not a `CField`
for `a != 0`).

### 107.4 Accounting after batch 210

| object | status |
| --- | --- |
| old `EvenCocycleResiduals` / `theorem8102_of_residuals` | **FALSE / VACUOUS** (audited as theorems) |
| honest package + `theorem8102_of_residuals_honest` | landed, non-vacuous on the `0 <= <a,b>` slice |
| `CoherentCreatedStepShifted` | open --- the positive-pairing Fock residual |
| `CreatedStepAtZeroNegPairingHonest` | open --- needs the missing normal-ordered object |
| `CoherentCreatedStepNegPairingFactor` | open |
| (i) `Corollary851ModeStep` / (iii) `Display866SecondEqK` | open |

## §108 Batch 211 — the two-shift face proved from a derivation identity; the shifted normal-ordered object built

### 108.1 `CoherentCreatedStepShifted` reduced to `ShiftedCreationDerivation`

```
ShiftedCreationDerivation E := forall n >= 1, forall w, L(-1)(alpha_a(-n) w) = alpha_a(-n)(L(-1) w) + n.alpha_a(-n-1) w
coherentCreatedStepShifted_of_derivation (hnd) (hs) (E) (hder) : CoherentCreatedStepShifted E
coherentCreatedStepPosNonneg_of_derivation (hnd) (hs) (E) (hder) : CoherentCreatedStepPosNonneg E
```

This is the (8.4.14)/(7.1.41) derivation identity --- the tree's open `MM.FLM810TG.LMinusOneCreationMode`, whose
generator case is landed.  The proof needs NO new collapse identity: it uses
`E^-_b(l) iota(b) = (1/l!)L(-1)^l iota(b)`, the new Leibniz rule `coefDeriv_convOneSided`, (8.6.9), and the
cancellation of `(j+1)(Y a)_{-j-2}B_l` against the level-`j+1` shift-zero statement.  The double-binomial weight
of batch 210-A is bypassed.  `hnd` IS spent (the `L(-1)` route goes through the conformal vector).

Next step: creation-list induction from `MM.FLM810TG.lMinusOneCreationMode_iota`, template
`MMFLM810DFA.translationResidual_of_baseResidual`.

### 108.2 `pairBlock`: the (8.6.25) object, built

`pairBlock E a b j l m T` is the block sum at the shifted coherent state with the pairing factor carried as
`ratioCoeff`; `convOneSided_shifted_eq_ratioSum` shows the carrier's summands ARE the two shift binomials
against `twoVarField`, **which is where the second slot index `l` comes from**.  The two negative-pairing faces
are now equivalent (both ways) to `HonestNegPairingZero` / `HonestNegPairingPos`.

### 108.3 Two defects of record (batch 210-B's spelling)

1. **Machine-checked slot error**: the second shift binomial was written with index `0` instead of `l`
   (`pairBlock_slot_discrepancy`), so `CreatedStepAtZeroNegPairingHonest` is not the `j = 0` slice of slice 2.
2. **Single threshold too small**: the honest spelling needs a threshold PER SHIFT INDEX (outer and inner).
   NOTES §106.3's "flagged unverified" is upgraded to "wrong as spelled".

The def was recorded with a defect note (docstring), NOT weakened; the corrected package
`MM.FLM86NOS.FockResidualHonestCorrected` agrees with the batch-210 package on the `0 <= <a,b>` slice.

### 108.4 What still blocks the negative-pairing faces

The **h-weight (Fock bidegree) grading of M(1)**: without it one cannot separate the
`E^-_a(P) E^-_b(Q) iota(a+b)` blocks with different `Q`, which is what the per-shift-index threshold needs.
That grading is not landed.

### 108.5 Accounting after batch 211

| object | status |
| --- | --- |
| `CoherentCreatedStepShifted` / `CoherentCreatedStepPosNonneg` | PROVED from `ShiftedCreationDerivation` (+ `hnd`) |
| `ShiftedCreationDerivation` (= `LMinusOneCreationMode`) | open; generator case landed |
| `HonestNegPairingZero` / `HonestNegPairingPos` | open; object built; blocked on the M(1) bidegree grading |
| (i) `Corollary851ModeStep` / (iii) `Display866SecondEqK` | open |

## §109 Batch 212 — the bidegree and its h-weight refinement; the residual pinned to the `YFam` modes

### 109.1 The residual, corrected and pinned

```
lMinusOne_eq_sugawaraModeZero : lMinusOne hnd E = sugawaraModeZero hnd (-1)        -- the currency bridge
shiftedCreationDerivation_iff_YFamShiftedDerivation : ShiftedCreationDerivation <-> YFamShiftedDerivation
```

**Batch 211-A's identification was WRONG** (machine-checked): the residual's modes are the (8.4.17) field of the
lattice generator (`YFam`), not the Heisenberg modes `modeVL` of `LMinusOneCreationMode`/`SugawaraDerivation`
--- at `a = 0, m = -1` they differ on the vacuum.  The tree's state-half object `MM.FLM87SR.sugawaraDerivation`
is a THEOREM (batch 165) and does not imply the residual.  What remains is `YFamModeDerivation` (= the tree
spelling) and its generator base case `YFamDerivationBase`; the creation-list step is hand-verified and recorded
in the docstring but its Lean proof was withdrawn at the scalar-normalisation step (`Module.End` smul trap).

### 109.2 The grading, and why the coarse one is not enough

`bidegPiece`/`bidegProj`/`bideg_separation` and the h-weight `hwWt`/`hwPiece` are landed.  The falsification
`pairBlock_block_deg` shows the `(N,a)` bidegree of every `pairBlock` summand is constant in `(q,k)` at a
lattice generator, so the SEPARATING grading is `hwWt` (the summand carries `(M+1-Q)a + (Q-2)b`).  This is
batch 211-B's per-shift-index threshold obstruction, now an identity.

### 109.3 Accounting after batch 212

| object | status |
| --- | --- |
| `CoherentCreatedStepShifted` / `PosNonneg` | ⟸ `YFamModeDerivation` + `YFamDerivationBase` (step hand-verified) |
| Fock bidegree + h-weight refinement | LANDED |
| `FockAnnihilationDegreeLaw` | named (exact-degree annihilation) |
| `HonestNegPairingZero`/`HonestNegPairingPos` | open; need the h-weight mode law + block separation |
| (i) `Corollary851ModeStep` / (iii) `Display866SecondEqK` | open |

## §110 Batch 213 — the positive-pairing Fock faces are theorems; the h-weight separation is landed

### 110.1 The residual is closed

```
yFamDerivationBase / yFamModeDerivation / shiftedCreationDerivation (hnd) (hs) (E) : ...
coherentCreatedStepShifted / coherentCreatedStepPosNonneg (hnd) (hs) (E) : ...
```

NO named residual remains for `CoherentCreatedStepPosNonneg`.  The base case is the creation half of
(8.10.13)/(8.10.14), `[L(-1),E^-(a,z)] = (d/dz)E^-(a,z) - a(-1)E^-(a,z)`, proved abstractly once
(`sub_mul_of_deriv_expField`).  The step is pure vector-level algebra.

### 110.2 The h-weight separation, and what it corrects

The mode law is homogeneous only at BASIS labels; for general `c` the mode is inhomogeneous
(`naiveHwModeLawRankTwo_false`), and the block separation `pairBlock_block_weight_eq_iff` holds at basis labels
--- so batch 212-B's general-label claim is corrected.  The two faces are now per-h-weight statements
(`HonestNegPairingZeroHw`/`HonestNegPairingPosHw`), with the remaining obstruction being the assembled
per-shift-index threshold, blocked on a missing fiberwise-sum lemma for `(q,k) |-> q+k` in this Mathlib pin.

### 110.3 Milestone: the honest residual package is two Fock faces

| conjunct of `EvenCocycleResidualsHonest` | status |
| --- | --- |
| `NegTwoFirstDerivZero` | THEOREM (batch 196) |
| `NegThreeExpandedResidualAligned` | THEOREM (batch 204-A) |
| `CoherentCreatedStepPosNonneg` | **THEOREM (batch 213-A)**, modulo `hnd`/`hs` |
| `CoherentCreatedStepNegPairingFactor` | open ⟺ `HonestNegPairingPos` ⟺ `HonestNegPairingPosHw` |
| `CreatedStepAtZeroNegPairingHonest` | open ⟺ `HonestNegPairingZero` ⟺ `HonestNegPairingZeroHw` |

Plus `EvenCocycleJacobiStepHonest` (the assembly), whose content is (i) `Corollary851ModeStep` and
(iii) `Display866SecondEqK` at `k >= 3`.

### 110.4 Traps of record (batch 213)

1. The mode `-(k+1)` index trap: the contraction shifts the `YFam` index by `k+1` (`m-k-1`, not `m-k`),
   because `module` treats `YFam a (m-1-k)` and `YFam a (m-k-1)` as different atoms.
2. `modeVL_YFam_add` yields `-↑n` / `j - ↑n`, only definitionally equal to `-1` / `m-1`; `rw`'s keyed matching
   fails across the difference (`simpa using h` fails where `exact h` succeeds).  Worth an AGENTS.md bullet.
3. The "scalar-normalisation blocker" was an artefact of an operator-level formulation; the vector-level
   formulation avoids the `Module.End` scalar API entirely.

## §111 Batch 214 — the boundary reduced to two Fock faces + the Jacobi step; the negative-pairing faces at per-`Q` thresholds

### 111.1 The reduced boundary (the campaign's current frontier)

```
EvenCocycleResidualsHonestReduced C h2 := CoherentCreatedStepNegPairingFactor E /\ CreatedStepAtZeroNegPairingHonest E
evenCocycleResidualsHonest_iff_reduced : EvenCocycleResidualsHonest C h2 <-> Reduced C h2
theorem8102_of_negPairingFock (hnd hs C h2 S hpd hW hfin) (hneg) (hzero) (hstep) : Theorem8102
EvenCocycleJacobiStepReduced C h2 : Prop   -- content (i) Corollary851ModeStep, (iii) Display866SecondEqK at k >= 3
```

Three of the five conjuncts of the honest package are THEOREMS at FLM's datum, so the residual is exactly: the two
negative-pairing Fock conjuncts + the Jacobi step.  At `hpd` the `0 <= <a,b>` non-vacuity slice is unavailable
(positive definite forms have `(x,-x)`), so the reduced hypothesis' satisfiability rests on the negative-pairing
faces themselves --- which is where the remaining work is.

### 111.2 The negative-pairing faces, corrected and assembled

```
pairBlock_eq_pairBlockIdxMax : pairBlock ... T = pairBlockIdx ... (l-T) (l+T) (fun Q => T - max 0 (Q-l))
HonestNegPairingZeroIdx / PosIdx     -- arbitrary windows and per-Q thresholds
honestNegPairing*_iff_diag           -- their diagonal slices are exactly batch 213-B's faces (both ways)
honestNegPairing*Idx_iff_hw          -- both ways to the per-h-weight inputs
```

The landed single-`T` forms are FALSE (falsification on batch 211-B's instance; coefficient core machine-checked
as `diagCoeff_pair_ne_one_one`), so the corrected (`Idx`) spelling is the honest one --- a genuine weakening.  The
operator-level refutation of the single-`T` forms still needs the linear independence of the distinct blocks.

### 111.3 Corrections of record

1. The "Mathlib's missing fiberwise-sum lemma" premise (batch 213-B / NOTES §110.2) is FALSE --- the lemmas exist
   and are used; what was missing was the explicit fiber/image of the window.
2. `AGENTS.md`'s "tools/build.sh is not usable on this machine" is stale.

### 111.4 Accounting after batch 214

| object | status |
| --- | --- |
| `theorem8102_of_negPairingFock` | landed --- Theorem 8.10.2 from the two negative-pairing Fock faces + the step |
| `HonestNegPairing*IdxHw` | open --- ONE named input per face |
| block linear independence | open (needed only for the refutation of the defective spelling) |
| (i) `Corollary851ModeStep` | open --- the Jacobi step's content |
| (iii) `Display866SecondEqK` at `k >= 3` | open --- the Jacobi step's content |
| six boundary inputs | as before |

## §112 Batch 215 — the general insertion step landed; the negative-pairing faces off-vacuum

### 112.1 The last two mathematical residuals of the campaign

    Fock  : HonestNegPairingZeroIdxHwNonvac E /\ HonestNegPairingPosIdxHwNonvac E   -- off-vacuum, <a,b> < 0
    Jacobi: (i) MM.FLM86C851.Corollary851ModeStep
            (iii) MM.FLM86D866T.Display866SeriesStep k                          -- k -> k+1 finite regrouping

Everything else in EvenCocycleResidualsHonestReduced and EvenCocycleJacobiStepReduced is either a theorem or
reduced to these.

### 112.2 What landed this batch

* the GENERAL OPE/insertion step powerFam_nestedExp (all lengths, only hs and 0 <= <a,b_j>) --- the k >= 3 case
  of (8.4.22)/(8.4.36) --- plus the l-fold (8.4.19) contraction and the series spelling with its both-way
  equivalence to Display866SecondEqK;
* the vacuum slice of the negative-pairing faces as a THEOREM, the both-way reduction to the off-vacuum residuals,
  the block-independence converse normOrdered2_blocks_no_cancel, and the OPERATOR-LEVEL refutation of the
  single-T ancestor spellings at the rank-one datum.

### 112.3 Traps and corrections of record

1. The operator-level refutation did not need the block independence (at the refutation datum C = 0 only the Q = 0
   block survives).
2. 0 <= <abar_i,abar_j> is load-bearing in (8.6.6): a negative pairing breaks the identity (falsification check),
   because pairSetK truncates.
3. Two attempted smaller negative-pairing reductions are non-reductions: basis-state restriction (witnesses not
   uniform in w) and single-shift collapse (separation needs i != j).
4. The coercion mismatch of zero (up-arrow 0 vs 0) in rw patterns, kin to the -n-vs-1 trap of batch 213-A.

### 112.4 Accounting after batch 215

| object | status |
| --- | --- |
| theorem8102_of_negPairingFock | landed |
| HonestNegPairing*IdxHwNonvac | open --- the off-vacuum (8.6.25)/(8.6.27) faces |
| Display866SeriesStep | open --- finite-index bookkeeping; OPE half proved; k = 3 follows from k = 2 |
| (i) Corollary851ModeStep | open |
| six boundary inputs | as before |

## §113 Batch 216 — (iii) closed; (i) refuted as spelled

### 113.1 Item (iii) is CLOSED

    display866SeriesStep : Display866SeriesStep bL form E k        -- every k
    display866SecondEqK_all : Display866SecondEqK at every k >= 1

The finite-index assembly PairIdx (k+1) = PairIdx k + Fin k is landed with all its bookkeeping, and the OPE half
(batch 215's powerFam_nestedExp) closes the step.  Item (iii) of the Jacobi ledger is therefore DONE.

### 113.2 Item (i) is false as spelled

    not_corollary851ModeStep_zero / _one : not (Corollary851ModeStep (ZAligned E) K)   -- K = 0, 1
    iterateOf_eq_zero_of_corollary851_succ : (C u v K -> C w v K) forces the K-th iterate of (w,v) to vanish

The window of Corollary851 is a TRUNCATION: the same-window step is strictly stronger than FLM (8.5.13).  The
corrected currency is Corollary851ModeStepExists (window-free), with the window-threaded (8.5.4) induction and the
composition with the batch-204-A corner landed --- ONE named input remains.

### 113.3 What the remaining (i) residual contains

1. the mode--field commutators [alpha_c(e), ZAligned E v n] at arbitrary v --- the ANNIHILATION modes
   alpha_b(n), n >= 0, are evaluated NOWHERE in the tree;
2. the (8.6.25)/(8.6.27) residue matching (powerFam_nestedExp is already a piece of it).

### 113.4 Accounting after batch 216

| object | status |
| --- | --- |
| (iii) Display866SecondEqK, every k >= 1 | **THEOREM** |
| (i) Corollary851ModeStepExists (window-free) | open --- the two pieces above |
| off-vacuum negative-pairing Fock faces | open |
| six boundary inputs | as before |

## §114 Batch 217 — (i) to three residuals; the negative-pairing faces in block form

### 114.1 Item (i), final shape

    Corollary851ModeStepExists Y <-> Left Y /\ Right Y                     -- Iff.rfl, per-half slices both ways
    corollary851ModeStepExists_iff_allStates : (i) <-> Corollary851AllStatesExists   -- both ways, modulo the lift
    ModeFieldCommInput : [alpha_c(e), Y(v)_n] = sum_j binom(e,j) Y(alpha_c(j)v)_{n+e-j}   -- PROVED at generators
                                                                                          and on their span

Residuals: ModeFieldCommInput off the generator span; ResidueInsertion8625; Corollary851SpanningToAllStates (a
genuine statement --- the landed (8.10.5) truncation is VECTORWISE).

### 114.2 The negative-pairing faces in block form

    HonestNegPairing*IdxHwNonvacBlock E   -- a FINITE set of shift indices, one truncation per index
    honestNegPairing*IdxHwNonvac_iff_block                                -- both ways (window scaffolding removed)
    lhsIdx_anchor_apply_of_neg : at l = 0 the LHS is eps(a,b).coherentFieldNat at iota(a+b), EVERY pairing

### 114.3 Corrections and traps of record

1. The 'annihilation modes are evaluated nowhere' claim is STALE (batch 192-A landed them at the assembled field).
2. E^-_2(r) is NOT 2^r h_r --- a first-draft refutation built on that was an arithmetic error.
3. rw with j := 0 yields the key -(up-arrow 0)-1, which does not match -1.
4. Re-verify an agent's files after its LAST write (two of this batch's files were still being written when the
   dispatcher's first rebuild ran, which failed; the final state builds clean).

### 114.4 Accounting after batch 217

| object | status |
| --- | --- |
| (iii) Display866SecondEqK, every k | THEOREM (batch 216) |
| (i) Corollary851ModeStepExists | open -> ModeFieldCommInput off-generator / ResidueInsertion8625 / SpanningToAllStates |
| Fock faces | open -> the four named block/anchor inputs |
| six boundary inputs | as before |

## §115 Batch 218 — (i) to one annihilation input; the Fock j = 0 face refuted

### 115.1 Item (i)

    modeFieldCommZeroHalf : the ZERO-MODE half of (8.6.9)/(8.6.10) at EVERY state   -- THEOREM
    modeFieldCommInput_iff_annStep : ModeFieldCommInput E <-> ModeFieldCommAnnStep E  -- BOTH WAYS

So (i)'s first residual is the ONE one-oscillator ANNIHILATION clause, with the (7.1.5) scalar defects, the ^+ half
vanishing, the two window laws and sum_scalar_term all landed.

### 115.2 The Fock j = 0 face is false in every spelling

At the rank-one negative datum, off the vacuum slice, the honest left side is `-(eps.eps).alpha_1(-1)iota(3)` while
the block form's right side is `(K.(eps.eps)).alpha_1(-1)iota(3)` with `K >= 0` a window cardinality.  The block,
per-Q and Idx spellings are all refuted (machine-checked, via the landed both-way reductions).  The corrected
residual is the FREE-coefficient form.

The methodological lesson (recorded): the check that missed it tested only the vacuum slice with `m <= -1`, where
the sign accident hid the defect.  Falsification scripts must probe the non-vacuum, non-generator regime too.

### 115.3 Accounting after batch 218

| object | status |
| --- | --- |
| (i) | ModeFieldCommAnnStep + ResidueInsertion8625 + Corollary851SpanningToAllStates |
| Fock j = 0 | the free form (refuted spellings replaced) |
| Fock j >= 1 | open (not refuted) |
| six boundary inputs | as before |

## §116 Batch 219 — the honest (8.6.25) coefficient; ModeFieldCommInput proved; the honest Fock package refuted

### 116.1 The honest coefficient (the answer to the Fock question)

    ZAligned E (YFam E a (-1) iota(b)) m w = sum_Q binomCoeff (C-Q) N . :Y(a,b):(C-Q,Q) w
    C = -m - <a,b> - 1,  N = -<a,b> > 0,   coefficient attached to the a-index P = C-Q

It is a BINOMIAL integer family, not the (8.5.37) fiber sum / cardinality that the campaign had used since batch
211-B.  At N = 1 it is C-Q, whose symmetric average over equal ordered pairs is C/2 = -m/2 --- exactly the
batch-218-B measurement.  Machine-checked for every mode m at w = iota(c).

### 116.2 Refuted spellings (do not consume)

    CreatedStepAtZeroNegPairingHonest / ...Corrected / FockResidualHonest / FockResidualHonestCorrected
    HonestNegPairingZero / HonestNegPairingZeroIdxHw / ...Nonvac / ...Block        (all cardinality-based)
    EvenCocycleResidualsHonest   -- and hence theorem8102_of_residuals_honest / theorem8102_of_negPairingFock
                                    are VACUOUS at the rank-one datum

### 116.3 The corrected residual

    HonestNegPairingZeroCoeff E   -- free support, DETERMINED coefficient binomCoeff (C-Q) N; attained at the datum
    FockResidualHonestCoeff := CoherentCreatedStepPosNonneg /\ HonestNegPairingPos /\ HonestNegPairingZeroCoeff

Slices 1 and 2 survive; slice 2 remains under suspicion (same pairBlock object).

### 116.4 Item (i): the mode-field commutator is PROVED

    modeFieldCommInput (hs) (E) : ModeFieldCommInput E     -- all states, (8.6.9)/(8.6.10)

Item (i) = ResidueInsertion8625 + Corollary851SpanningToAllStates (+ the bridge from the mode-field commutator to
the (8.5.13) mode step, which is the (8.6.4)--(8.6.27) residue matching itself).

### 116.5 Accounting after batch 219

| object | status |
| --- | --- |
| ModeFieldCommInput (all states) | THEOREM |
| HonestNegPairingZeroCoeff | open (the corrected Fock j=0 residual) |
| HonestNegPairingPos (j>=1) | open, under suspicion |
| general-N coefficient / l>=1 | open (needs the (8.6.27) dictionary) |
| ResidueInsertion8625 / Corollary851SpanningToAllStates | open |
| six boundary inputs | as before |

## §117 Batch 220 — slice 2 refuted; the residual is a reindexing identity

### 117.1 Fock: the corrected face and its single gap

    HonestNegPairingPos                              -- REFUTED (rank-one, (j,l,m,w) = (1,0,3,iota(1)))
    FockResidualHonestCoeff                          -- REFUTED (its 2nd conjunct)
    honestBlockCoeffJ a b j m Q = binomCoeff (blockFirstIdx j 0 m <a,b> - Q) (j + (-<a,b>).toNat)
    HonestNegPairingCoeff                            -- the corrected face (attained at the datum)
    FlatBinomialExpansion                            -- the ONE remaining identity (vertex-algebra-free)

### 117.2 Item (i): the matching is unnamed

ResidueInsertion8625 is the FINITENESS half only (proved); the (8.6.25)/(8.6.27) matching remains open and must be
named.  Corollary851SpanningToAllStates reduces to Corollary851SpanningEventualWindows (window synchronization).

### 117.3 Accounting after batch 220

| object | status |
| --- | --- |
| ModeFieldCommInput | THEOREM (batch 219) |
| HonestNegPairingCoeff (both slices) | open <-- FlatBinomialExpansion |
| (8.6.25)/(8.6.27) matching | open, unnamed |
| Corollary851SpanningEventualWindows | open |
| six boundary inputs | as before |

## §118 Batch 221 — the flat identity proved; the generator slice of the Fock residual proved; the matching named and half-proved

### 118.1 The honest identity (corrected window)

    flatBinomialExpansion : sum_{l<=n} C(s,l)((n-l)!)^-1 . flatMul (iterDeriv (n-l) X) Y (l+TOT-n)
                             = sum_{p in Icc 0 TOT} C(p+s,n) . (X p * Y (TOT-p))

The batch-220 `Finset.range (TOT.toNat+1)` spelling is FALSE at `TOT < 0` (machine-checked).

### 118.2 The Fock residual's current shape

    honestNegPairingCoeff_at_generators : the l = 0 slice at w = iotaVL c   -- THEOREM
    HonestNegPairingCoeff                                                  -- open at arbitrary w
    ResidueMatching8625NegSecondSlot                                       -- open (the l >= 1 slot)
    residueMatching8625Nonneg                                              -- THEOREM (half the matching)
    modeStepExists_of_insertion                                            -- the bridge, PROVED

### 118.3 Ledger after batch 221

| object | status |
| --- | --- |
| FlatBinomialExpansion | THEOREM (Icc 0 TOT form) |
| HonestNegPairingCoeff, l = 0 at generators | THEOREM |
| HonestNegPairingCoeff, arbitrary w | open |
| ResidueMatching8625NegSecondSlot | open |
| (8.6.25)/(8.6.27) matching | named; nonneg half THEOREM; bridge to the mode step PROVED |
| ModeFieldCommInsertionStep8625 / Corollary851SpanningEventualWindows | open |

## §119 Batch 222 — the boundary re-hung (non-vacuous); the Fock half named; the lattice factor proved

### 119.1 The corrected package (the new delivery frontier)

    EvenCocycleResidualsCorrected C h2 := ResidueMatching8625 E /\ Corollary851SpanningEventualWindows
        /\ exists Ins, IterateInsertion8625 E Ins /\ ModeFieldCommInsertionStep8625 E Ins
    theorem8102_of_correctedResiduals / theorem8102_of_correctedNormal : Theorem8102
    corollary851AllStatesExists_of_corrected : the package -> all-states (8.5.13)
    evenCocycleResidualsCorrected_iff_normal                                   -- BOTH WAYS

The four proved conjuncts are discharged by landed theorems; theorem8102_of_residuals is formally DEAD (its
package contains the refuted CoherentCreatedStep and is unsatisfiable at hpd).

### 119.2 The Fock side

    honestNegPairingCoeff_on_latticeSpan        -- THEOREM (the lattice factor F{L})
    honestNegPairingCoeff_iff_atStates          -- BOTH WAYS; the Fock half is the residual
    honestNegPairingCoeff_iff_finiteDimUniform  -- the uniformity is FORCED, not assumed

### 119.3 Accounting after batch 222

| object | status |
| --- | --- |
| boundary theorem (corrected package) | LANDED, non-vacuous |
| HonestNegPairingCoeff on the lattice factor | THEOREM |
| HonestNegPairingCoeffAtStates (Fock half) | open, the exact residual |
| ResidueMatching8625NegSecondSlot | open |
| Corollary851SpanningEventualWindows | open |
| ModeFieldCommInsertionStep8625 | open |
| EvenCocycleJacobiStepCorrected (assembly) | open |
| six boundary inputs | as before |

## §120 Batch 223 — the assembly built; the ledger halves again

### 120.1 The assembly's residual

    stepCorrected_of_spanningEventual : Corollary851SpanningEventualWindows (ZAligned E)
        -> NegativeResidual E -> EvenCocycleJacobiStepCorrected C h2
    crossBracket_eq_crossIterate_of_nonneg_of_grounded : every l >= 0 face is a translate of the grounded l = 0 face
    faceWindows_zAligned : (8.10.8)'s finiteness = the landed (8.10.5)

So `l > 0` is NOT residual, and `NegativeResidual E` (the `l < 0` faces at non-generator pairs) is the new named
obligation --- batch 202's faces were only the generator `0 <= <a,b>` chamber (a correction of record).

### 120.2 The insertion conjunct is redundant

    evenCocycleResidualsCorrected C h2 <-> ResidueMatching8625 E /\ Corollary851SpanningEventualWindows (ZAligned E)
    ModeFieldCommInsertionStep8625 E (honestIns8625 E) <-> Corollary851SpanningEventualWindows (ZAligned E)

### 120.3 Accounting after batch 223

| object | status |
| --- | --- |
| assembly step | reduced to the windows conjunct + NegativeResidual |
| l >= 0 faces, every state/pairing | THEOREM (the translate engine) |
| l = -c faces, generator 0 <= <a,b> chamber | THEOREM (batch 202) |
| NegativeResidual (l < 0 at non-generator pairs) | open --- the normal-ordering engine there |
| Corollary851SpanningEventualWindows | open |
| HonestNegPairingCoeffAtStates / ResidueMatching8625NegSecondSlot | open (Fock) |
| six boundary inputs | as before |


**HANDOVER**: the state of the campaign as of batch 223 (delivered reductions, the remaining book formulas, the refuted-spelling blacklist, the module map and the verification discipline) is summarised in `HANDOVER.md` at the repository root.

## §121 Batch 224 — residual #1 sharpened: the window synchronization is the grounded one-mode step

### 121.1 The sharpened residual and the theorem corner

    GroundedPairWindow Y u v : exists K, Corollary851 Y u v K and iterateOf Y u v i = 0 for all i >= K
    GroundedModeStepAtStates Y : grounded at (D, D') -> grounded at (alpha_c(-r) u_D, u_D') and (u_D, alpha_c(-r) u_D')
    groundedLatticeCorner_evenCocycle : GroundedLatticeCorner (ZAligned E)        -- THEOREM
    spanningEventual_zAligned_iff_groundedStep :
        Corollary851SpanningEventualWindows (ZAligned E) <-> GroundedModeStepAtStates (ZAligned E)

The corner is unconditional at the even cocycle (batch 204-A's lattice identity with the PROVED negative
pairing MM.FLM86NTP.residueNegativeAligned_evenCocycle, plus (8.5.11) through the iterate); the equivalence
is both ways: forward because a one-mode successor of a (8.5.4) pair is again a (8.5.4) pair, backward by
the (8.5.4) list induction in the grounded currency.  So residual #1 carries exactly ONE piece of content:
the grounded one-mode step, i.e. the (8.6.23)/(8.6.24) cross contraction at the descendant pair.

### 121.2 The delivery re-hung

    groundedZeroFace_of_groundedStep / stepCorrected_of_groundedStep / theorem8102_of_matching_and_groundedStep

### 121.3 Accounting after batch 224

| object | status |
| --- | --- |
| residual #1 | = GroundedModeStepAtStates (ZAligned E) (both ways; the window-free form is strictly weaker) |
| (8.5.4) lattice corner | THEOREM (unconditional at the even cocycle) |
| NegativeResidual | open --- the normal-ordering engine at non-generator pairs |
| HonestNegPairingCoeffAtStates / ResidueMatching8625NegSecondSlot | open (Fock) |
| six boundary inputs | as before |

## §122 Batch 225 — the window synchronization is absorbed: residual #1 IS the display

### 122.1 The bridge between the grounded and the threshold currencies

    groundedPairWindow_iff_aboveThreshold : (forall i >= N, iterateOf Y u v i = 0)
        -> (GroundedPairWindow Y u v <-> forall K >= N, Corollary851 Y u v K)
    pairWindowAboveThreshold_mono

### 122.2 The display gives the descendant windows outright

    groundedPairWindow_modeVL_left/right_of_commutatorInsertion :
        CommutatorInsertion8625 E (honestIns8625 E) -> the (8.10.5) threshold of the descendant pair is a grounded window
    groundedModeStep_of_commutatorInsertion :
        CommutatorInsertion8625 E (honestIns8625 E) -> GroundedModeStepAtStates (ZAligned E)     -- NO hypotheses
    DescendantsGroundedWindows / descendantsGrounded_of_commutatorInsertion

The threshold window is never raised, so the 223-A WindowSyncAtStates obstruction is bypassed rather
than solved: the display's own vanishing hypothesis is the honest window.

### 122.3 The accounting, both ways

    groundedModeStep_iff_commutatorInsertion_of_evenCocycle
    spanningEventual_of_commutatorInsertion_of_evenCocycle

### 122.4 Accounting after batch 225

| object | status |
| --- | --- |
| residual #1 | = CommutatorInsertion8625 E (honestIns8625 E) (both ways at an even cocycle) |
| window synchronization (223-A WindowSyncAtStates) | ABSORBED by (8.10.5) + the display |
| (8.5.4) lattice corner | THEOREM (unconditional at the even cocycle) |
| NegativeResidual | open — the normal-ordering engine at non-generator pairs |
| HonestNegPairingCoeffAtStates / ResidueMatching8625NegSecondSlot | open (Fock) |
| six boundary inputs | as before |

## §123 Batch 226 — residual #1 in the per-pair display currency

### 123.1 The display at a pair, and the per-pair bridge

    DisplayAtPair Y u v : forall K, (forall i >= K, iterateOf Y u v i = 0) -> Corollary851 Y u v K
    DisplayAllStates Y := forall u v, DisplayAtPair Y u v
    displayAllStates_iff_commutatorInsertion8625 :
        DisplayAllStates (ZAligned E) <-> CommutatorInsertion8625 E (honestIns8625 E)
    displayAtPair_iff_groundedPairWindow_of_threshold :
        (threshold N) -> (DisplayAtPair Y u v <-> GroundedPairWindow Y u v)
    displayAtPair_lattice_evenCocycle : the display at the lattice pairs is a THEOREM at the even cocycle

The pair alone selects the display: the inert (8.6.4) labels `c, r` are dropped, and the per-pair bridge
identifies the display with the grounded window (batch 224's currency) on the nose.

### 123.2 The step, and the residual

    DisplayModeStep Y     -- the display is inherited by the two one-creation-mode descendants
    displayModeStep_iff_groundedModeStep_of_evenCocycle
    honestDisplay_iff_displayModeStep_of_evenCocycle :
        DisplayAllStates (ZAligned E) <-> DisplayModeStep (ZAligned E)
    displayAllStates_iff_spanningEventual_of_evenCocycle

### 123.3 Accounting after batch 226

| object | status |
| --- | --- |
| residual #1 | = DisplayModeStep (ZAligned E) (the (8.6.23)/(8.6.24) cross contraction; label-free display currency) |
| window synchronization | ABSORBED (batch 225) |
| (8.5.4) lattice corner / display at lattice pairs | THEOREM (unconditional at the even cocycle) |
| NegativeResidual | open |
| HonestNegPairingCoeffAtStates / ResidueMatching8625NegSecondSlot | open (Fock) |
| six boundary inputs | as before |

## §124 Batch 227 — residual #1 IS the source's (8.6.3)

### 124.1 The carriers coincide at one window

    corollary851_iff_sourceUnshifted / corollary851_iff_sourceStatement
    corollary851_iff_theorem861SourceStatement

The spine's factor-free carrier IS the landed corollary851Residue (rfl), and the source's operator-valued
grading factor is inert (its sector sum is the identity).

### 124.2 Residual #1 at a labelled pair, and at all pairs

    displayAtPair_iff_forall_theorem861SourceStatement
    displayAllStates_iff_sourceStatements_of_evenCocycle :
        DisplayAllStates (ZAligned E) <-> (the source's (8.6.3) at every labelled (8.6.2) pair, at every
        window above the vanishing bound)

### 124.3 Accounting after batch 227

| object | status |
| --- | --- |
| residual #1 | = FLM (8.6.3) in the source's carrier (both ways; `MMFLM86SourceBridge`) |
| the spine's named next step (8.6.3) | THE SAME statement (machine-checked) |
| window synchronization / labels | absorbed (batches 225/226) |
| (8.5.4) lattice corner / display at lattice pairs | THEOREM (unconditional at the even cocycle) |
| NegativeResidual | open |
| HonestNegPairingCoeffAtStates / ResidueMatching8625NegSecondSlot | open (Fock) |
| six boundary inputs | as before |

## §125 Batch 228 — the Fock half is ONE one-mode step; its lattice base is a theorem

### 125.1 The face at a family, and the step

    FaceAtFamily E a b j m n D              -- the (8.6.25) face at one finite family of (8.5.4) states
    honestNegPairingCoeffAtStates_iff_familyFaces
    FockStep E                              -- the face at a family is inherited when a LIST of creation
                                            -- modes is appended to each state (one-mode: fockStep_imp_oneMode)
    faceAtFamily_lattice                    -- THEOREM (the union of the canonical windows)
    honestNegPairingCoeffAtStates_iff_fockStep : HonestNegPairingCoeffAtStates E <-> FockStep E

The step's proof is the mode action on the normal-ordered blocks `blockOfShift` (the recorded gap), the
left side being the PROVED (8.6.9)/(8.6.10) mode--field commutator at the inserted state and the
coefficients being compared by the landed flat binomial expansion.

### 125.2 Accounting after batch 228

| object | status |
| --- | --- |
| residual #1 | = the source's (8.6.3) (batches 225--227) |
| Fock half (residual #3) | = FockStep E (one-mode step; lattice base a THEOREM) |
| l >= 1 second slot | open (ResidueMatching8625NegSecondSlot) |
| NegativeResidual | open |
| six boundary inputs | as before |

## §126 Batch 229 --- the two Chapter-8 theorems delivered with one registered `sorry` each

Owner's directive ("未证明部分使用sorry") turned the residuals #1/#2 into named holes of named
theorems rather than descriptions:

* `Definitions/MMFLM810Theorem861Stated.lean` (`MM.FLM810T861S`): `displayAllStates_evenCocycle` is
  the registered `sorry` (residual #1); `theorem861_source`/`theorem861_all_pairs`/
  `theorem861_residual` are FLM Theorem 8.6.1 at the boundary field, composed from landed theorems.
* `Definitions/MMFLM88Theorem889Stated.lean` (`MM.FLM88T889S`): `theorem889_jacobi` is the
  registered `sorry` (residual #2 at its `l < 0` part); `theorem889_delta`,
  `remark8811_res_z0`/`_res_z1`, `remark8812_res_z2` are Remark 8.8.11's faces; and
  `theorem861_of_theorem889` (the engine link) is PROVED with the three standard axioms.
* Guards: the sorry-consuming declarations print `[propext, sorryAx, Classical.choice, Quot.sound]`;
  `README.md`'s `sorry` map gained a two-row table for them; `Probe/forbidden_tokens.py` reads
  `sorry=1` for exactly these two files.  Acceptance: `FinalCheck.lean` GREEN with the two imports.

## §127 Policy: per-batch `sorry` is the sanctioned marker of unfinished work (2026-09-22)

Owner's decision: **each batch may leave `sorry`s for the parts that are not yet proved** --- it is
the sanctioned way to deliver a statement whose proof is still open --- provided every `sorry` is
registered in `README.md`'s `sorry` map **and** tied to the open task/residual it belongs to in
`ROADMAP.md` / `HANDOVER.md`.  The declaration that consumes it carries a `#guard_msgs` footprint
printing `[propext, sorryAx, Classical.choice, Quot.sound]`, so the hole is visible in the build.
**"No `sorry` anywhere" is the completion criterion of the whole project** (the formalization of
monstrous moonshine), not a per-batch invariant.  `axiom`, `admit` and `native_decide` remain
forbidden.  The policy text now lives in `AGENTS.md` (ground rules + "Before any change is
considered done"), `README.md` (top + `sorry` map), `ROADMAP.md` ("Working agreement") and
`HANDOVER.md` (§0 and §5).  The first holes under it are batch 229-A's two statement-only
deliveries (`MM.FLM810T861S.displayAllStates_evenCocycle`, `MM.FLM88T889S.theorem889_jacobi`), tied
to residual #1 and residual #2.

---

## §128 Batch 230 --- documentation restructured: roles fixed (README = index, ROADMAP = plan, NOTES = log, HANDOVER = handover)

Owner's directive: `AGENTS.md` stays as the conventions file; `README.md` is the project description
and master index; `ROADMAP.md` is the plan ("extend when a task is added, compress when it is
completed, no redundant information"); this file is the work log (per-batch results and lessons);
`HANDOVER.md` is the handover manual for the long-running FLM campaign.

Landed as a pure documentation move --- nothing deleted:

* the ROADMAP's **batch pile (batches 151--229, ~470 KB) moved verbatim into Appendix A** below.
  `ROADMAP.md` now has a fixed two-part shape: **front matter = Index + "1. Project status"**
  (the four assumed inputs' status table and the FLM campaign's five-residual frontier) **+ "2. How
  to read this file" + "3. Working agreement"**; **"Part II -- the branches"** then carries the
  streams in order S, T, M, A, the long campaign (moved after the streams) and the FLM campaign.
  Completed work is one row of a branch table, never a batch narrative; the FLM branch's open
  residuals are the task entries `[~]` R1--R4 / `[ ]` R5.  Plan here, record in this file;
* two stale pointers in `ROADMAP.md` ("Current ordering ... just before the verification protocol")
  were corrected --- those sections no longer existed --- and it now points at the campaign-status
  section and `HANDOVER.md` §6;
* `README.md` gained a **Documentation map** (the master index: which file has which role);
* the three auxiliary docs at the root (`FLM-CH8-MAP.md`, `FLT-ASSETS.md`, `FLT-L1-VERIFY.md`) were
  moved into `refs/` (the auxiliary-evidence directory, alongside `refs/README.md`), and the living
  pointers were updated (`README.md`'s documentation map and `ROADMAP.md`'s M1 block); the verbatim
  records in Appendix A keep the paths as they were written;
* this file gained the role statement at the top and the Appendix A pointer; `HANDOVER.md` §7 and
  its head were rewritten to the same roles;
* the per-batch `sorry` policy (owner's standing instruction, 2026-09-22) was written into
  `AGENTS.md` (ground rules + the "before anything is done" list), `README.md` (top + `sorry` map),
  `ROADMAP.md` (working agreement), `HANDOVER.md` (§0 and §5) and `FinalCheck.lean`'s header;
  `Probe/forbidden_tokens.py` documents it too.  See §127.

---

## Appendix A --- dispatcher batch records, batches 151--229

Moved **verbatim** out of `ROADMAP.md` in batch 230 (the plan keeps only the compressed status
tables).  Each entry records, for its batch: the files landed, the exact statements, the brief errors
and the accounting supersessions.  The topic-wise analysis and lessons are in `§§64--127` above.

### Batch 151, final acceptance

`Probe/agentbuild.sh Definitions/MMFLM85HrelClose.lean` -> `--- OK ---`, exit 0; the file passes
`Probe/forbidden_tokens.py` at `sorry=0 admit=0 axiom=0 native_decide=0`; all 9 in-file guards
matched verbatim, every footprint `[propext, Classical.choice, Quot.sound]`, none touching `sorryAx`.
`FinalCheck.lean` acceptance build: `Probe/b151_final_accept.log`.

**The dispatcher repaired three defects in the landed file before acceptance**, and they are worth
recording because each is a distinct failure mode of a *dispatched* agent's product:

1. `normalOrderedFam_label_add_at` failed to synthesize `Module.Free ℤ L` (the ambient instance is
   not derivable from the explicit `bL : Module.Basis κ ℤ L` parameter) -- **and Lean reported the
   consequence, not the cause**: the declaration's axiom footprint came out as
   `[propext, sorryAx, Classical.choice, Quot.sound]`, i.e. an *unproved* declaration with no `sorry`
   anywhere in the file text.  `Probe/forbidden_tokens.py` passes such a file; only the
   `#guard_msgs` axiom guards catch it.  **This is the second independent reason a dispatched agent's
   file must be compiled by the dispatcher, not read.**  Fix: `haveI : Module.Free ℤ L :=
   Module.Free.of_basis bL`.
2. Two imports were missing (`MMFLM85Action` for `MM.FLM85Ac.factorFamily`, and
   `MMFLM85LabelLinearity` for `MMFLM85LL.factorFamily_add`): the file had been written against the
   *author's* import closure, which the interrupted run never rebuilt.
3. **A name-resolution defect that survived both fixes, and whose cause is not yet known**:
   `MM.FLM85Ac.factorFamily` resolves as a *simple* identifier under `open FLM85Ac` in a minimal probe
   with the same imports (measured: `Probe/b151_probe4.lean`, exit 0) but **not** inside this file's
   body, where Lean reports `unknown identifier factorFamily` and suggests `autoImplicit`.  It is
   fixed by fully qualifying the two occurrences; the *cause* is recorded as unknown rather than
   explained away, because a silently-unresolvable `open`ed name is the kind of thing that will recur.

**What batch 151 leaves open, precisely.**  `Hrel` is closed, so `generalVertexOperator_existsUnique`
now produces the general vertex operator with **no hypothesis** and `MM.FLM810F.theorem8102_of_hrel_modelBase`
has lost its last free input on the relation side.  What is *not* done:

1. the four `ActionRemainingFields` conjuncts (Jacobi, `L(0)` grading, translation, Virasoro) for the
   corrected action prescription -- the honest assembly boundary of `MM.FLM810CA.theorem8102_of_remainingFields`;
2. **Theorem 8.6.1's display**, which by `MM.FLM86TDA.residue863_iff_corollary851` is equivalent to
   `MM.FLM85Co.Corollary851` at the states of (8.6.2), and by
   `MM.FLM86TDA.display863_of_jacobiIdentity` follows from a Jacobi identity plus the finiteness
   `hvan`.  The two remaining questions there are (a) the **concrete** field on `V_L`
   (`annihExp · actVL · creationExpVL`, all landed) and its two-point product identity, and (b) the
   **uniform-in-`(m,n)`** vanishing `hvan`, which the VOA axioms supply only per component.  Both are
   stated with their exact types in `Probe/briefs/B151-W3-vacuum851.md` and
   `Probe/briefs/B151-W4-assembly.md`.

---

### Batch 152, final acceptance — **Theorem 8.6.1's display is a theorem, and its last input is one named `Prop`**

`Definitions/MMFLM86UniformVanishing.lean` (460 lines) and
`Definitions/MMFLM86DisplayVOASplit.lean` (319 lines) both build `--- OK ---`; both pass
`python3 Scratch2/forbidden_tokens.py` at `sorry=0 admit=0 axiom=0 native_decide=0`; the 14 + 11 in-file
`#guard_msgs` axiom guards all match verbatim with footprints `[propext, Classical.choice, Quot.sound]`
or narrower and none touching `sorryAx`; `FinalCheck.lean` acceptance build is `--- OK: FinalCheck.lean
---` (`/tmp/finalcheck.log`), with the four new guards appended.

**The display (8.6.3) is landed as a theorem.**  `MM.FLM86DVS.display863_fixed_window` proves, at a
**fixed** window `K` (no `∃K`, no extra hypothesis beyond the two named inputs),

    `MM.FLM86TDA.Display863 form E Y u' v' a b K`,

from exactly two inputs: the faithful (8.10.8) `MM.FLM810JF.FaithfulJacobi Y u v` — which is *verbatim*
the `jacobi` field of the landed `MM.FLM810.VertexOperatorAlgebra` — and the finiteness parenthetical of
(8.5.26) at `K`.  `display863_of_voa_self` is the transport-free form for a structure's own field;
`display863_of_remainingFields` is the Section 8.10 assembly form, and it consumes **only the
`JacobiAxiom` conjunct** of the four `ActionRemainingFields`, none of `hpd`, `heven`, `hrank`, `hW`,
`hfin`.

**The finiteness parenthetical is one line from the axiom (8.10.5)**, and that corrects the campaign's
own framing of it.  `MM.FLM86UV.exists_uniform_bracketTerm_vanishing` proves the uniform-in-`(m,n)`
vanishing of `MM.FLM85Br.bracketTerm` from `A.componentTruncation` alone — no grading, no homogeneity, no
weight: *the source's parenthetical is (8.10.5) at the pair `(a,b)`.*  Batches 144--151 carried it as
"the VOA axioms supply truncation only per component, not uniformly in `(m,n)`" (see batch 151's §2
above and NOTES §64.4); that is **wrong** for the pair-indexed form the display needs — it is right only
for the *operator*-valued `MM.FLM88.JacobiIdentity` window, which the display does not consume.
`exists_iterate_uniform_vanishing` gives the `ℤ`-indexed form `∃K, ∀ i ≥ K, iterateOf Y a b i = 0`.

**So Theorem 8.6.1 now rests on exactly one input, and it is named.**  Reading the display's right-hand
side as the `Res_{z₀}` face (`display863_iff_residue863` + `residue863_iff_corollary851`) the theorem is
`MM.FLM85Co.Corollary851 (toVertexOp Y) u v K` at the states of (8.6.2), from `FaithfulJacobi`.  What is
*not* claimed: the paper's proof of (8.6.3) via (8.6.11)--(8.6.27) is not reproduced; the display is
reached through the Jacobi axiom instead.  The concrete field on `V_L` is not needed, and neither is the
two-point product identity that batches 146--151 were aimed at.

**The concrete residual.**  `FaithfulJacobi Y u v` for the corrected action prescription is
`FaithfulJacobi`-on-generators, which `MM.FLM810RR.jacobiAxiom_of_stateTargets` already reduces to
`MM.FLM810RR.JacobiOnStateTargets bL form Y` (`Definitions/MMFLM810RemainingReduction.lean:166`).  So
Theorem 8.6.1 is now conditional on **that one landed `Prop`** and nothing else.

**A landed definition is refutable, and this is the batch's second finding.**
`MM.FLM85.WeightGrading.IsHomogeneousOp S T p` requires `S.wt (T a) = S.wt a + p` also at `a = 0`, where
it reads `S.wt 0 = S.wt 0 + p`; hence `MM.FLM86UV.isHomogeneousOp_eq_zero` gives `p = 0` for **every**
operator `T`, and `not_isHomogeneousOp` refutes the predicate at every `p ≠ 0`.  Every landed lemma
taking an `IsHomogeneousOp` hypothesis is therefore **vacuous** — including
`MM.FLM85Br.bracketTerm_vanishing_of_grading` and the batch-152 `*_on` statements that generalise it,
and the `hOp` hypotheses of `MM.FLM85Br.component_eq_zero_of_toNat_le`,
`weightOfComponents_of_isHomogeneousOp` and the whole Corollary 8.5.6 chain.  The satisfiable repair is landed here as `MM.FLM86UV.IsHomogeneousOpOn`
(the same statement with the `T a = 0` case dropped, where `WeightGrading.wt_eq_of_mem` has no content),
with `IsHomogeneousOp.isHomogeneousOpOn` recording that it generalises the landed predicate, and the
uniform-vanishing statements reproved under it.  The batch's brief had asked for a theorem *from*
`IsHomogeneousOp`; the agent measured that the hypothesis is unsatisfiable instead of proving a vacuous
statement, and that measurement is worth more than the theorem would have been.

**Cost note (measured).**  Two dispatched agents were launched at 22:58 and both were killed at the
owner's instruction at ~23:21, having written their files but not their reports; the dispatcher built,
guarded, extended (`display863_fixed_window`), wired into `FinalCheck.lean` and accepted them.  Of the
~23 minutes, roughly 12 were spent queueing on the build lock held by a stale `lean --server`.

---

### Batch 153, reconnaissance — **the residual is not a proof gap, it is a wrong field**, and Theorem 8.6.1 is therefore not one lemma from done

Owner's directive for this session: *finish Theorem 8.6.1*, dispatching work to fresh subagents in
batches.  The batch opened with two reconnaissance agents (dispatch rule of `AGENTS.md`: fresh agents,
self-contained briefs, no forking) whose reports are `Probe/B153_R1_PROVED.md` and
`Probe/B153_R2_VOA.md`.  Their findings, all re-verified here by reading the cited lines:

**(1) The display is not the gap; the field is.**  By `MM.FLM86TDA.display863_iff_corollary851` and
`MM.FLM810RR.jacobiAxiom_of_stateTargets`, Theorem 8.6.1's display is closed once
`JacobiOnStateTargets` holds for the concrete field.  **It cannot hold**, because two of the four
sibling fields of the same boundary are *refuted* at exactly the field the target quantifies over:

* `MM.FLM810ModesL0.lean:370` `l0OnWeightGenerators_forces_zero` and `:398`
  `not_l0OnWeightGenerators_modelBase` — `L0OnWeightGenerators` is **false** for the field of
  `MM.FLM85AP.actionPrescriptionTA` at the (8.4.17) model base as soon as a lattice vector of nonzero
  even norm exists (for the Leech lattice: it does);
* `MMFLM810TranslationStateObstruction.lean:271` `not_translationOnStateData_modelBase` —
  `TranslationOnStateData` is false there as well.

Both are consequences of **one definitional defect**: the (8.4.7) Heisenberg field
`MM.FLM85Cp.heisFam a = fun m => modeVL a (−m−1)` has `modeVL a 0 = 0` at the power `m = −1`, i.e. it
omits the (7.1.33)/(7.1.41) zero mode `a(0)`, which in FLM acts on the lattice factor by the pairing
`a(0)·ι(b) = (a,b)ι(b)`.  **So the campaign's "last named `Prop`" is not an open problem — it is an
unsatisfiable one, at the only field the repository has for it.**  This supersedes the closing sentence
of batch 152's ledger entry above, which read the residual as "one landed `Prop` and nothing else": the
`Prop` is landed, and *false*.

**(2) The repair is landed and unconditional, but it is not wired into the assembly.**  Batch 80's
files restore the zero mode and are complete on their own terms:

* `MM.FLM85ZM.heisFamZero` (`Definitions/MMFLM85ZeroModeField.lean:116`) — (8.4.7) with `a(0)` at the
  power `−1`;
* `MM.FLM85ZP.actionPrescriptionTAZero` + **unconditional** existence--uniqueness
  `existsUnique_actionPrescriptionTAZero` (`MMFLM85ZeroPrescriptionSpanning.lean`), i.e. (8.5.1) with
  (8.5.6) for the corrected prescription — the relation-spanning input is *the same proposition* as
  the landed one, because the relation module never mentions the Heisenberg field;
* `MM.FLM810ZMU.Yzero` with `Yzero_spec`/`Yzero_unique` (`MMFLM810ZeroModeL0Uncond.lean:82`, `:87`,
  `:107`) — the unique concrete `Y`, henceforth **the** field of the construction;
* `MM.FLM810ZMW.L0OnWeightGenerators_zero` (`MMFLM810L0General.lean:301`) — (8.10.12) at the vacuum
  generators for that field, where the landed prescription gave `0`.

But `MM.FLM810CA.theorem8102_of_remainingFields` still takes `actionPrescriptionTA`, and **there is no
assembly line for `actionPrescriptionTAZero`** — no corrected `ActionRemainingFields`, no corrected
`theorem8102_of_generatorFields`.  That is the structural blocker, and it is mechanical.

**(3) Every `l ≠ 0` face of the Jacobi identity is untouched**, and the objects FLM uses for it —
the localisation `F[z₁,z₂]_S` and the maps `ι_{i₁⋯iₙ}` of (8.10.36)--(8.10.42) — do not exist in the
repository (`Definitions/MMFLM88Jacobi.lean:189` records this).  The `l = 0` face, which is what the
display is, *is* equivalent to the named open `MM.FLM85Co.Corollary851`
(`Definitions/MMFLM88Jacobi.lean:588` `corollary851_iff_jacobiIdentity_zero_face`).  The two window
conjuncts are **free**: `crossWindowVectorwise_of_truncation`/`iterateWindowVectorwise_of_truncation`
need only `ComponentTruncation`, which *is* proved for the concrete field
(`MMFLM85ActionTruncationTA.componentTruncation_of_actionPrescriptionTA`).

**(4) There is exactly one `VertexOperatorAlgebra` literal in the repository**
(`Definitions/MMFLM810Axioms.lean:923`), its state space is the real `V_L`, and **all five of its
fields are hypotheses**; `grep -rn "instance.*VertexOperatorAlgebra"` is empty.  The vendored
`VertexAlg` Borcherds identity is a *class field* with no construction upstream and is not connected to
`MM`'s `VL` (different carrier, no shared import); its one concrete field is refuted.  The
`IsHomogeneousOp` weight route is dead (batch 152).  **Reconnaissance verdict: (8.10.8) is not
derivable from anything landed, and the honest cost of the remaining chapter is ~3.5--6.5 kLoC.**

**(5) What this means for the plan.**  Theorem 8.6.1 is *not* one lemma from done.  Two routes remain,
and both are now being worked as batch 153's proof tasks on disjoint files:

* **P1 (`Definitions/MMFLM86ExpProduct.lean`)** — the concrete exponential two-point product on `V_L`
  (`Y(ι(a),z₁)Y(ι(b),z₂) = ε(a,b)·:Y(ι(a),z₁)Y(ι(b),z₂):·(z₁−z₂)^{(a,b)}` and its commutator corollary).
  This is FLM's *own* engine for Theorem 8.6.1 — the display is the `Res_{z₀}` face of the Jacobi
  identity, and for the lattice states that face is this product identity.  It is finite, explicit, and
  independent of the §8.10 rationality machinery.
* **P2 (`Definitions/MMFLM810CorrectedZeroAssembly.lean`)** — the corrected assembly line for `Yzero`:
  the four-field package and the `theorem8102` boundary restated at the zero-mode prescription, so that
  the field the refutations are about is no longer the field the assembly consumes.

This batch's reconnaissance is itself the deliverable of the round: it changes the target from "prove
one `Prop`" to "repair the field, then prove the product identity", and it is recorded because the
repository's own ledger had the residual wrong.

**(6) The product-identity ledger, measured (`Probe/B153_M2_PRODUCTLEDGER.md`).**  A second measurement
agent tabulated every `def … : Prop` matching `Product|Diag|Contraction|Expand|Ordinary|Statement|
Obligation|Window` across `Definitions/MMFLM85*.lean` and `MMFLM86*.lean` (44 objects), every
conclusion-position occurrence and every consumer, plus all 58 refutation-shaped declarations.  The
three findings that change what may be dispatched:

* **(8.5.12) is open through *three* named `Prop`s, not one.**  `ProductExpansion8512`
  (`MMFLM8512Done.lean:475`) has a single conditional producer (`MMFLM84Diag.lean:507`) whose
  hypotheses are `Iterate8436` (`MMFLM84Diag.lean:440`, whose only conclusion-position occurrence is
  the `Iff.rfl` re-spelling `iterate8436_iff`), `StateVanishesBelow` (`MMFLM85Corollary.lean:510`,
  whose only producer needs `StateFamilyStatement`, `MMFLMLatticeState.lean:567`, **which has no
  producer at all**) and `ModeFamHeisShift` (`MMFLM8512Heis.lean:221`, `Iff.rfl` only).  The same
  agent measured that the residual for the Heisenberg-shaped reading is `Iterate8436` at `k ≥ 1` plus
  `StateFamilyStatement`, and that `ModeFamHeisShift` is a *separate mis-shaping defect*
  (`MMFLM8512Heis.lean:192`) rather than a third independent input.
* **The grading chain is vacuous, and this was verified here line-by-line.**  `MM.FLM88`'s
  `corollary854_of_jacobiIdentity_grading` (`:553`) and `MM.FLM85Co`'s
  `corollary855_of_corollary851_grading` (`:744`) both take
  `hOp : ∀ n, S.IsHomogeneousOp (component Y n a) (S.wt a - n - 1)`, i.e. the predicate batch 152
  refuted (`MM.FLM86UV.not_isHomogeneousOp`).  So the two "grading" routes to Corollaries 8.5.4/8.5.5
  are **vacuous**, and any plan reaching `Corollary851` through them must be re-stated with
  `MM.FLM86UV.IsHomogeneousOpOn` — whose non-vacuous uniform statement
  `bracketTerm_vanishing_of_gradingOn_uniform` (`MMFLM86UniformVanishing.lean:271`) is already landed,
  with the blanket `_uniform` form (`:287`) documented as the vacuous one it generalises.
* **For the concrete `V_L` field the unconditional assets are exactly three**: the field itself
  (`MMFLM85PermSpanning.existsUnique_actionPrescriptionTA:272`, `MMFLM85HrelClose.generalVertexOperator_existsUnique:251`,
  `MMFLM810ZeroModeL0Uncond.Yzero_spec:87`), `ComponentTruncation`
  (`MMFLM85ActionTruncationTA.componentTruncation_of_actionPrescriptionTA:63`) — which is what makes
  both Jacobi window conjuncts free — and `L0OnWeightGenerators_zero` (`MMFLM810L0General.lean:301`),
  the latter for `Yzero` only.  **No product statement for the concrete field is unconditional**, and
  the abstract convolution's additivity *fails* without local finiteness
  (`MMFLM85ConvAdditive.not_conv_add_right_unconditional:315`).  This is the honest starting line for
  the exponential product identity (task P1).

**(7) The concrete exponential API, measured (`Probe/B153_M1_EXPAPI.md`).**  A third measurement agent
mapped the exponential layer and, decisively, **reduced the whole of Theorem 8.6.1's empty-oscillator
case to one already-named `Prop`**:

* The concrete exponentials are `MM.FLMLatSt.creationExpVL`
  (`Definitions/MMFLMLatticeState.lean:199`, the creation half `E^-`) and `MM.FLM84CM.annihExp`
  (`Definitions/MMFLM84Completion.lean:682`, the annihilation half `E^+`).  All five contraction lemmas
  that move a mode or the zero mode past a creation exponential are landed
  (`MM.FLM84CT.modeVL_comm_creationExpVL:375`, `:436`, `MMFLM84TwoVarProduct.lean:600`,
  `MM.FLM84CT.hZeroVL_comm_creationExpVL:639`, `hZeroVL_contraction:659`, `MM.FLMLatSt.creationExpVL_mul_actVL:333`,
  `annihExp_mul_creationExpVL_eq_sum_of_subset` `MMFLM84RatioWindowClose.lean:813`).  **No
  normal-ordered product formula for two creation exponentials exists**, the only product statement
  being the commutation `creationExpVL_comm` (`MMFLM84TwoVarProduct.lean:624`).
* **The object mismatch that has been costing the campaign is now named**: `actionPrescriptionTA` folds
  oscillator factors built from `heisFam a m = modeVL a (−m−1)` (`MMFLM85Compat.lean:213`), while the
  (8.4.36) display and its product expansion use `modeFam Y a m = Y (iotaVL a) (−m−1)`
  (`MMFLM8512Done.lean:403`).  The two are identified **only** by the `Prop`
  `MM.FLM8512H.ModeFamHeisShift` (`MMFLM8512Heis.lean:221`) — which has no producer anywhere.  This is
  the second, independent mis-shaping defect of the same construction, alongside the missing zero mode.
* **The reduction.**  `theorem842Concrete_iff_corollary851` (`MMFLM8512Done.lean:805`) with
  `theorem842Concrete_iff_apply` (`:793`) identifies `Theorem842Concrete Y E a b (form a b) K` with
  `Corollary851 Y (iotaVL a) (iotaVL b) K`, and `stateOf_one_eq_iotaVL` is `rfl`, so this *is*
  `Display863Unshifted form E Y 1 1 a b K` (`MMFLM86TheoremDisplay.lean:163`) — the empty-oscillator
  case of (8.6.3)'s second sentence.  **So `Theorem842Concrete` at the lattice generators is exactly
  the `(1,1)` case of Theorem 8.6.1**, and it is one named `Prop`.
* **What does *not* lift it**: `theorem861Source_of_coeffFamily_and_spanning`
  (`MMFLM86SpineGap.lean:632`) is proved, but its spanning hypotheses are families `A, B` with their
  own coefficient conditions; the pure lattice generators do not span `V_L = M(1) ⊗ F{L}`, so the
  `(1,1)` case does **not** lift to general `u', v''` by it.  The spanning input is Remark 8.3.9, not
  landed, and no landed lemma moves a Fock creation operator past `iterateOf`/`commutatorFamily`.
* **The remaining mathematics, priced in four printed steps**: (a) (8.5.12)'s `k ≥ 1` lines, i.e.
  (8.5.5) at the Fock states `E_k·ι(a+b)`, plus the `ModeFamHeisShift` dictionary; (b) the (8.4.35)
  commutator identity with its contraction factor; (c) the `z_1 = z_0 + z_2` shifted resummation
  (`MMFLM86MixedProduct.jointNO:345` lands the shift object; its identification with the iterate is
  `DiagOrdinary`, unproved); (d) the Fock half of (8.5.5).  Not needed: the window
  (`Corollary851.canonical:536`) and the (8.6.1) symmetry hypothesis.

### Batch 153, acceptance — **the corrected assembly boundary is landed, and the "repaired field" turned out to be a third defect**

Two proof tasks were dispatched on disjoint files and both are **accepted** after the dispatcher's own
verification (rebuild from a deleted `.olean`, token check, axiom footprints).  Neither file edits an
existing declaration; both are wired into `FinalCheck.lean` in this batch.

**P1 — `Definitions/MMFLM86ExpProduct.lean`** (349 lines, namespace `MM.FLM86EP`, 7 guards).
`creationExpVL_mul_comm` (a restatement of the strictly stronger landed
`MM.FLM84TV.creationExpVL_comm`, `MMFLM84TwoVarProduct.lean:624`); the concrete two-point product at
the lattice generators, `Yzero_iotaVL_mul_Yzero_iotaVL :
Yzero (iotaVL a) m * Yzero (iotaVL b) n = E.eps a b • (actVL E (a+b) * (creationExpVL a m * creationExpVL b n))`,
with the state form `Yzero_iotaVL_apply_iotaVL` ((8.5.11) at datum `p = 0`); and the commutator family
`commutatorFamily_Yzero_iotaVL = E.eps a b • X − E.eps b a • X`,
`…_eq_c : … = ((E.c a b − 1) * E.eps b a) • X`.

**P2 — `Definitions/MMFLM810CorrectedZeroAssembly.lean`** (299 lines, namespace `MM.FLM810CZA`, 7
guards).  The corrected Section 8.10 assembly line at `Yzero`: `componentTruncationZero` ((8.10.5) —
**reproved**, see below), `vacuumOperatorZero`/`creationRegularZero`/`creationPropertyZero`
((8.10.6)/(8.10.7), which cost only applications), and the two boundaries
`actionRemainingFieldsZero_of_generators`, `theorem8102_of_remainingFieldsZero`,
`theorem8102_of_generatorFieldsZero`.  `ActionRemainingFieldsZero` was **not created** — the landed
`MM.FLM810AFT.ActionRemainingFields` is about an arbitrary `VertexOp` and applies verbatim — and the
four generator reductions (`l0Grading_of_weightGenerators`, `translationAxiom_of_stateData`,
`virasoroAxiomsOf_of_stateData`, `jacobiAxiom_of_stateTargets`) also apply verbatim.

**Finding 1 — a wrong-prescription statement with a prescription-free proof.**  `MM.FLM85ATT`'s
`componentTruncation_action_modelBase` / `componentTruncation_of_actionPrescriptionTA` and
`MM.FLM810AFT`'s `vacuumOperator_action_modelBase` / `creationRegular_action_modelBase` /
`creationProperty_action_modelBase` all carry a compatibility hypothesis **for
`actionPrescriptionTA`**, although their proofs use only the underlying `TruncatedField` datum and its
factor list.  So they do **not** apply to `Yzero`, and `MMFLM810AFT`'s docstring — which calls them
fields "of the corrected object" — overstates what the statements say.  (8.10.5) had to be ported (~40
lines).  The bridges that *are* generic in the factor list (`MM.FLM85TAV.actionFold_vacuumNonnegative`,
`…_eq_cutOff`, `…_zero`) transferred with no work.

**Finding 2 — the zero mode is not the only defect, and the corrected field still fails (8.10.13).**
`MM.FLM810TR.not_translationOnStateData_modelBase`
(`Definitions/MMFLM810TranslationStateObstruction.lean:271`) proves, unconditionally, that
`TranslationOnStateData` is **false** for `vertexOpOfField (Yzero … (modelBaseField E weight))` at the
conformal vector whenever `(form a b : F) ≠ 0` and `actVL E a (iotaVL b) ≠ 0`.  So
`theorem8102_of_generatorFieldsZero` at that base is vacuous: **the obstruction is the
vacuum-supported (8.4.17) base field, not the missing `a(0)`.**  This corrects the batch's own opening
framing, and it means the construction carries **three** stacked defects, not one:

1. the missing (7.1.33)/(7.1.41) zero mode in `heisFam` — repaired by `heisFamZero`/`actionPrescriptionTAZero`/`Yzero`;
2. the missing grading shift `z^{a(0)}` in the base field — `Yzero`'s model base is creation-half only,
   and P1 measured the honest assembled (8.4.17) field `YFam` to be the model base **at a
   pairing-shifted index** (`Yzero_iotaVL_apply_iotaVL_eq_YFam`), which is the same shift seen from
   the other side;
3. the missing dictionary `MM.FLM8512H.ModeFamHeisShift` (`MMFLM8512Heis.lean:221`) identifying the
   prescription's factors `heisFam a m = modeVL a (−m−1)` with the field's modes
   `modeFam Y a m = Y (iotaVL a) (−m−1)` (`MMFLM8512Done.lean:403`) — a `Prop` with no producer.

**Consequence for the plan, and it is the point of the batch.**  The object the next tasks must be
stated about is **`MM.FLM84LF.YFam`** (`Definitions/MMFLM84LatticeField.lean:720`), the assembled
three-factor field `E^-(z)·(az^∂)·E^+(z)` of (8.4.17) — the only landed object that carries the shift
and the annihilation half — and **not** `Yzero` (creation half only) or `modelBaseField` (no shift).
P1's own `## What this file does not prove` block names the landed next steps: `fullPlus_contraction`,
`annihExp_mul_creationExpVL_eq_sum_of_subset` (with `ratioCoeff`/`ratioWindow`),
`YFam_iotaVL_eq_exp`, `Corollary851`, `Display8625One`.

**Also corrected in this batch's record**: `Yzero` is at `MMFLM810ZeroModeL0Uncond.lean:81` and
`Yzero_unique` at `:100` (the batch's own brief had `:82` and `:107`); the two batch-79 refutations are
`MM.FLM810MD.exists_not_l0OnWeightGenerators` (identity base) and
`MM.FLM810MD.exists_not_translationOnStateData` (counterexample base `translationTestBase`) in
`Definitions/MMFLM810ModesL0Trans.lean`, **not** at the (8.4.17) model base; and the corrected `hL0` of
the assembly boundary is already a theorem (`MM.FLM810ZMW.L0OnWeightGenerators_zero`,
`MMFLM810L0General.lean:301`), so that boundary needs only `hjac`/`htrans`/`hvir`.

**Dead ends, named so that no future batch re-dispatches them**: `IsHomogeneousOp` and everything
consuming it, `UniformBracketWindow` (`MMFLM86DisplayVOASplit.lean:80`), `Display8625`,
`CrossContraction` (`MMFLM86CrossContraction.lean:166`, refuted at `:392`), `Contraction8612Ratio`,
`ResidueUMulCompatibilityStatement`, `OperatorDisplay8620Statement`, `RelSpanningStatement`
(refuted `MMFLM85RelSpanning.lean:513`, superseded by the proved `PermSpanningStatement`), and
`MM.FLM86Q86.QuadrantResidueStateFree` (refuted `:526`).


---

### Batch 154 — **the honest field's two-point product is landed, and the swap symmetry is the way past its sign restriction**

**Q1 — `Definitions/MMFLM86YFamProduct.lean`** (351 lines, namespace `MM.FLM86YP`, 11 guards; verified
here by a rebuild from a deleted `.olean`, the token check, and the guards).  It attacks the two-point
product of the **honest** field `MM.FLM84LF.YFam` — the assembled (8.4.17) object carrying both the
grading shift `z^{a(0)}` and the annihilation half `E^+`, which batch 153 identified as the field the
remaining work must be stated about (not `Yzero`, which is creation-half only).

Landed: `YFam_mul_eq_twoVarField` (`YFam E a m * YFam E b n = twoVarField E a b (−m−1) (−n−1)`, the
bridge to the landed batch-147 two-variable carrier); `YFam_mul_eq_sum_ratioCoeff` and
`YFam_mul_eq_convWith2_contrFam` (the (8.4.22)/(8.4.25) expansion, `= convWith2 (normOrdered2 a b)
(contrFam ⟨a,b⟩) …`); `YFam_mul_YFam_iotaVL`/`_eq_exp` (the unconditional state form — the **inner**
factor evaluated on `ι(c)`, since **no landed lemma evaluates `YFam` on a general state**, which is the
file's first named residual); `contraction_coeff_eq_binom`
(`contrFam N (N−j) j = ((−1)^j · binomCoeff N j) • 1`, the coefficientwise `(z₁−z₂)^N`); and a
non-vacuity witness at `⟨1,−2⟩ = −2 < 0` (`YFam_mul_rankOne_negTwo_vacuum_ne_zero`).

**The restriction, measured, and it is a real one.**  `MM.FLM86PE.twoVarField_apply_eq_convWith2_contrFam`
(`Definitions/MMFLM86ProductExpansion.lean:462`) carries `hN : 0 ≤ form a b`: for `⟨a,b⟩ < 0` the
contraction family is not locally finite (`not_locallyFinite2_contrFam_of_neg`), every
`ratioCoeff ⟨a,b⟩ j` is nonzero, and the landed `MM.FLM86RX.threshold_holds` supplies only a
per-`(P,Q)`-per-state threshold, not an identity of operator families.  The brief's warning about this
was correct and the agent verified it rather than assuming it.

**The way past it, and the batch's second task.**  The product is symmetric under exchanging the two
states *and* the two variables: that swap carries `(a,b)` to `(b,a)`, whose pairing is the same number,
so the negative-pairing case **is** the positive-pairing case of the swapped pair.  The landed
`MM.FLM86TN.normOrdered2_swap` (`MMFLM86TwoVarNorm.lean:547`,
`swap2 (normOrdered2 E b a) = E.c b a • normOrdered2 E a b`) is exactly this statement for the
normal-ordered factor, and its three ingredient exchange laws are landed (`creationPair2_swap:451`,
`annihPair2_swap:457`, `midBlock2_swap2:464`, `swap2_convWith2:343`).  What is missing is the same
statement for `twoVarField` itself; task **R1** (`Definitions/MMFLM86TwoVarSwap.lean`) lands it and
transports the expansion along it, so that the `⟨a,b⟩ ≥ 0` restriction is removed in a named,
honest form.  **This is the classical resolution of the sign issue and it is bounded work** — it is
recorded here because the alternative reading ("the Leech lattice has negative pairings, so the
campaign is stuck") is wrong.

**Q2 — `Definitions/MMFLM86IterateIota.lean`** (349 lines, namespace `MM.FLM86II`, 6 guards; verified
here by a rebuild from a deleted `.olean` plus the token check).  It attacks the *other* side of
`MM.FLM85Co.Corollary851` — the iterate summands of (8.5.24) at the lattice generators, where FLM's
(8.5.11) is a theorem (`MM.FLM84LF.YFam_iotaVL`, `YFam_iotaVL_eq_exp`).  Landed:
`iterateStateFamily_lattice`, `iterateOf_lattice_eq_exp`, `iterateState_lattice_boundary`,
`iterateOf_lattice_boundary`, `bracketTerm_lattice`, `corollary851Residue_lattice`,
`stateFamilyStatement_lattice` (**this discharges, at a field carrying (8.5.2), a predicate
`MMFLMLatticeState.lean`'s header lists as open for a general vertex operator**),
`stateVanishesBelow_lattice`, `iterateOf_lattice_eq_zero_of_window`, `corollary851_lattice_iff`,
`Corollary851.canonical_lattice`, `iterateOf_lattice_boundary_ne_zero` (the threshold is **sharp**),
and (8.4.41)/(8.4.42) with every hypothesis discharged.

**The window at the lattice generators is the computed number `(−⟨a,b⟩).toNat`, with no (8.10.5)
hypothesis.**  `MM.FLM86UV.exists_iterate_uniform_vanishing` was checked and does **not** apply as-is
(its hypothesis is `ComponentTruncation` for the *whole* field, and it yields only `∃K`, not the
explicit threshold).

**The one hypothesis it had to carry, and it is a type fact rather than a gap in the mathematics.**
`iterateOf` consumes a **state-linear** map `Y : M →ₗ[F] CoefFamily F M`, whereas `YFam E a` is a
coefficient family; so the file carries the source's (8.5.2) as
`MM.FLM86II.YFamOnGenerators … Y := ∀ a, Y (iotaVL a) = YFam E a`.  **The dispatcher has since
identified a landed inhabitant**: with `base := MM.FLM85TAV.modelBaseField E weight`,
`MM.FLM810TC.Yzero_iotaVL_eq_modelBase` (`MMFLM810TranslationCorrected.lean:111`) gives
`Yzero (iotaVL a) = MM.FLM810F.modelBase E weight a`, and `modelBase` is by definition
`fun a => (latticeElement E a a w).YCoeff` (`MMFLM810Fields.lean:311`), which on a lattice generator
is exactly what `MM.FLM84LF.YFam_iotaVL` evaluates — so `Yzero` inhabits `YFamOnGenerators`.  Batch
155's task **S1** (`Definitions/MMFLM86YzeroIsYFam.lean`) lands that chain, which makes the whole
iterate side of Corollary 8.5.1 at the lattice generators unconditional.

**Which side is closed, exactly.**  The **RHS** (the residue) at the lattice generators is in closed
form for `⟨a,b⟩ ≥ 0` (residue `0`) and for `⟨a,b⟩ = −1` (`ε(a,b)·YFam E (a+b) (m+n)`), and its top
summand is closed for every pairing; for `⟨a,b⟩ ≤ −2` the lower summands need the outer evaluation
`Y(u_i)` at a Fock-descendant created state — the landed named gap
`MM.FLM85IS.CreatedStepResidual` / `MM.FLM86SF.Display8610List` (8.5.5)/(8.6.10), not landed and not
assumed.  The **LHS** is a defined expression in the assembled field alone
(`YFam E a m * YFam E b n − YFam E b n * YFam E a m`) but is not evaluated; that evaluation is the
two-point product Q1 addresses, and batch 155's task **S2** (`Definitions/MMFLM86CommIota.lean`)
assembles the commutator from Q1 plus the `z₁ ↔ z₂` swap of R1.

**Brief errors worth recording** (six, all measured by the agent): the namespace is `MM.FLM85It`
(neither `MM.FLM85Iterate` nor `MMFLM85It`); `iterateOf YFam a b i` is ill-typed (see above);
`YFam E a • (iotaVL b)` uses the wrong operation; `iterateComponent` is not a landed identifier (the
landed names are `MM.FLM85It.iterateOf` and `MM.FLM85Br.bracketTerm`); the vanishing lemmas used are
`MM.FLM85Co.StateVanishesBelow` / `MM.FLMLatSt.stateVanishesBelow_of_stateFamilyStatement`, not in
`FLM85It`; and the driver prints `--- OK: <path> ---`, not `--- OK ---`.

### Batch 155, part 1 — **`YFamOnGenerators` is REFUTED at `Yzero`: the dispatcher's own proposed identification was wrong, and the agent proved it**

Task S1 was dispatched to land the identification sketched in batch 154's entry — that
`MM.FLM810ZMU.Yzero` at the (8.4.17) model base inhabits the iterate side's hypothesis
`MM.FLM86II.YFamOnGenerators`, i.e. `∀ a, Y (iotaVL a) = YFam E a`.  **It came back with a
machine-checked refutation instead**, `Definitions/MMFLM86YzeroIsYFam.lean` (249 lines, namespace
`MM.FLM86YZF`, 8 guards; verified here by a rebuild from a deleted `.olean` plus the token check):

    not_YFamOnGenerators_modelBaseField (E) (weight) :
      ¬ YFamOnGenerators bL form E (Yzero bL form (modelBaseField E weight))

**The witness, and the root cause.**  `Yzero bL form base (iotaVL a) = (base a).fam` and the base is
`Yzero_iotaVL_eq_modelBase`'s `MM.FLM810F.modelBase E weight a`, i.e. `YCoeff` of the lattice element.
`YCoeff` is indexed by the **ordinary power** `z^m` ((8.4.20)/(8.4.24); `YCoeff m = 0` for `m < 0`),
whereas `YFam` is a **(8.5.15) `CoefFamily`**, whose index `m` is the coefficient of `z^{−m−1}`.  The
landed dictionary between the two conventions is `MM.FLM84TV.reflectFam A m = A (−m−1)`.  At the
index where the two are compared the values differ:
`(Yzero … (ι(0))) 0 (vacuumVL) = ι(0) ≠ 0` (`Yzero_iotaVL_zero_vacuum`) against
`YFam E 0 0 (vacuumVL) = 0` (`YFam_apply_zero_vacuum`, independently
`MM.FLM84LF.rankOne_YFam_zero_eq_zero`).  So this was never a missing lemma: **the two families are
different components**, and no choice of landed base puts `YFam E a` at `Yzero base (ι(a))`, because
the base interface `MM.FLM85TA.TruncatedField`/`ofNonnegative` is ordinary-power indexed.

**A second refutation, sharper.**  `not_iterateStateFamily_lattice_Yzero` shows the batch-154 *iterate
conclusion itself* is **false** at `Yzero`: `∀ a b i, (Yzero … (ι(a))) i (ι(b)) =
latticeState E a b ⟨a,b⟩ (−i−1)` fails at `i = 0`, `b = 0`, where it reads `iotaVL a = 0`.  So the 16
conditional results of `Definitions/MMFLM86IterateIota.lean` do not merely lack a witness — their
hypothesis is refuted at the only landed candidate, and **instantiating them at `Yzero` is impossible**.

**What survives, exactly (all landed in the same file, all sharp and conditional-free):**

* `Yzero_iotaVL_apply_iotaVL : (Yzero … (ι(a))) m (ι(c)) = YFam E a (−m−1−⟨a,c⟩) (ι(c))` — the true
  relation between the two: the `reflectFam` dictionary **composed with the state-dependent shift**
  `⟨a,c⟩` (the `z^{a(0)}` factor) and the `E^+` factor (which acts only off the vacuum sector).
* `Yzero_iterateStateFamily_latticeState : (Yzero … (ι(a))) i (ι(b)) = latticeState E a b ⟨a,b⟩ (i + ⟨a,b⟩)`.
* `Yzero_iotaVL_vacuum_eq_reflectFam` — on the **vacuum sector** the brief's identification *is* true
  through `reflectFam`, which is why the error was easy to make.

**The repair, named.**  The iterate hypothesis must be restated for the family the field actually
carries: `Y (ι(a)) = reflectFam (YFam E a)`, using the base
`fun a => ofNonnegative (reflectFam (YFam E a))` — whose lower-truncation datum `TruncatedField` is
owed (it should follow from `YFam_eq_zero_of_gt`, `MMFLM84LatticeField.lean:870`) — and then the whole
iterate machinery of `MMFLM86IterateIota.lean` becomes available with an inhabited hypothesis.  **This
is a change to batch 154's file, not to S1's**, and it is batch 156's first task.

**The meta-lesson, and it is the fifth of its kind in this campaign**: the dispatcher's brief asserted
an identification from three landed lemmas (`Yzero_iotaVL_eq_modelBase`, `modelBase = YCoeff`,
`YFam_iotaVL`) without checking the **index convention** of the two objects — the same class of error as
the `vacuity` of `IsHomogeneousOp` (batch 152) and the `RelSpanningStatement` retraction (batch 149),
and the reason `AGENTS.md` requires that a brief's key step be compiled before the brief is dispatched.
The agent measured the claim false instead of proving a false theorem, which is the outcome the
repository's rules ask for.

### Batch 155, part 2 — **the `z₁ ↔ z₂` swap does NOT remove the sign restriction, and the file as handed back does not build**

Task **R1** (`Definitions/MMFLM86TwoVarSwap.lean`, namespace `MM.FLM86TS`) was dispatched on the
dispatcher's own claim (recorded in batch 154's entry) that the `⟨a,b⟩ ≥ 0` restriction of the
(8.4.25) expansion is removable by the classical swap, "because exchanging the two states and the two
variables carries `(a,b)` to `(b,a)`, whose pairing is the same number".  **The premise is false, and
the agent refuted it machine-checked.**

**The correct exchange law, at `0 ≤ ⟨a,b⟩`, and it is not what the brief said.**
`MM.FLM86TS.twoVarField_swap2`:

    swap2 (twoVarField E b a) = (E.c b a * (-1)^⟨a,b⟩) • twoVarField E a b

— the cocycle **times the sign of the transported binomial**, not the cocycle alone.  (The sign comes
from `swap2_contrFam`, the swap of the contraction factor itself.)  With it,
`twoVarField_sub_swap2` gives the coefficientwise form of locality: the two orders differ by the
constant `1 − E.c b a · (−1)^⟨a,b⟩`, i.e. **the commutator is not a delta-function term in this
carrier** at `0 ≤ ⟨a,b⟩`.

**The restriction is NOT removed, and this is the batch's finding.**  §3 of the file is a rank-one
counterexample at `⟨a,b⟩ = −2 < 0` (`a = −2`, `b = 1`, `F = ℚ`, `L = ℤ`):

* `rankOne_swap2_twoVarField_negTwo_zero`: the exchanged product at `(m,k) = (0,−2)` on the vacuum is
  `ε(−2,1) • ι(−1) ≠ 0`;
* `rankOne_twoVarField_negTwo_zero_zero`: the un-exchanged product there is `0`;
* `not_swap2_twoVarField_rankOne (C)`: therefore `swap2 (twoVarField E (−2) 1) ≠ C • twoVarField E 1 (−2)`
  for **every** `C : ℚ`; `not_exists_swap_factor_rankOne` is the existential form.  **No constant factor
  whatsoever** — not the brief's `E.c b a`, not the corrected `E.c b a · (−1)^⟨a,b⟩` — makes the
  exchange law hold at negative pairing.

**Why the premise fails, in one sentence.**  The swap exchanges the two states *and* the two variables
at once, so it compares the two different *expansion orders* of the same rational function; at
`⟨a,b⟩ < 0` the contraction factor is a genuine series rather than the polynomial `contrFam`
(which is why `locallyFinite2_contrFam` needs `0 ≤ ⟨a,b⟩`), and the two expansion orders are not
related by a constant at the coefficient level.  This is the same phenomenon the repository already
records for the two residue faces of the Jacobi identity
(`MM.FLM810Route.res_z1_face_swap`, `Definitions/MMFLM810JacobiRoute.lean`), now measured on the
product itself.

**A dispatcher measurement that was right about a state of the file and wrong about the file.**
The dispatcher rebuilt the file while the agent was still rewriting it and hit

    Definitions/MMFLM86TwoVarSwap.lean:216:0: error: (deterministic) timeout at `whnf`,
    maximum number of heartbeats (400000) has been reached

for `twoVarField_swap2`, cascading to `unknown constant 'MM.FLM86TS.twoVarField_swap2'` and three
`#guard_msgs` mismatches, one printing the self-referential footprint
`[propext, Classical.choice, Quot.sound, twoVarField_sub_swap2]`; raising the budget to `2000000`
heartbeats did not help either.  **The final file is 315 lines, builds in 2.3 seconds and passes its
guards**, and the agent's own measured advice explains the difference: a `convWith2`-level proof of the
exchange law — `rw` with `convWith2_congr`, or `simpa using congrFun (congrFun …)`, which drags the
`LocallyFinite2` **proof terms** through the whole term — exceeds 2 000 000 heartbeats, while the
**pointwise `Finset.range` route** (state-level expansion plus `Finset.sum_range_reflect`) elaborates
in seconds.  The dispatcher's verdict is therefore corrected in place: the earlier claim that the file
"does not build" described a mid-edit revision, not the deliverable.  **The lesson that survives is the
one worth keeping**: a rebuild by the dispatcher must be run against a *settled* file — and the agent,
for its part, must not leave the tree in a state where the dispatcher cannot tell.
The landed `MM.FLM86TN.convWith2_proof_irrel` (`Definitions/MMFLM86TwoVarNorm.lean:238`) is available
for any future proof that genuinely needs to rewrite under `convWith2`.

### Batch 156 — four fronts opened toward Theorem 8.10.2

Owner's directive: continue without check-ins until **Theorem 8.10.2** (that `V_L` is a vertex operator
algebra) is complete.  The corrected assembly line is already landed
(`MM.FLM810CZA.theorem8102_of_generatorFieldsZero`, batch 153), so what Theorem 8.10.2 needs is the
**four fields** for the right field: Jacobi (8.10.8), `L0` (8.10.12), translation (8.10.13) and
Virasoro (8.10.9) — plus, for the full Jacobi identity, the rationality layer of §8.10.  Four tasks were
dispatched on disjoint files:

* **T1 `Definitions/MMFLM86GenFamTrunc.lean`** (`MM.FLM86GT`) — restate the iterate hypothesis in the
  indexing the field actually carries (`Y (ι(a)) = reflectFam (YFam E a)`, base
  `ofNonnegative (reflectFam (YFam E a))`, lower truncation from `YFam_eq_zero_of_gt`), and prove that
  restated hypothesis **inhabited** by the unique extension of the corrected prescription.  This is
  what makes batch 154's iterate side usable again after
  `MM.FLM86YZF.not_YFamOnGenerators_modelBaseField` killed its original hypothesis.
* **T2 `Definitions/MMFLM86CommIota.lean`** (`MM.FLM86CI`) — the **left-hand side** of Corollary 8.5.1
  at the lattice generators: the commutator family as an explicit two-variable expression, in closed
  form at `0 ≤ ⟨a,b⟩` via the batch-155 exchange law `twoVarField_swap2`/`twoVarField_sub_swap2`, and
  the sharpest supportable statement at negative pairing (the `TwoVar` carrier cannot express
  `contrFam N` for `N < 0`).
* **U1 `Definitions/MMFLMIotaVar.lean`** (`MM.FLMIotaVar`) — FLM (8.10.36)–(8.10.42), the
  **several-variable `ι` maps**, which do not exist in the repository and are the prerequisite of
  **every** `l ≠ 0` face of the Jacobi identity: Prop. 8.10.3–8.10.5 and Prop. 8.10.7 are all stated in
  that language.  The `n = 1` case is landed in `Definitions/MMFLMLocalization.lean` (§1–§3, with
  `iotaOne_eq_iotaPlus` identifying it with `MM.FLM81.iotaPlus`); the recursive step (8.10.40)/(8.10.41)
  is not.  The brief asks for a precise `n = 2` file over a half-finished general one.
* **V1 `Definitions/MMFLM86CreatedStep.lean`** (`MM.FLM86CS`) — the **created-state step**: extend the
  proved lowest case (`MM.FLM85IS.iterateProductStepLevelAt_zero_created_nil_componentField`, i.e.
  (8.5.23) at `l = []`, `j = 0`) to every coefficient `j`, then to the first non-vacuum created state
  `l = [x]`.  `CreatedStepResidual` is **equivalent** to the display
  (`createdStepResidual_iff_display8610List`), so this is on the critical path and is not a weakening —
  the brief says so explicitly to prevent the obvious circularity.

**Process note for this batch, and it is a risk taken knowingly**: four agents were launched at once,
and `AGENTS.md`'s build-hygiene rule is that concurrency is limited by **memory** (≈4.2 GB per `lean`
process) and that two concurrent builds "manufacture fake timeouts".  A serialising build lock
(flock on a lock file in `Probe/`) is the right fix and is recorded here as the next process
improvement; the batch is being watched for timeouts that are artefacts rather than real failures.

**T2 ACCEPTED — `Definitions/MMFLM86CommIota.lean`** (421 lines, namespace `MM.FLM86CI`, 12 guards;
verified by the dispatcher: rebuild from a deleted `.olean` `--- OK ---`, token check at zero,
`FinalCheck.lean` acceptance build `--- OK ---`).  The left-hand side of Corollary 8.5.1 at the lattice
generators is now in closed form, and the batch produced **two** results that change the map.

**(a) The commutator vanishes at the intended cocycle, unconditionally.**
`commutator_YFam_eq_twoVarField_sub_swap2` expresses the commutator through
`twoVarField a b − swap2 (twoVarField b a)` for **every** pairing (no sign hypothesis), and
`commutator_YFam_eq_smul_mul_of_nonneg` makes the `0 ≤ ⟨a,b⟩` case explicit: the commutator is the
**constant** `1 − E.c b a · (−1)^⟨a,b⟩` times the product — **no delta term** in this carrier.  The new
mathematical content is `evenCocycle_c_mul_neg_one_zpow`: for the source's own even-lattice cocycle
`MM.FLMLattice.EvenCocycle` (from `C.c_eq_signPow`), that constant is **zero**:

    (C.toCentralExtension h2).c b a * (-1)^(form a b) = 1        (h2 : (2 : F) ≠ 0, form symmetric)

hence `commutator_YFam_eq_zero_of_even`: at the honest fields and the intended cocycle,
`[YFam a m, YFam b n] = 0` for `0 ≤ ⟨a,b⟩`, with **no `hY` and no `h851`**.  *This is the compatibility
`c(a,b) = (−1)^⟨a,b⟩` that the landed `CentralExtension` does not assume and `EvenCocycle` supplies —
and it is exactly the hypothesis Theorem 8.6.1's (8.6.1) is stated under.*

**(b) The batch-154 iterate-side headline is vacuous at the landed rank-one datum, and this is now a
theorem.**  `not_YFamOnGenerators_and_corollary851_rankOne` refutes `hY ∧ h851` at the rank-one datum
with `a = b = 1`: there `⟨1,1⟩ = 1 ≥ 0`, the constant is `1 − c(1,1)(−1)^1 = 2 ≠ 0` for the trivial
cocycle, and the product is nonzero on the vacuum (`YFam 1 (−2)(YFam 1 (−1) vac) = ε(1,1)•ι(2)`), so
`(hY, h851)` would force `2 • (nonzero) = 0`.  Consequently
`MM.FLM86II.commutatorFamily_lattice_eq_zero_of_nonneg` is **not** "the `0 ≤ ⟨a,b⟩` case of (8.4.41)"
in landed generality: its hypothesis pair is empty at that datum.  The regression the brief asked for
therefore *failed*, in the useful direction — and the failure isolates the cocycle/form compatibility as
the real dividing line, rather than the pairing's sign alone.

**(c) The missing range is now exactly `⟨a,b⟩ ≤ −2`.**  For `N < 0` the contraction family is not
locally finite, so `convWith2` has no instance: no finite coefficient expansion of (8.4.25) exists in
this carrier.  The exchange law is false there (`MM.FLM86TS.not_swap2_twoVarField_rankOne`) and the
commutator is genuinely nonzero (`commutator_rankOne_negTwo_ne_zero`:
`−ε(1,−2) • ι(−1)` at `(a,b) = (1,−2)`, `(m,n) = (−1,1)`, vacuum).  So the negative-pairing case is
**not** a defect of the formalization — it is a real phenomenon that the finite-support carrier cannot
express, and the only route to it is the residue sum of (8.5.13) itself, i.e. `Corollary851` with its
outer evaluation at the created states (the `CreatedStepResidual`/(8.6.10) gap that task V1 is
attacking).  `⟨a,b⟩ = −1` **is** in closed form (8.4.42), but only under `hY ∧ h851`.

**Where Theorem 8.10.2 now stands, field by field** (updated in batch 156, and this is the ledger entry
that matters for the remaining work):

| field | status |
| --- | --- |
| (8.10.5) truncation | **proved** for `Yzero` (`MM.FLM810CZA.componentTruncationZero`); |
| (8.10.6) vacuum, (8.10.7) both creation halves | **proved** for `Yzero` (`vacuumOperatorZero`, `creationRegularZero`, `creationPropertyZero`); |
| **(8.10.12) `L0` grading** | **PROVED** — `MM.FLM810ZMW.LOf_conformalVectorZero_on_weightPiece` (`MMFLM810L0General.lean:273`) and `L0OnWeightGenerators_zero` (`:301`), from `hnd`, symmetry of `form` and `(base 0).fam = identityField`.  **This is not an open field**, contrary to the batch-153 reading; |
| **(8.10.13) translation** | **false at the model base** (`MM.FLM810TR.not_translationOnStateData_modelBase`); the repaired base (`reflectFam`-indexed) is being tested by task W1; |
| **(8.10.9) Virasoro** | reduced, conditionally, to **two named `Prop`s** by `MM.FLM87ZBC.virasoroOnStateData_of_named` (`MMFLM87ZeroModeBracketClose.lean:129`): `MM.FLM87ZB.sugawaraModeZero_eq_fockSugawaraVLZero` and `MM.FLM87ZB.fockSugawaraVLZeroBracket`, whose honest form is `MM.FLM87SA.FreeBosonSugawaraBracketZero` (`MMFLM87SugawaraAux.lean:218`) — FLM (8.7.9) for the corrected modes, the paper's `L(n)` bracket.  Its header records that it is **not refuted**: it is about the honest inverse-Gram Sugawara vector; |
| **(8.10.8) Jacobi** | **open**, and the only field with no landed reduction to a finite list.  The `Res_{z₀}` face is Theorem 8.6.1's display (batch 152); the `l ≠ 0` faces need the several-variable `ι` maps (task U1) and the rationality propositions (8.10.3)–(8.10.7); the `l = 0` face at negative pairing needs a carrier that can express a contraction series. |

So the remaining work for Theorem 8.10.2 is **translation + Virasoro + Jacobi**, and each of the three
now has either a landed reduction (Virasoro: two named `Prop`s; translation: the base repair) or a
named infrastructure prerequisite (Jacobi: U1's `ι` maps).  `L0` is done.

**U1 ACCEPTED — `Definitions/MMFLMIotaVar.lean`** (852 lines, namespace `MM.FLMIotaVar`, 7 guards;
verified by the dispatcher: rebuild from a deleted `.olean` `--- OK ---`, token check at zero,
`FinalCheck.lean` acceptance build `--- OK ---`).  FLM (8.10.37)–(8.10.42) now exist in the repository
for `n = 2`, and they did not exist at all before this batch: the `l ≠ 0` faces of the Jacobi identity
were blocked on exactly this.

**The carrier decision, which is the batch's structural finding.**  FLM's `F[[z_1,z_1^{-1},z_2,z_2^{-1}]]`
is **not a ring in FLM's use**: the landed `no_ringHom_monomial` shows no ring homomorphism into the
product model can send `z_1 − z_2` to a unit, although `z_1 − z_2 ∈ S` by (8.10.36) and the localisation
`F[z_1,z_2]_S` inverts it.  So `ι` is honestly an **`F`-linear map into the product space** (FLM's own
target), while each *ordering* does live in a ring: expanding first in negative powers of `z_{i_1}` makes
the outer exponents well-founded (`OrderDual ℤ`), and the inner ones are Laurent.  Hence the carrier
`IotaVar F := HahnSeries (OrderDual ℤ) (HahnSeries ℤ F)` (a field), with two variable assignments and
**both** incarnations landed: ring homs `iota12Ring`/`iota21Ring` into the carrier, and the `F`-linear
`iota12`/`iota21` into FLM's target `AllSeriesN F 2`, which is what Prop. 8.10.3–8.10.5 consume.

**Landed**: `innerEmbed` (= the landed `polyToSeriesOne`) with injectivity; `polyToOnePoly_injective`
(proved here — the landed file lacked it); `outerEval` with its support formula and injectivity;
`emb12`/`emb21` with injectivity and generator values; `iota12Ring`/`iota21Ring` via `IsLocalization.lift`
with `iota{12,21}Ring_algebraMap` and injectivity; `invX`; the values of both readings on the five
generators `z_1, z_2, z_1−z_2, z_1^{-1}, z_2^{-1}`; and the series
`invSeries = Σ_{m≥0} z_2^m z_1^{−m−1}` with `mul_invSeries` (telescoping), `iota12_invShift = expand12`,
`iota21_invShift = expand21`, `iota12_sub_iota21_invShift = deltaRatio`,
`shift_iota12_invShift`, and `iota12_invShift_ne_iota21_invShift`.

**Injectivity is proved, not assumed** (the source's "it is clear that the maps `ι_{i_1⋯i_n}` are
injective"), at all three levels: the polynomial embeddings, the ring homs out of the localisation
(`IsLocalization.lift_injective_iff` plus `IsLocalization.injective` with the landed
`S_le_nonZeroDivisors`), and the readings.  Given that the repository's `MM.FLM81.KerThetaStatement`
exists precisely because such statements can fail, this is worth recording as a proved item rather
than a citation.

**The sharp negative fact that bounds what the `l ≠ 0` faces can say.**  The delta series is supported on
the whole antidiagonal — unbounded above in the outer exponent — so it is **not an element of the
carrier**: the difference of the two expansions exists only in FLM's target, while in the carrier the two
elements differ by a sign (`± invSeries`).  This is the `n = 2` form of FLM's Remark 8.10.1, and it is the
same phenomenon batch 155 measured on the product (the two expansion orders are not related by a
constant), now located in the `ι` language and given a carrier.

**What the file does not prove** (its own `## What this file does not prove`): general `n`; FLM's
(8.10.41) partial-fraction form of the recursion; the *general* delta statement — which needs the landed
but unproved (8.1.9) `MM.FLM81.ThetaShiftStatement` — beyond the `f = (z_1−z_2)^{-1}` case and the five
generators; (8.10.45) beyond generators; the kernel characterisation; and any VOA content.  The brief's
own claim that "the two expansions agree exactly on Laurent polynomials" is `n = 1` only; for `n = 2` the
landed fact is the opposite (`expand12 ≠ expand21`), and this is recorded as a brief error.

**V1 ACCEPTED — `Definitions/MMFLM86CreatedStep.lean`** (254 lines, namespace `MM.FLM86CS`, 4 guards;
verified by the dispatcher: rebuild from a deleted `.olean` `--- OK ---`, token check at zero,
`FinalCheck.lean` acceptance build `--- OK ---`).  It was dispatched to extend the proved lowest case of
the created-state step to every coefficient `j`, and it came back with a **refutation** instead — the
fourth appearance of the same defect.

* **Landed**: `zeroFactorLevel_zero_componentField`, the `j = 0` level of (8.5.12)/(8.5.23) at
  `iteratedCreation _ _ []` — batch 123's theorem restated at this interface.
* **Refuted**: `not_modelBasePositiveModesKillVacuum` at the landed model base (rank one,
  `F = ℚ`, `L = ℤ`, trivial cocycle), from the unconditional
  `rankOne_component_neg_two_ne_zero : (componentField (Yzero … (modelBaseField …))) (iotaVL 1) (−2) vacuumVL ≠ 0`.
  **Mechanism**: the landed base is `YCoeff = E^-(z)·(a z^∂)` — creation half only — so its positive
  modes are `actVL E a * creationExpVL a m`, and batch 133's `creationExpVL_ne_zero`
  (`MM.FLM85FS`) makes them non-vanishing on `ι(1)` at **every** `m ≥ 0`.  Hence FLM (8.10.7) fails at
  the model base, and no §8.5 normal-ordering computation can repair it: **the missing factor is the
  annihilation half `E^+` of (8.4.17)**, exactly as `MMFLMY87.lean`'s header records.
* `PositiveModesKillVacuum` / `ModelBasePositiveModesKillVacuum` / `CreatedStepResidualLevels` name the
  residual exactly.
* Two brief errors of record: the level is **not** vacuous at `j ≥ 1` — its right-hand side is
  `conv (shiftedField Y a j) A`, which is `1` at `A = vacuumField` and `0` at `A = 0`, so it carries
  content at every `j` — and target 2 (`l = [x]`) is target 1 one step out
  (`iterateAt_singleton_reduce` is `rfl`), hence refuted wherever target 1 is.

**This is now the decisive convergence of batch 156**: W1 (translation at the repaired base), T1 (the
iterate hypothesis at the `reflectFam`-indexed base) and V1 (the created-state step) have independently
arrived at the **same** repair — the base field of the (8.5.5) prescription must be the honest assembled
(8.4.17) family `MM.FLM84LF.YFam`, with its `z^{a(0)}` shift and its `E^+` half, rather than the landed
`YCoeff`/`modelBase` which has neither.  Three separate fields of Theorem 8.10.2 (translation, the Fock
half of (8.10.7), and the created-state step) fail at the old base and are expected to hold at the new
one.

**W1 ACCEPTED — `Definitions/MMFLM810TranslationBase.lean`** (438 lines, namespace `MM.FLM810TB`, 5
guards; verified by the dispatcher: rebuild from a deleted `.olean` `--- OK ---`, token check at zero,
`FinalCheck.lean` acceptance build `--- OK ---`, all 20 declarations at
`[propext, Classical.choice, Quot.sound]`).  It was dispatched to test the dispatcher's own
"`reflectFam`-indexed base" repair for (8.10.13), and it **corrected the diagnosis** while landing the
right object.

**Landed.**  `baseRepaired (E) : L → TruncatedField F L` with `fam := reflectFam (YFam E a)` — the honest
(8.4.17) assembled field in the ordinary-power convention — its `trunc` from
`MM.FLM84LF.YFam_eq_zero_of_gt`, its unfolding `baseRepaired_fam`, and
`baseRepaired_zero : (baseRepaired E 0).fam = FLM84.identityField` (8.5.3), with the factor-level chain
that gets there.  `l0OnWeightGenerators_baseRepaired` gives the `L0` field at it from `hnd` and symmetry
of `form` alone.

**The verdict on (8.10.13) at the repaired base is a reduction, not a verdict — and it is sharp.**
`translationOnStateData_baseRepaired_base_state_iff`: for every `a` and `p`, the component `m = −p−1` of
`TranslationOnStateData` at `D = ⟨[], a⟩` is **equivalent** to the honest (8.4.23) at the honest family,

    (actionFold [heisFactor a] (baseRepaired E a)).fam p = deriv (baseRepaired E a).fam p

and `translationOnStateData_baseRepaired_forces_deriv` is the one-way whole-predicate form.  So the
repair **removes the model base's obstruction** — the (8.4.16) creation half is nonempty there, and the
tail at `p = −1`, `⟨a,c⟩ = −1` is `actVL E a (α_a(−1)ι(c)) = ε(a,c)·α_a(−1)ι(a+c)` by the landed
`actVL_modeVL_comm` — but it does **not** close (8.10.13): the residual is the honest (8.4.23)
`deriv (YFam E a) = 8α_a(z)Y(a,z)8`, which is **not landed** (the landed (8.4.23) is the model-base
identity, shown by `MM.FLM810TC.deriv_modelBase_ne_flat_add_tail` to be the creation half alone).
Oscillator states `D` with `k ≥ 1` are untouched.

**The diagnosis is corrected, and the correction matters.**  The dispatcher had claimed the model base was
`reflectFam (YFam E a)` mis-indexed, i.e. a pure indexing artefact.  **False**: `modelBase = gOp·E^-` is
the *vacuum-sector/flat* model of (8.4.17), missing the middle `a z^∂` and the `E^+` half; it agrees with
the honest family only on the vacuum sector (`YFam_vacuumVL_eq_YCoeff`) and differs as a family —
`baseRepaired_ne_modelBase`, a landed witness.  Also corrected: `ofNonnegative (reflectFam (YFam E a))`
does **not** typecheck (the honest family has genuine negative powers, bounded below only), so the
`TruncatedField` was built directly; `Yzero_iotaVL_apply_iotaVL` is `MM.FLM86EP.*`
(`MMFLM86ExpProduct.lean:209`), not `FLM86YZF`; the landed model-base refutation carries two explicit
nonvanishing hypotheses, so "unconditionally false" was an overstatement; and `L0` does not test the
repair at all, since `L0OnWeightGenerators_zero` takes `(base 0).fam = identityField` as a hypothesis and
holds for an arbitrary base.

**Batch 156's convergence, now with the correct content**: W1, T1 and V1 agree that the base must be the
honest `YFam` family — but W1 shows that this repair is a *reduction to the honest (8.4.23)*, not a
discharge of translation.  The next task on this front is therefore **the honest (8.4.23) at `YFam`**
(`deriv (YFam E a) = 8α_a(z)Y(a,z)8`), which is a concrete landed-target computation on the assembled
three-factor field.

**T1 (in flight) — the key finding, and it is a Lean trap, not a mathematical one.**  Task T1
(`Definitions/MMFLM86GenFamTrunc.lean`) has confirmed the mathematics of the repair: `genFam E a m =
YFam E a (−m−1)` is the correct reindexing, and the restated hypothesis **is inhabited** by `Yzero` at
the base `fun a => ofNonnegative (genFam E a) hgen`, because
`actionPrescriptionTAZero base ⟨[], a⟩ = actionFold [] (base a) = base a` and
`stateOfVL ⟨[], a⟩ = iotaVL a` are both `rfl`, so the uniqueness statement of
`MM.FLM85ZP.existsUnique_actionPrescriptionTAZero` applies with **no** hypothesis failing.

The prerequisite `hgen : ∀ m < 0, genFam E a m = 0` reduces to the **sharpened** statement
`YFam E a k = 0` for all `k ≥ 0` — which is **not** the landed one-sided
`MM.FLM84LF.YFam_eq_zero_of_gt` (`:870`, vanishing *above* a bound), and is not landed anywhere.  The
agent reduced it to a complete proof shape (`annihFam` is supported at the single index `−1`,
`annihFam_eq_ite`, so `YFam E a m v = convWith creationFam midFam hB m v`, and the support set of
`z^{a(0)}` can be taken involution-closed so that each summand vanishes), and every step compiled except
the final arithmetic.

**The blocker is `omega`, and it is now recorded in `AGENTS.md` as the most expensive trap in this
pin.**  `omega` **silently drops** linear `ℤ` hypotheses and goals containing subtraction, unary
negation or a numeral offset: `example (j : ℤ) (h : (0:ℤ) < j + 1) : (0:ℤ) < j := by omega` reports
*"No usable constraints found"*, and `linarith` fails it too.  It succeeds when the goal is
syntactically a hypothesis or when no negation/offset occurs.  The workaround is to hand-build the
`Int` chain or restate the arithmetic; **do not conclude a goal is unprovable from an `omega`
failure.**  Also recorded there: `Finset.mem_of_eq`, `lt_of_not_le` and `Int.lt_of_le_of_ne` do not
exist in this pin, and `MM.FLM85TA.TruncatedField`/`ofNonnegative` live in
`Definitions/MMFLM85TrueAction.lean` — a file name that does **not** match its namespace, so a `grep`
for the namespace in file *names* finds nothing.

**T1 CLOSED, NOT LANDED — and the repair it was testing does not work: the honest field has genuine poles.**
Task T1 was to restate the iterate hypothesis in the indexing the field carries, with the base
`fun a => MM.FLM85TA.ofNonnegative (genFam E a) hgen` where `genFam E a m = YFam E a (−m−1)`
(`Definitions/MMFLM85TrueAction.lean:38`/`:151`, **not** a `MMFLM85TA*.lean` file).  It landed no file:
the prerequisite `hgen : ∀ m < 0, genFam E a m = 0` — equivalently `YFam E a k = 0` for **all** `k ≥ 0` —
resisted, and the dispatcher's own reading of the landed value lemmas says why.

**The prerequisite is false, and the reason is structural.**  `MM.FLM84LF.YFam_iotaVL` (`:770`) gives
`YFam E a m (ι(c)) = latticeState E a c ⟨a,c⟩ (−m−1)`, and `latticeState` is, by its definition
(`Definitions/MMFLMLatticeState.lean:391`),
`actVL E a (creationExpVL a (m − p) (actVL E c ω))` — i.e. a coefficient of the creation exponential
**shifted by the exponent `p = ⟨a,c⟩`**.  At `m = 0`, `c = 0` this is the constant term of
`creationExpVL`, which is `1` (`creationExpVL_zero`), so

    YFam E a 0 (vacuumVL) = actVL E a (vacuumVL) = iotaVL a  ≠ 0

(`actVL_vacuumVL`), whereas the prerequisite demands `0`.  The same computation at `c = a` reads
`YFam E a 0 (ι(a)) = E.eps a a • ι(2a) = iotaVL (2a) ≠ 0`.  So **no** choice of landed base puts a
nonnegatively-supported family at `Yzero base (ι(a))` — the honest (8.4.17) field is bounded below but
genuinely carries negative powers of `z`, which is exactly why the FLM (8.5.5) prescription has to
accept a **negative lower bound**, i.e. why `MM.FLM85TA.TruncatedField` is a *lower truncation* datum
rather than a nonnegativity datum.

**A second brief error, of the same class as the batch-155 refutation.**  The brief suggested deriving
`m < 0 → genFam E a m = 0` from the landed `MM.FLM84LF.YFam_eq_zero_of_gt` (`:870`).  That lemma is
**one-sided** — it vanishes *above* a bound, index → +∞ — and says nothing about vanishing at
nonnegative indices.  This is the fourth brief in this campaign whose key step was asserted without
compiling it, and the agent was right to refuse to write a file around it.

**What T1 did settle, and it is worth keeping.**  `MM.FLM85ZP.existsUnique_actionPrescriptionTAZero` is
stated for an **arbitrary** base, so it applies verbatim to any base of the shape above; and the chain
`Yzero_spec base ⟨[], a⟩` → `stateOfVL ⟨[], a⟩ = iotaVL a` → `actionPrescriptionTAZero base ⟨[], a⟩ =
actionFold [] (base a) = base a` is **definitional** at every step.  So the *machinery* for restating the
iterate hypothesis works; what fails is the choice of `genFam` as the base.

**The direction this leaves for the iterate side.**  Since the honest field's lower bound is negative,
the base interface must be fed a lower-truncated family with a genuinely negative bound (or the
prescription must be restated over a two-sided indexing).  That is a change to the *interface*, not to
the iterate machinery, and it is recorded here rather than attempted again with the same brief.

**CORRECTION OF RECORD (dispatcher, batch 156 close).**  The entry above retracting the `genFam` base
repair is **wrong in its main claim**, and the primary source settles it.  FLM printed 198, (8.5.1) and
(8.5.8), quoted verbatim from the page image:

    8.5.1  the correspondence iota(a) -> Y(a,z) for a in L extends uniquely to a well-defined linear map
    8.5.8  lim_{z->0} Y(u,z) iota(1) = u,  "Here of course the limit has algebraic meaning because
           Y(u,z) iota(1) involves only nonnegative integral powers of z."

So **the base field of (8.5.5) must vanish at every negative exponent**, and the honest assembled field
`MM.FLM84LF.YFam` — being the restriction of the general vertex operator to the lattice states — is
exactly the object the base must be.  The dispatcher's "refutation" computed `YFam E a 0` by composing
the three factors in the **wrong order**: `convCoef` is `Σ_j A (m−1−j) (B j w)`, so with `YFam =
convWith (convWith creationFam midFam) annihFam` the outer application is `creationFam … ∘ annihFam`,
and at `m = 0` the term is `creationFam a 0 (midFam a (−1) (annihFam a (−2) ω))`.  The correct
verification of the truncation is therefore **task T1's own proof shape**, which reduced it to
`∀ m < 0, genFam E a m = 0` via an involution-closed support argument and failed only on the `omega`
arithmetic trap — not on mathematics.  `Definitions/MMFLM86GenFamTrunc.lean` is **not** refuted and
should be re-dispatched, with the `omega` workaround budgeted explicitly (see `AGENTS.md`).

*The meta-lesson, and it is the fifth of its kind in this campaign: the dispatcher computed a value by
hand from three landed lemmas and did not compile it.  A hand computation is a hypothesis exactly as a
brief's claimed route is.*

---

### Batch 157 — **the owner's directive: align the definitions with the book first, then prove Theorem 8.10.2**

Owner's instruction (2026-09-20, before leaving the session): *"do the definitional alignment first,
otherwise all we do is waste compute proving things false; I want a proof of Theorem 8.10.2, and the
alignment must follow the book."*  The goal is re-stated accordingly (revision 5): **align, then
prove**.

**The alignment audit, with the source quoted (`Probe/FLM_8p4_8p5_quoted.txt`, PDF pages 244--258 =
printed 189--203):**

| # | the book | the repository | verdict |
| --- | --- | --- | --- |
| 1 | **(8.4.7)** `a(z) = Σ_{n∈ℤ} a(n) z^{−n−1} = z^{−1}a^old(z)` — the sum runs over **all** of `ℤ`, and (8.4.13)/(8.4.14) put `a(0)` in the normal ordering | `MM.FLM85Cp.heisFam a m = modeVL a (−m−1)` (`MMFLM85Compat.lean:213`), and `modeVL a 0 = 0` (`Definitions/MMFLMLatticeModule.lean:796` composed with `MM.FLM84.mode_zero`, `MMFLMFock.lean:398`) | **zero mode missing** |
| 2 | **(8.4.17)** `Y(a,z) = 8e^{∫(a(z)−a(0)z^{−1})} a z^∂8 = X(a,z) z^{−(a,a)/2}` — **three** factors | the assembly's base `MM.FLM810F.modelBase E w` (`MMFLM810Fields.lean:311`) is `(latticeElement …).YCoeff = gOp · E^−` — the **creation half only** | **`a z^∂` and `E^+` missing** |
| 3 | **(8.5.1)** the correspondence `ι(a) ↦ Y(a,z)` extends **uniquely**; **(8.5.8)** `lim_{z→0} Y(u,z)ι(1) = u`, "because `Y(u,z)ι(1)` involves only nonnegative integral powers of `z`" | the base interface `MM.FLM85TA.TruncatedField`/`ofNonnegative` (`MMFLM85TrueAction.lean:38`, `:151`) demands vanishing below `0`, but no landed theorem says the honest base satisfies it | **never proved** |

**The correction of record that this audit forced.**  Batch 156's closing entry had retracted the
`genFam` base repair as "refuted".  **That retraction is itself wrong**, and (8.5.8) is why: the base
field *must* vanish at negative exponents, and the honest `YFam` does.  The dispatcher's hand
computation that produced the "refutation" composed the three factors in the wrong order —
`convCoef A B hB m w = Σ_j A (m−1−j) (B j w)` (`MMFLM85ConvProduct.lean:256`) applies **`B` first** —
and T1's own proof shape (involution-closed support of `z^{a(0)}`) was right; it failed only on the
`omega` trap.  Both the false refutation and its correction are now in the record.

**Batch 157 tasks (dispatched):**

* **D0 — reconnaissance** (`Probe/B157_D0_IMPACT.md`): the migration impact map — every consumer of
  `heisFam`, `modeVL`, `modelBase`, `YCoeff`, `TruncatedField`, `ofNonnegative`, classified as
  definition site / proof consumer / doc-comment only, with the theorems that would become **false**
  under the alignment identified by name (that set is the migration's cost), the already-landed repaired
  layer (`MM.FLM85ZM.*`, `MM.FLM85ZP.*`, `MM.FLM810ZM*`, `MM.FLM810ZMU.*`, `MM.FLM84LF.*`) listed, and a
  dependency-ordered migration plan.
* **D1 — proof** (`Definitions/MMFLM85BaseFieldAligned.lean`, namespace `MM.FLM85BA`): alignment step 1,
  the base field as the book defines it — `genFam E a m := YFam E a (−m−1)`, **the lower truncation
  (8.5.8) as a theorem** (`∀ m < 0, genFam E a m = 0`, the lemma task T1 could not finish), the
  `TruncatedField` package `baseAligned`, its identity at `a = 0` (8.5.3), and the inhabitance of the
  restated generator hypothesis by `Yaligned := Yzero (baseAligned E)`.  The brief carries the `omega`
  workaround explicitly.
* **X1 — proof** (`Definitions/MMFLM84DerivYFam.lean`, namespace `MM.FLM84DY`, in flight from batch
  156): the honest (8.4.23) `deriv (YFam E a) = 8α_a(z)Y(a,z)8`, which stands on the correct field
  already and is the residual of translation (8.10.13).

**Standing rules for the rest of this campaign, restated so that any later batch can be audited against
them:** every brief's key step is a hypothesis until compiled; every handed-back file is rebuilt by the
dispatcher from a deleted `.olean` before acceptance; every negative report is re-measured before it is
believed; and every claim that turns out false is corrected **in place** in this ledger rather than
quietly dropped.

**D0 ACCEPTED — `Probe/B157_D0_IMPACT.md` (410 lines)**; the dispatcher verified its two structural
claims against the sources before acting on them.

**Correction to this batch's own audit, and it is good news.**  Batch 157's table above listed the base
interface as a deviation ("the base interface requires vanishing below `0`").  **That is only half
right**, and D0 measured it: `MM.FLM85TA.TruncatedField` (`Definitions/MMFLM85TrueAction.lean:38`)
requires **lower truncation only** —

    trunc : ∀ (w : VL F L) (m : ℤ), ∃ N : ℕ, ∀ i : ℕ, N ≤ i → fam (m - (i : ℤ)) w = 0

— which is exactly FLM (8.5.6)'s local finiteness and is **base-agnostic**; the nonnegativity demand
lives in the separate constructor `ofNonnegative` (`:151`), which is the cut-off artefact.  So **the
interface needs no change**: the alignment is a change of *which family* is fed to it, not of the
interface.  This removes the largest feared item from the migration.

**The verdict, and its numbers.**  A **seam change with a wide but shallow tail**, not a layer rebuild:
2 declaration edits (`MM.FLM85Cp.heisFam`, `MMFLM810F.modelBase`) + ~20--25 files touched, of which only
~5 need genuinely new proofs — because **both honest replacements already exist**
(`MM.FLM85ZM.heisFamZero`, `MM.FLM810TB.baseRepaired`).  Occurrence counts (word-boundary `grep -c` over
`Definitions/` plus `FinalCheck.lean`): `heisFam` 183 hits in 22 files but only **7 statement sites**;
`modeVL` 943/85 but ~8 statement sites; `modelBase` 153/26; `YCoeff` 193/34; `TruncatedField` 272/42
**with 0 edits needed**; `YFam` 253/17; `Yzero` 213/23; `heisFamZero` 195/23.

**The insulation that makes the tail shallow.**  `splitMinus_heisFamZero_eq`
(`Definitions/MMFLM85ZeroModeField.lean:219`) and `opOpenBox_heisFamZero_eq` (`:233`) say the corrected
and defective fields have the *same* creation half and the *same* open-box product; so everything routed
through `splitMinus`/`opOpenBox` needs **no re-proof**.  The genuinely `splitPlus`-side breakages are
seven, and every one has a landed counterpart:
`MM.FLM85Ap.splitPlus_heisFam_apply` (`MMFLM85Action.lean:290`, false at `m = −1`) and
`splitPlus_heisFam_neg` (`:296`, false at `k = 0`) against
`MM.FLM85ZM.splitPlus_heisFamZero_apply`/`_neg_one`/`_neg`
(`Definitions/MMFLM85ZeroModeField.lean:162`/`:150`/`:175`);
`MM.FLM87FC.tailSummand_eq_normalOrderedHeisPair` (`MMFLM87SugawaraFoldClose.lean:136`, false at
`k' = 0`) and `actionFold_eq_normalOrderedHeisFinsum` (`:169`) against
`MM.FLM87ZN.tailSummandZero_eq` (`Definitions/MMFLM87ZeroModeNormalOrder.lean:197`) and
`actionFoldZero_eq_normalOrderedHeisFinsumZero` (`:236`);
`MM.FLM845CN.annihPair_oscField_neg_one_eq_zero` (`MMFLM845ContractionNeg.lean:165`, a theorem *about*
the defect); and `MM.FLM85Ap.modeVL_zero` (`MMFLM85ActionRelations.lean:87`, only if `modeVL` is itself
repaired — see below).

**The riskiest single step, named.**  The `modelBase → baseRepaired` substitution
(`Definitions/MMFLM810Fields.lean:311`).  `modelBase` **ignores the central extension `E`** — `YCoeff`
never uses it — whereas `baseRepaired` genuinely uses `E` through `actVL` inside `midFam`.  So every
`modelBase`-statement re-elaborates with `E` as a real parameter, and every `Hrel`/`Prescription`
hypothesis stated *at* `modelBase` (`MMFLM810Fields.lean:477`/`:565`, `MMFLM810ActionFields.lean:123`,
`MMFLM85IterateStep.lean:421`--`:492`, `MMFLM84VacuumField.lean:330`/`:422`, `MMFLM810Creation.lean:122`,
`MMFLM810Truncation.lean:80`) becomes a hypothesis about a **provably different family**
(`MM.FLM810TB.baseRepaired_ne_modelBase`, `Definitions/MMFLM810TranslationBase.lean:250`) and must be
re-proved.  D0's recommendation, which the dispatcher adopts: **do the `heisFam`-side rewiring as one
self-contained batch, and leave the `modelBase → baseRepaired` substitution to a separate one.**

**One recommendation the dispatcher declines, with the reason.**  D0 suggests **not** repairing
`modeVL` itself (`modeVL a 0 = 0`), because that would falsify `MMFLM85ActionRelations.lean:87` and
re-elaborate all 34 `YCoeff` files "for no gain".  That is right about the cost but wrong about the goal:
FLM (7.1.41) gives `a(0)` its own action `hZeroVL` **on the lattice factor**, and `modeVL a n` is defined
as the Fock mode tensored with the identity, so `modeVL` is the right object **away from `n = 0`** and the
zero mode is the separate operator `hZeroVL`.  The alignment therefore does not need `modeVL` repaired —
it needs the **field** `a(z)` to carry both pieces, which is exactly what `heisFamZero` does (it is
`hZeroVL` at `m = −1` and `modeVL a (−m−1)` elsewhere).  `modeVL_zero` stays a true theorem about
`modeVL`, and no `YCoeff` file is re-elaborated.  Recorded because the distinction between the *mode
family* and the *field* is what the whole alignment turns on.

**D2 dispatched on the strength of this report** (`Definitions/MMFLM85CanonicalField.lean`, namespace
`MM.FLM85CF`, additive only): make the corrected (8.4.7) field canonical, land the agreement/difference
ledger (including the `k = 0` case where the landed `splitPlus_heisFam_neg` is false, side by side with
its landed counterpart), restate the insulation facts that make the tail shallow, and mark the defective
declarations deprecated — from the new file, without editing any existing one.

**D1 ACCEPTED — `Definitions/MMFLM85BaseFieldAligned.lean`** (331 lines, namespace `MM.FLM85BA`, 5
guards; verified by the dispatcher: rebuild from a deleted `.olean` `--- OK ---`, token check at zero,
`FinalCheck.lean` acceptance build `--- OK ---`).  Alignment step 1 is done, and it settles the
truncation question that batches 156 and 157 argued about — **both sides were partly right.**

**Landed.**  `genFam E a m := YFam E a (−m−1)`, with `genFam_eq_reflectFam` (definitional bridge to the
landed `MM.FLM84TV.reflectFam`), `genFam_iotaVL : genFam E a m (ι c) = latticeState E a c ⟨a,c⟩ m`,
`genFam_zero_label` (at `a = 0`: `1` at `m = −1`, `0` elsewhere), and `genFam_vacuumVL_eq_modelBase`
(the vacuum sector, where the defective base agrees).  `baseAligned E : L → TruncatedField F L` with
`fam = genFam E a`, its truncation taken **per vector** from `MM.FLM84LF.YFam_eq_zero_of_gt`,
`baseAligned_zero` (8.5.3, from the full `YFam_zero_label` — **not** from `modelBase_zero`, which is
vacuum-only), and — importantly — `baseAligned_fam_eq_baseRepaired`, which is **`rfl`**: the object this
batch built is **definitionally the `baseRepaired` of batch 156**.  Finally `Yaligned := Yzero (baseAligned E)`
with `Yaligned_iotaVL : Yaligned E (ι a) = genFam E a` (all three bridges `rfl`, landed as
`stateOfVL_base_state`, `actionPrescriptionTAZero_base_state`).

**The truncation question, resolved by the source.**  The statement this batch's predecessor tried and
the dispatcher twice mis-stated — `∀ m < 0, genFam E a m = 0`, equivalently `∀ k ≥ 0, YFam E a k = 0` —
is **false as an operator identity**, and is now refuted **unconditionally**
(`genFam_lower_truncation_false_rankOne`, at rank one: `genFam E 1 (−1) ι(−1) = ε•ι(0) ≠ 0`, the zero
mode).  What is true, and is now a theorem, is FLM (8.5.8) read correctly — **at the state `ι(1)`**:

    genFam_vacuum_creates : genFam E a 0 (vacuumVL) = iotaVL a        (the limit of (8.5.8))
    genFam_lower_truncation_vacuum : ∀ m < 0, genFam E a m (vacuumVL) = 0

and, at a general lattice state, the **sharp threshold**

    genFam_lower_truncation_iotaVL : ∀ m < ⟨a,c⟩, genFam E a m (ι c) = 0

with the threshold attained (`genFam_iotaVL_pairing_eq`: `ε(a,c)·ι(a+c)`), plus the per-vector
`∃K` form `genFam_eq_zero_of_lt` — FLM (8.10.5) at the base.  So the honest field's lower bound is
**`⟨a,c⟩`, not `0`**, which is exactly the `z^{−(a,a)/2}` of (8.4.17); the dispatcher's hand computation
of batch 156 got the composition order wrong, and its "refutation" of the *base* was therefore wrong,
while its instinct that "an operator-level statement fails" was right for the *operator* form only.
Both halves are now in the record, and the sharp threshold is the aligned statement.

**The brief's central claim was false a third time** (item 2 conflated the operator identity with the
state-level (8.5.8)), its `ofNonnegative` route was impossible (that constructor's hypothesis *is* the
refuted statement), and its (8.5.3) citation was insufficient (vacuum-only).  It also missed that
`genFam`/`baseAligned` already existed as the landed `baseRepaired`.  The `omega` trap did **not** fire:
the file contains no live `omega`/`linarith` — the two dangerous `ℤ` steps are hand-built
(`Int.add_lt_add_right`, `neg_lt_neg_iff`, `sub_lt_zero`, `exists_nat_gt`) exactly as `AGENTS.md` now
prescribes, and that is the first payoff from that recorded measurement.

**Consequence for the migration.**  The base substitute is now **landed twice over** (`baseRepaired`,
`baseAligned`, definitionally equal) and carries (8.5.3), the correct truncation, and the inhabitance of
the restated generator hypothesis.  What remains of the alignment is: (i) the `heisFam`-side rewiring of
the seven `splitPlus`-stated breakages (self-contained, counterparts landed); (ii) the
`modelBase → baseAligned` substitution at the ~8 hypothesis sites, with `E` becoming a real parameter —
the risky step, to be done as its own batch; and (iii) `Definitions/MMFLM85CanonicalField.lean` (task D2,
in flight) which marks the defective declarations and lands the agreement/difference ledger.

**X1 ACCEPTED — `Definitions/MMFLM84DerivYFam.lean`** (341 lines, namespace `MM.FLM84DY`, 5 guards;
verified by the dispatcher: rebuild from a deleted `.olean` `--- OK ---`, token check at zero,
`FinalCheck.lean` acceptance build `--- OK ---`).  **This batch is the strongest empirical vindication
of the owner's alignment directive so far**, because the task turned out to require the corrected field
for its statement even to be *true*.

**Landed, with no hypotheses.**  At every power and every lattice generator:

    actionFold_iotaVL : (FLM85TA.actionFold [heisFactor a] (baseRepaired E a)).fam m (ι c)
                     = deriv (baseRepaired E a).fam m (ι c)

— the honest (8.4.23) at `YFam` on the (8.5.4)/(8.5.11) generators — together with its two halves
`headOp_baseRepaired_iotaVL`, `tailOp_baseRepaired_iotaVL`, the (8.4.1) ODE
`deriv_creationExpVL : deriv (creationExpVL a) = flatMul (splitMinus (heisFam a)) (creationExpVL a)`
(new), and `baseRepaired_fam_iotaVL`.  **Residual**, named exactly: the general state `w = x ⊗ ι(c)`,
`x ≠ 1`, needs the (8.4.16) Leibniz rule for the truncated one-factor action; its head half is landed
(`MM.FLM810WR.headOp_deriv`), its **tail half is not** (`MMFLM810HeadLeibniz`'s header records it
missing).  The agent also flagged, correctly, that that header's weight count is a *hypothesis* rather
than a theorem — after reindexing the tail weights do add up — so the tail Leibniz is likely provable.

**THE CORRECTION, and it is the alignment's own argument stated by a proof.**  The brief's literal target
`deriv (YFam E a) = opOpenBox (heisFam …) (YFam E a)` is **false**, for two independent reasons:

1. **index convention**: `deriv` is power-indexed (`m ↦ (m+1)·s_{m+1}`) while `YFam` is (8.5.15)-indexed
   (`YFam E a m` is the coefficient of `z^{−m−1}`), so `deriv (YFam E a)` is not the derivative of the
   field at all; the `baseRepaired`/`reflectFam` carrier is the correct one;
2. **the zero mode is load-bearing**: the identity needs the factor's family to be `heisFamZero`
   (`MM.FLM85ZM`), whose `z^{−1}` coefficient is `hZeroVL` — the (8.4.14) zero mode — because that term
   is exactly what cancels the head at `p = −1` on `ι(c)` with `⟨a,c⟩ = −1`.  **With the landed
   `heisFam a` (zero mode deleted, `modeVL a 0 = 0`) the identity is FALSE** there: head `≠ 0`,
   tail `= 0`, `deriv = 0`.  The relevant landed (8.4.23) is therefore
   `MM.FLM85ZM.deriv_YCoeff_eq_opOpenBox_heisFamZero` (`Definitions/MMFLM85ZeroModeField.lean:242`),
   **not** `MM.FLM84DI.deriv_YCoeff_eq_opOpenBox_heisFam` (`Definitions/MMFLM84DerivId.lean:146`).

*This is the owner's point, proved: a formaliser following the old definitions would have landed a false
theorem and then spent batches "proving" things about it.  The alignment is not cosmetic.*

**Two smaller corrections, both recorded in place.**  W1's docstring
(`Definitions/MMFLM810TranslationBase.lean:48--50`) mis-states the cancellation — the tail is `−head`,
not equal to it, and the identity at `p = −1` reads `0 = 0` rather than cancelling two terms of size
`2×`; and the page-offset note in this ledger is off by one for a zero-indexed reader
(`r.pages[246]` is printed 192; `pdftoppm -f 247` is needed).  The `AGENTS.md` `le_or_lt`-absent trap
fired once and cost one build round — `by_cases` plus `not_le.mp` is the fix, as recorded there.

**D2 ACCEPTED — `Definitions/MMFLM85CanonicalField.lean`** (299 lines, namespace `MM.FLM85CF`, 7
guards; purely additive; verified by the dispatcher: rebuild from a deleted `.olean` `--- OK ---`,
token check at zero, `FinalCheck.lean` acceptance build `--- OK ---`).  Alignment step 2 is done: the
corrected (8.4.7) field now has a **name that says what it is**, the defective one has a migration
pointer, and the defect itself is a **theorem**.

**Landed.**  `canonicalField a := MM.FLM85ZM.heisFamZero a` (an `abbrev` whose body *cites* the landed
field rather than restating it, docstring quoting (8.4.7)); the agreement/difference ledger
`canonicalField_apply_of_ne` (agrees with `heisFam` for `m ≠ −1`), `canonicalField_neg_one`
(`= hZeroVL (ofLattice a)`), `canonicalField_eq_heisFam_of_ne`,
`canonicalField_ne_heisFam_neg_one`; the insulation facts restated
(`splitMinus_canonicalField_eq`, `opOpenBox_canonicalField_eq`, and the general
`opOpenBox_congr_of_splitMinus_eq`) with (8.4.23) verbatim as
`deriv_YCoeff_eq_opOpenBox_canonicalField`; and the defect as a proposition,
`splitPlus_defect_at_zero : splitPlus (canonicalField a) (−1) = hZeroVL (ofLattice a) ∧
splitPlus (heisFam a) (−1) = 0`, with `splitPlus_canonicalField_neg_of_pos` showing the landed
`splitPlus_heisFam_neg` is true for exactly `k ≥ 1`, and `splitPlus_defect_witness` exhibiting a label
where the defective value kills `ι(b)` and the canonical value is the eigenvalue times `ι(b)`.

**A measurement that closes a door, recorded in `AGENTS.md`.**  The `deprecated` attribute **cannot**
be applied to a declaration in another module in this pin: both
`attribute [deprecated (since := "…")] MM.FLM85Cp.heisFam` and the two-name form fail with *"Cannot add
attribute '[deprecated]' to declaration '…' because it is in an imported module"*
(`Lean/Attributes.lean`, `ParametricAttribute.add` rejects `(env.getModuleIdxFor? decl).isSome`).  So
`heisFam`, `modelBase` and `YCoeff` **cannot be made to warn on use** from a new file; the available
routes are an edit to their own files or the `## Migration` block plus `#check`-level pointer lemmas
that D2 landed (`canonicalField_pointer`, `baseRepaired_pointer`, `YFam_pointer`, `YCoeff_pointer`).
This is why the alignment's remaining work must be done by **rewiring statements**, not by making the
old names complain.

**A brief error that had to be caught by the compiler.**  The brief (and D0's prose) invited the
generalisation that the two fields' halves coincide as families.  That is true for `splitMinus` and
`opOpenBox` — and **false for `splitPlus`**: at `m = −1` the corrected value is `hZeroVL (ofLattice a)`
and the defective value is `0`, so `splitPlus (canonicalField a) ≠ splitPlus (heisFam a)`.  The agent
landed the false family equality, saw it fail, and replaced it with the correct `k ≥ 1` statement —
which is exactly the behaviour `AGENTS.md` asks for.  Also corrected: `MMFLM87SugawaraFoldClose.lean`'s
statement sites are `:134` and `:167` (D0's `:136`/`:169` are proof lines), and the zero-mode witness
needs a label `b` with `⟨ā,b̄⟩ ≠ 0` to be nonvanishing.

**A guard-measurement correction.**  Batch 156 recorded "a 126-character guard message prints on one
line, so the wrap threshold is above 120".  This batch's guard for
`deriv_YCoeff_eq_opOpenBox_canonicalField` is **121** characters and **wrapped**.  So the wrapping is not
a function of length alone; `AGENTS.md` now says the only safe route is empirical — add the guard with a
one-line expectation, build, and read the diff that `guard_msgs` prints.

**Z1 ACCEPTED — `Definitions/MMFLM85SplitPlusAligned.lean`** (375 lines, namespace `MM.FLM85SP`, 17
declarations, 5 guards; purely additive; verified by the dispatcher: rebuild from a deleted `.olean`
`--- OK ---`, token check at zero, `FinalCheck.lean` acceptance build `--- OK ---`).  Alignment step 3:
**every defective `^+`-side statement now has its corrected counterpart, in one place, with the defect
itself as a proposition.**

**Landed.**  For the two `splitPlus` statements: `splitPlus_canonicalField_apply` (cited from
`MM.FLM85ZM`), the `m = −1` correction `splitPlus_heisFam_apply_defect_at_neg_one` (defective instance
`= modeVL a 0`, corrected `= hZeroVL (ofLattice a)`, and `modeVL a 0 = 0`), the `k ≥ 1` and `k = 0`
halves `splitPlus_canonicalField_neg`/`_neg_one`, the validity range
`splitPlus_canonicalField_eq_heisFam_of_pos`, the `k = 0` correction
`splitPlus_heisFam_neg_defect_at_zero`, and `canonicalField_neg_one_ne_defective_zero` (the two
operators genuinely differ whenever `⟨a,b⟩ ≠ 0`).  For the Sugawara pair:
`tailSummandZero_eq_canonicalField` and `actionFoldZero_eq_normalOrderedHeisFinsumZero_canonicalField`
(cited from `MM.FLM87ZN`), with `tailSummand_defect_at_zero`.  For the contraction: **a counterpart that
did not exist before** — `annihPair_canonicalField_latticeField`, the **unguarded** (8.4.19)/(8.4.25)
form at every `m < 0` (built from the landed `MM.FLM84CT.hZeroVL_contraction` at `m = −1` together with
`annihPair_oscField_latticeField` at `m < −1`), plus `annihPair_canonicalField_neg_one`,
`heisFam_eq_oscField` (`rfl`) and `annihPair_heisFam_unguarded_false` (cited from `MM.FLM84CN`).  And for
the Leibniz head half: `headOp_deriv_canonicalField`, which needed no correction — `headOp_deriv` is
quantified over arbitrary families and holds verbatim at the corrected field.

**The consumption table, which is what a rewiring batch needs** (live mentions, comment-stripped, over
`Definitions/` plus `FinalCheck.lean`; **every consumer uses the defective name in a proof only — none
states it**): item 1 → 3 consumers in 2 files; item 2 → **7 consumers in 4 files**; item 3 → 1, no
external consumer; item 4 → 2, with `MM.FLM87ZD.sugawaraMode_apply_eq_sum_pair` the only out-of-file one
guarded in `FinalCheck.lean:5641`; item 5 → 4, none external; item 6 (`modeVL_zero`, **not** defective) →
11 files / 26 sites with two homonyms; item 7 → 1.  **Riskiest single rewire: item 2**, because
`KillsAbove`/`Factor.nil` is built on `splitPlus X (−k−1)` at *every* `k`, so the true hypothesis `0 < k`
breaks `exists_killsAbove_heisFam`, `…heisFamZero` and both fold lemmas.  Suggested order:
3 → 4 → 1 → 5 → 2.

**Brief errors of record** (all `grep`-verified by the agent): the namespaces are `MM.FLM85Ac` for items
1--2 (not `Ap`) and `MM.FLM85AR` for item 6; item 5's file is `Definitions/MMFLM84ContractionNeg.lean`
(there is no `MMFLM845ContractionNeg.lean`), its statement is at `:163`, namespace `MM.FLM84CN`, and its
counterpart is **not** `annihPair`-named but `MM.FLM84CT.hZeroVL_contraction`
(`Definitions/MMFLM84Contraction.lean:659`); item 7 is **not** defective, so the brief's title calling all
seven defective was wrong; the brief's seven are **not** B157 §3a's seven —
`MM.FLM85LL.heisFam_add` (`Definitions/MMFLM85LabelLinearity.lean:244`, one consumer
`factorFamily_add:257`, corrected twin `MM.FLM85ZP.heisFamZero_add`,
`Definitions/MMFLM85ZeroPrescription.lean:118`) was dropped; and B157's second `modeVL_zero` homonym is
`MM.FLM810ZMT` (`Definitions/MMFLM810ZeroModeTwoFactor.lean:100`), not `MM.FLM810ZM`.  The
`MMFLM85LabelLinearity`/`heisFam_add` item is added to the rewiring list by this entry.

**Guard measurement, third data point.**  A 118-character message printed on one line; a 121-character
message wrapped.  Combined with batch 156's 126-on-one-line, this confirms `AGENTS.md`'s revised rule:
the threshold is not a function of the message length alone, so the route is empirical — add the guard
with a one-line expectation, build, and read the diff.  The dispatcher has now hit this three times and
each cost exactly one build round.

**Y1 ACCEPTED (as a reduction) — `Definitions/MMFLM84LeibnizTail.lean`** (241 lines, namespace
`MM.FLM84LT`, 4 guards; verified by the dispatcher: rebuild from a deleted `.olean` `--- OK ---`,
token check at zero; to be wired into `FinalCheck.lean` in the next batch together with its siblings).
The tail half of (8.4.16) is **not** a theorem yet — but its exact shape is now **known**, and that is
the batch's result.

**The true shape of the tail Leibniz rule** (the agent's derivation, which the dispatcher checked):

    deriv (tailOp X Y) m = headOp X.fam (deriv Y) m + tailOp (dFactor X) Y m

The mechanism: `tailWindow X Y m M = Σ_{k≤M} Y (m+k+1) (X⁺_{−k−1} w)`, and `deriv s m = (m+1) • s (m+1)`
puts the extra weight *inside* `Y`; with `j = k+1` the sum becomes `Σ_{j≥1} (m+j+1) • Y (m+j+1) (X⁺_{−j} w)`
and, for `j ≥ 1`, `X⁺_{−j} = splitPlus (dFactor X).fam (−j)` (from the landed
`MM.FLM84.deriv_splitPlus`, `Definitions/MMFLM84NormalOrder.lean:235`).  So **the two halves trade the
outer index `k = 0`**: the `k = 0` summand becomes the *head* term, and the rest is the *tail at the
shifted factor*.  The **naive** form `tailOp (dFactor X) Y m + tailOp X (deriv Y) m` is **false** — it is
off by exactly the creation half, which is landed as the conditional
`MM.FLM84LT.tailOp_deriv_sub`.  The brief's guessed product rule
`tailOp m (x ⊗ ιc) = (tailOp m x) ⊗ ιc + …` **does not exist** in the definitions.

**Landed**: `tailLeibnizResidual` (the `Prop` above, with its derivation in the module docstring),
`tailOp_deriv_sub` (the conditional difference identity), `derivActionFamilyResidual`,
`actionFamily_modeVL` and `generalStateDeriv` (the two inputs of the general-state (8.4.23), as named
`Prop`s), `generalStateDeriv_iotaVL` (**its generator case IS batch 157's
`MM.FLM84DY.actionFold_iotaVL`, verbatim** — so the (8.4.23) at general states is now one named
`Prop` away), plus `negOne_smul` and `modeVL_vacuumVL_eq_zero`.

**A false intermediate, measured and recorded so that no future batch repeats it**: "the corrected field
vanishes at index `m+1` for `m ≤ −2`" is **FALSE** — at `m = −2` the index `m+1 = −1` is the (7.1.33)
zero-mode index, where `heisFamZero a (−1) = hZeroVL (ofLattice a) ≠ 0`, and for `m ≤ −3` it is the
annihilation mode, which kills the vacuum but is **not** the zero operator ((7.1.5)/(7.1.32) are
*vector-level* statements).  Any proof of the tail rule must do the reindexing **pointwise on the
vector**, which is exactly where the mechanisation budget was lost.

**A dispatcher measurement, offered to the next attempt.**  The residual is a finite reindexing of
`Finset.range`, and the dispatcher verified the *mathematics* of the reindexing by hand at
`tailWindow`'s definition (`Definitions/MMFLM85Action.lean:400`):

    deriv (tailWindow X.fam Y m M) w
      = Σ_{k≤M} (m+k+2) • Y (m+k+2) (X⁺_{−k−1} w)
      = (m+1) • Y (m+1) (X⁺_{−1} w) + Σ_{k≤M−1} (m+k+3) • Y (m+k+3) (X⁺_{−k−2} w)

whose two pieces are `headOp X.fam (deriv Y) m` at `i = 0` and the tail at the shifted index — so the
residual **is** provable, and the work is `Finset.sum_range_succ'` plus `Int`-cast normalization.  The
dispatcher's own throwaway probe reached the final arithmetic goal and no further; it is recorded here
rather than left as a claim, and the file was deleted.

**B1 ACCEPTED — `Definitions/MMFLM810TheoremAligned.lean`** (299 lines, namespace `MM.FLM810TA2`, 4
guards; verified by the dispatcher: rebuild from a deleted `.olean` `--- OK ---`, token check at zero,
`FinalCheck.lean` acceptance build `--- OK ---`).  **Theorem 8.10.2 is now stated at the book's field**,
and one of its four inputs is discharged outright.

**The landed boundary** (verbatim type):

    theorem8102_of_generatorFieldsAligned : ... →
      JacobiOnStateTargets bL form (vertexOpOfField (Yaligned bL form E)) →
      L0OnWeightGenerators form (vertexOpOfField (Yaligned bL form E)) om →
      TranslationOnStateData bL form (vertexOpOfField (Yaligned bL form E)) om →
      VirasoroOnStateData bL form (vertexOpOfField (Yaligned bL form E)) om r →
      Theorem8102 form r

with `theorem8102_of_generatorFieldsAligned_l0` the same at `om := conformalVector bL form hnd` and
**no `L0` hypothesis**.  Also landed in the same file: (8.10.5)--(8.10.7) at `baseAligned`
(`componentTruncationAligned`, `vacuumOperatorAligned`, `creationRegularAligned`,
`creationPropertyAligned`) and the `VacuumNormalized` spelling of (8.10.6).

**The four inputs, measured:**

| input | status |
| --- | --- |
| `hL0` (8.10.12) | **DISCHARGED OUTRIGHT** — `l0OnWeightGenerators_aligned`, from the *base-generic* `MM.FLM810ZMW.L0OnWeightGenerators_zero` plus `MM.FLM85BA.baseAligned_zero`; the `L0` field needs only `hnd` and symmetry of `form`; |
| `hvir` (8.10.9) | **REDUCED** to the named `Prop` `MM.FLM87SA.FreeBosonSugawaraBracketZero` (FLM (8.7.9)) by the landed `MM.FLM87SA.virasoroOnStateData_of_freeBosonBracketZero`; |
| `htrans` (8.10.13) | **OPEN** — the landed links are the base-state iff `MM.FLM810TB.translationOnStateData_baseRepaired_base_state_iff` (at `D = ⟨[], a⟩`, equivalent to the honest (8.4.23)) and `MM.FLM84DY.actionFold_iotaVL` (at the lattice generators); |
| `hjac` (8.10.8) | **OPEN** — `JacobiOnStateTargets` has no producer for any field in the repository; |

plus the four inherited hypotheses `hpd`, `hrank`, `hW`, `hfin` of `MM.FLM810.theorem8102_of_fields`.
**Theorem 8.10.2 is not unconditional**, and the file says so.

**Where the brief was wrong, and two of these were the dispatcher's own errors:**

1. **The dispatcher claimed the corrected assembly line takes the base as a parameter.  It does not**:
   `MM.FLM810CZA.theorem8102_of_generatorFieldsZero` fixes every hypothesis at
   `Yzero (modelBaseField E weight)` (`Definitions/MMFLM810CorrectedZeroAssembly.lean:228`--`:244`), so
   the boundary had to be **re-proved** at `Yaligned`, not instantiated.  This is the sixth brief in the
   campaign whose key step was asserted without compiling it.
2. **The dispatcher's `L0` route was needlessly indirect** (and omitted that the field needs `hnd` and
   form symmetry): `L0OnWeightGenerators_zero` is base-generic, so `baseAligned_zero` alone suffices and
   no transport from `baseRepaired` is needed.
3. **The dispatcher overstated translation**: `MMFLM84LeibnizTail.lean` lands its residuals as
   **unproved** `def … : Prop`s and states that nothing there decides `TranslationOnStateData`; there is
   **no** landed reduction of the predicate to a residual — only the base-state iff and the
   lattice-generator case.  (This entry corrects batch 158's own Y1 paragraph accordingly: the tail
   Leibniz rule's *shape* is known and the residual is provably a finite reindexing, but **no reduction
   of `TranslationOnStateData` to it is landed**.)
4. **The dispatcher named the wrong Virasoro route**: `MM.FLM87ZBC.virasoroOnStateData_of_named` needs
   two further named statements that that file's own §4b measures untenable; the direct
   `MM.FLM87SA.virasoroOnStateData_of_freeBosonBracketZero` (bracket only, base as a parameter) is the
   route that works.

**T1 ACCEPTED — `Definitions/MMFLM84TailLeibnizProof.lean`** (280 lines, namespace `MM.FLM84TLP`, 9
guards; verified by the dispatcher: rebuild from a deleted `.olean` `--- OK ---`, token check at zero,
`FinalCheck.lean` acceptance build `--- OK ---`).  **The (8.4.16) Leibniz rule is now unconditional —
and batch 158's residual was false.**

**The true rule** (a plain product rule, with the tails pairing as `k ↦ k+1`, **not** `k ↦ k`):

    tailLeibniz_tailOp : deriv (tailOp X Y) m = tailOp (dFactor X) Y m + tailOp X (deriv Y) m
    actionFamily_deriv : deriv (FLM85Ac.actionFamily X Y) m
                       = FLM85Ac.actionFamily (dFactor X) Y m + FLM85Ac.actionFamily X (deriv Y) m

the second being the **full (8.4.16) Leibniz rule, unconditional**, with head half = the landed
`MM.FLM810WR.headOp_deriv_factor` and tail half = the new `tailLeibniz_tailOp`.  Cores:
`tailWindow_deriv_window` and `tailOp_deriv_pointwise`.

**Batch 158's `MM.FLM84LT.tailLeibnizResidual` is REFUTED**, unconditionally
(`not_tailLeibnizResidual`): the residual asserted `deriv (tailOp X Y) m = headOp X.fam (deriv Y) m +
tailOp (dFactor X) Y m`, and `tailLeibnizResidual_iff` shows that is **equivalent to**
`tailOp X (deriv Y) m = headOp X.fam (deriv Y) m` for all `X, Y, m` — i.e. to the annihilation and
creation halves of `8X(dY)8` coinciding, which is false.  The counterexample is the corrected factor
`heisFactor (bL i)`, the lower-truncated family `testFamily` (identity at power 1, zero elsewhere),
`m = 0` and the vacuum: `tailOp_vacuumVL` gives `0` while `headOp X.fam (deriv testFamily) 0 =
splitMinus X.fam 0 = modeVL (bL i) (−1)`, which creates a nonzero state.  **So the dispatcher's
"verification by hand", offered to T1 as the plan, was wrong at exactly the step it flagged as
"plumbing": splitting off the `k = 0` summand does not produce `headOp`; it trades `tailOp X (deriv Y)`
for `headOp X.fam (deriv Y)`, and those are different sums.**  The correct shift is `k ↦ k+1`, and with
it the two tails' weights sum to `m+1`.

**Consequences, recorded so that nothing downstream uses the false premise.**
`MM.FLM84LT.tailLeibnizResidual`, and everything conditional on it — `MM.FLM84LT.tailOp_deriv_sub`,
`MM.FLM84LT.derivActionFamilyResidual`, and `MM.FLM84LT`'s §"what this file does not prove" framing —
**rest on a false premise** and must not be used; the replacements are `MM.FLM84TLP.tailLeibniz_tailOp`
and `MM.FLM84TLP.actionFamily_deriv`.  `MM.FLM84LT.derivActionFamilyResidual` additionally names the
**uncorrected** fold (`FLM85Ac.actionFamily`) where the corrected head over a `TruncatedField`
(`FLM85TA.actionFamily`) is needed, so the (8.10.13) reduction must be restated for the corrected fold —
that is the next task on the translation front, and it is now a **restatement over proved halves**
rather than a proof search.

**The lesson, and it is the third of its kind in two batches.**  A disjoint-propagation of the same
residual has now been asserted false by two different agents (`MM.FLM810HeadLeibniz`'s header and
`MM.FLM84LT`'s), corrected once by a third, and the dispatcher itself endorsed the false form.  The rule
that caught it is the one in `AGENTS.md`: **a claimed route is a hypothesis until compiled** — the
dispatcher's hand derivation, however detailed, was still a hypothesis, and T1 was right to compile it
rather than build on it.

**T2 ACCEPTED — `Definitions/MMFLM810TranslationGeneral.lean`** (524 lines, namespace `MM.FLM810TG`, 8
guards; verified by the dispatcher: rebuild from a deleted `.olean` `--- OK ---`, token check at zero,
`FinalCheck.lean` acceptance build `--- OK ---`).  The translation field of Theorem 8.10.2 is now fully
**reduced**, and the reduction identified the input that is genuinely missing — it is not the Leibniz
rule.

**Target 1 — LANDED: the corrected-fold Leibniz rule, and the object it needed.**  The second argument
of `MM.FLM85TA.actionFamily` is a `TruncatedField`, so `deriv Y` cannot be passed: the missing object is
the **derivative field**, and no truncation hypothesis is needed (`deriv` preserves lower truncation,
bound `+1`):

    noncomputable def dField (Y : TruncatedField F L) : TruncatedField F L := ⟨deriv Y.fam, …⟩
    headOp_deriv_field      : deriv (FLM85TA.headOp X.fam Y) m
                            = FLM85TA.headOp (dFactor X).fam Y m + FLM85TA.headOp X.fam (dField Y) m
    actionFamily_deriv_field : deriv (FLM85TA.actionFamily X Y) m
                            = FLM85TA.actionFamily (dFactor X) Y m + FLM85TA.actionFamily X (dField Y) m
    actionFold_deriv_field   : the one-factor fold version
    actionFold_deriv_dFold   : ∀ l Z m, deriv ((FLM85TA.actionFold l Z).fam) m = (dFold l Z).fam m

**Target 2 — the honest reduction, and the real input.**  `MM.FLM84LT.generalStateDeriv` is **not**
reachable from `actionFold_iotaVL` plus a creation-list induction: **`{ι(c)}` does not span `V_L`**, and
the induction additionally needs a fact about the Heisenberg modes at an arbitrary vector.  Landed
instead:

    baseStateTranslation_iff_generalStateDeriv :
      (∀ E, BaseStateTranslation bL form hnd E) ↔ MM.FLM84LT.generalStateDeriv bL form
    translationOnStateData_iff_dFold :
      ⟨the (8.10.13) component at D, m⟩
        ↔ Yzero base (L(−1) v_D) (−m−1) = (dFold (datumFactorsZero D) (baseRepaired E D.a)).fam (−m−1)
    def LMinusOneCreationMode (hnd) (E) : Prop        -- the named state half
    lMinusOneCreationMode_iota                          -- its hypothesis-free instance at ι(c)

So **no state of `TranslationOnStateData` is newly decided**: the base states are *equivalent* to the
open `generalStateDeriv`, and the oscillator states are equivalent to the state half
`[L(−1), α_b(−n)] = n·α_b(−n−1)` at an arbitrary vector — landed only at `ι(c)`
(`MM.FLM810TRD.LOf_conformalVectorZero_neg_one_modeVL`).  The file says this plainly.

**Where the brief was wrong, and one of these is a trap now in `AGENTS.md`.**  (i)
`MM.FLM810WR.headOp_deriv_factor` is the **flat** head (`FLM85Ac.headOp`, window `m`), **not** the
corrected head (`FLM85TA.headOp`, window `Y.bound w m`) — so "both halves are already proved" was
false for the corrected fold, and a new head half was required.  (ii) The creation-list-induction route
is unavailable for the reason above.  (iii) Three new API traps were measured and are recorded:
`Finset.sum_apply` does not fire on `(∑ f) w` for `LinearMap`s (use `LinearMap.sum_apply`);
`Module.End`'s `LinearMap.instSMul` defeats `Int.cast_smul_eq_zsmul` (prove scalar identities at the
vector level); and `rw` cannot apply an operator-level lemma under `c • (f w)` (use a vector-level
lemma or `congrArg (· w)`).

**Where this leaves Theorem 8.10.2**: `hL0` discharged; `htrans` reduced to the state half
`LMinusOneCreationMode` (plus the now-proved Leibniz rule); `hvir` reduced to FLM (8.7.9); `hjac` open.

**U1 (batch 160) FAILED — no report, and the cause is a scheduling error by the dispatcher, not the
mathematics.**  The task was to prove the state half `MM.FLM810TG.LMinusOneCreationMode` — the last
input of the translation field — by reducing it to the Fock factor.  The agent died mid-proof, leaving a
single partial lemma (`actionFold_two_neg_one_modeVL_apply`, 249 lines, no guards); the file built, but
it proves none of the target.  Its file was **deleted** so that a re-dispatch does not collide with a
stale revision, and the partial result is recorded here rather than kept: it is an intermediate step of
a proof that has to be redone as a whole.

**The scheduling error.**  Batches 159 and 160 were dispatched back to back at a point where the session
had been running long; the dispatcher did not confirm that the previous batch's reports had been read
and recorded before opening the next, and `AGENTS.md`'s own dispatching rule says **one batch at a time**
unless the files are disjoint.  A subagent that dies without a report leaves the tree in an unknown
state, which is exactly what happened: the file had to be inspected, built, judged incomplete (one
lemma, no guards, not the target) and removed.  The rule that follows is cheap and is now the practice:
**before dispatching a batch, check `git status` and the previous batch's reports; never open a new
batch on top of an unfinished one.**

**The mathematics is unaffected**: the state half is still the last input of `htrans`, the plan
(Fock-factor reduction, since `modeVL` is the Fock mode tensored with the identity) still stands, and
the task is re-dispatched with a tighter scope.

**U2 ACCEPTED — `Definitions/MMFLM87SugawaraStateHalf.lean`** (299 lines, namespace `MM.FLM87SH`, 6
guards; verified by the dispatcher: rebuild from a deleted `.olean` `--- OK ---`, token check at zero,
`FinalCheck.lean` acceptance build `--- OK ---`).

**The Fock-level state half was already landed, and in a place no brief had named.**
`MM.Heisenberg.L_commutator_alpha` (`Definitions/MMFockVirasoro.lean:649`) and its `m = −1` instance
`L_neg_one_commutator_alpha` (`:1572`) give `[L(−1), α(−k)] = k·α(−k−1)` on the Heisenberg Fock space —
**and the brief's suggested homes (`MM.FLM87FZ`/`SA`/`ZB`) do not carry it**; `MM.FLM87FZ` in fact
records that its operator-level `finsum` is `0`.  This is the third time a task has found its target
already landed under a different name, which is worth a rule: **before proving an identity, grep the
whole `Definitions/` for its statement shape, not just the files whose names suggest it.**

**Landed new.**  On the repository's own Fock carrier `M1 F L` (the transport is forced: the
repository's Sugawara is built from the (8.4.16) folds and acts on `V_L`, not on `M1`, so the only
`L(−1)` on `M1` is the transported `MM.FLM87ST.fockSugawara`):

* `fockSugawara_comm_mode` — `[fockSugawara m, mode s a] = ((−s : ℤ) : F) • mode (m+s) a` for every
  `a : H F L`;
* **`fockSugawara_neg_one_creation`** — the target at `m = −1`: `[fockSugawara (−1), mode (−(k:ℤ)) a] =
  (k : F) • mode (−(k:ℤ)−1) a`, for `k : ℕ` (no `k ≥ 1` needed: `k = 0` is `0 = 0`);
* `fockSugawara_neg_one_creation_ofLattice` — the same at `a = ofLattice c`;
* **`sugawaraMode_neg_one_creation_basis`** — a new `V_L` corollary for the landed (8.4.7)-factor
  Sugawara, from the landed `MM.FLM87FC.sugawaraFoldIdentity` (`MMFLM87SugawaraFoldClose.lean:320`,
  which closed the old `SugawaraFoldIdentity` gap) together with
  `MM.FLM87ZBK.fockSugawaraVL_comm_modeVL` (`MMFLM87ZeroModeBracket.lean:507`); this composition was
  not landed before.

**The exact remaining residual for `htrans`.**  The corrected (`Yzero`) state half is **not** assembled:
the file lands the named `Prop`

    ZeroPartCreationCommutes : ∀ k i,
      MM.FLM87ZD.zeroPart (−1) * modeVL (bL i) (−(k:ℤ))
        − modeVL (bL i) (−(k:ℤ)) * MM.FLM87ZD.zeroPart (−1) = 0

together with `correctedStateHalf_of_zeroPart`, which derives `CorrectedStateHalf` (the
`sugawaraModeZero` form, basis labels) from it via the landed decomposition
`sugawaraModeZero n = fockSugawaraVL n + zeroPart n` (`MMFLM87FockZero.lean:225`, `:260`).  Closing
`ZeroPartCreationCommutes` is `MM.FLFock.mode_commutator` at `(−1, −k)` (which vanishes for `k ≥ 1`)
plus `hZeroVL_mul_modeVL_comm` — a `V_L` tensor-factor computation, deliberately left out of this file.
To finish `LMinusOneCreationMode` from there one still needs the `hY` identification
(`MM.FLM87SA.LOf_eq_sugawaraModeZero`) and the step from basis labels to arbitrary `b ∈ L`
(`MM.FLM85LL.modeVL_add` plus integral combinations).  **Translation is therefore two small steps and
one tensor-factor computation from closed.**

**V1 ACCEPTED — `Definitions/MMFLM810StateHalfClose.lean`** (300 lines, namespace `MM.FLM810SHC`, 9
guards; verified by the dispatcher: rebuild from a deleted `.olean` `--- OK ---`, token check at zero,
`FinalCheck.lean` acceptance build `--- OK ---`).  **The state half of (8.10.13) is closed**, and the
translation field's remaining content is now a single named field identity.

**Landed.**  `zeroPartCreationCommutes` (so `MM.FLM87SH.ZeroPartCreationCommutes` **is a theorem**),
`zeroPart_comm_creation` at an arbitrary lattice label, `correctedStateHalf` (from the landed
`correctedStateHalf_of_zeroPart`), the general form `correctedStateHalf_general :
[sugawaraModeZero (−1), modeVL b (−(k:ℤ))] = (k:F) • modeVL b (−(k:ℤ)−1)` for **every `b : L`**, and
— the headline —

    lMinusOneCreationMode : MM.FLM810TG.LMinusOneCreationMode bL form hnd E

so the named state half is a theorem at arbitrary `b`, arbitrary vector, with no symmetry hypothesis.
The bridge that made it work is a new `vacuumNormalized_baseRepaired` (from `baseRepaired_zero` plus
field extensionality) feeding the landed `LOf_eq_sugawaraModeZero`.

**The fourth "already landed under an unguessed name" case, and the rule is now firm.**  The route the
brief proposed (`MM.FLFock.mode_commutator` at `(−1,−k)` plus `hZeroVL_mul_modeVL_comm`) was
**superseded**: the target already existed as `MM.FLM87ZBK.zeroPart_comm_modeZ`
(`Definitions/MMFLM87ZeroModeBracket.lean:257`), for **all** `(n,m)` and all labels, so neither
computation was needed and `k = 0` is covered too.  The rule, now stated three times in this ledger and
once in `AGENTS.md`: **before proving an identity, grep all of `Definitions/` for its statement shape**,
not the files whose names suggest it.

**The translation field, exactly.**  `TranslationOnStateData` is **not** a theorem, and the brief's hope
that the state half would yield it is **wrong**: `translationOnStateData_iff_translationResidual`
reduces the predicate to a **field** identity,

    TranslationResidual (bL) (form) (hnd) (E) : Prop :=
      ∀ D m, Yzero (baseRepaired E) (sugawaraModeZero (−1) (stateOfVL D)) (−m−1)
               = (dFold (datumFactorsZero D) (baseRepaired E D.a)).fam (−m−1)

whose two missing halves are named and independent of the state half: (i) the **base case**
`D = ⟨[], a⟩`, *equivalent* to the unproved `MM.FLM84LT.generalStateDeriv`
(`MM.FLM810TG.baseStateTranslation_iff_generalStateDeriv`); (ii) the **creation intertwining**
`Yzero (α_b(−n) w) = actionFamily (factor n b) (Yzero w)` at arbitrary `w`
(`MM.FLM810TRS.translationFieldHalfStatement`, unproved; the only landed instance is
`MM.FLM810TR.Yzero_modeVL_eq_actionFamily`, the single case `α_a(−1) ι(a)` at the model base).

**A tested trap, recorded.**  The Finset/`sum_repr` route for basis-labels-to-arbitrary-`b` is a trap in
this pin: `Module.End F (VL F L)` carries `LinearMap.instSub`/`LinearMap.instSMul`, so
`sub_eq_iff_eq_add'`, `smul_sub`, `Finset.sum_sub_distrib` and `sub_self` do **not** fire, and
`rw [modeVL_add]` rewrites only the first index it matches (the explicit `n` binder leaves one
metavariable).  The working route is `Submodule.span_induction` on `bL.span_eq` in the **addition** form,
recovering the subtraction form at the vector level.

---

### Batch 163 — the owner's standing instruction: **do not stop until Theorem 8.10.2 is complete**

Owner's instruction (2026-09-20, on leaving the session): *"Theorem 8.10.2 没完成不要结束"* — the campaign
is **not** to be concluded until Theorem 8.10.2 is done, definitions aligned to the book.  Two tasks were
dispatched, on disjoint files, against the two remaining reducible inputs:

* **W1 `Definitions/MMFLM810CreationIntertwine.lean`** (`MM.FLM810CI`) — the **creation intertwining**
  `Yzero (α_b(−n) w) = actionFold [factor n b] (Yzero w)` at arbitrary `w`, i.e. the **field half** of
  `MM.FLM810SHC.TranslationResidual`, generalising the single landed case
  `MM.FLM810TR.Yzero_modeVL_eq_actionFamily` (the case `α_a(−1) ι(a)` at the model base).  Together with
  the base case (`MM.FLM84LT.generalStateDeriv`, equivalent to the base-state translation) this closes
  `htrans`.
* **W2 `Definitions/MMFLM87SugawaraBracketClose.lean`** (`MM.FLM87SBC`) — FLM **(8.7.9)**,
  `MM.FLM87SA.FreeBosonSugawaraBracketZero`: the Sugawara modes' Virasoro bracket with the forced
  central charge `dim 𝔥`, i.e. the **whole remaining content of `hvir`**.  The brief records that the
  two-statement route `MM.FLM87ZBC.virasoroOnStateData_of_named` is **dead** (that file's own §4b
  measures its second statement untenable) and that the Fock-level bracket may already be landed in
  `Definitions/MMFockVirasoro.lean` — which is exactly the file that carried the Fock-level
  `L_commutator_alpha` that batch 161 needed.

**Standing rules for every remaining batch** (restated so that any later reader can audit the campaign):
grep all of `Definitions/` for a statement's **shape** before proving it (four targets have already been
found landed under unguessed names); never use a refuted object (`heisFam`, `modelBase`,
`tailLeibnizResidual`, `IsHomogeneousOp`, `Display8625`, `CrossContraction`, `Contraction8612Ratio`,
`ResidueUMulCompatibilityStatement`, `OperatorDisplay8620Statement`, `RelSpanningStatement`,
`QuadrantResidueStateFree`); verify every `file:line` with `grep -n`; budget discipline (a named residual
beats an unfinished file); and the dispatcher rebuilds every handed-back file from a deleted `.olean`
before accepting it.

**Where Theorem 8.10.2 stands at this entry** (the exact inventory, all names landed and verified):
`hL0` **discharged**; `htrans` = the named `MM.FLM810SHC.TranslationResidual`, whose two halves are the
base case (`generalStateDeriv`) and W1's intertwining; `hvir` = the named
`MM.FLM87SA.FreeBosonSugawaraBracketZero` (W2); `hjac` = `M JacobiOnStateTargets`, open, with the `n = 2`
`ι` maps landed (`MM.FLMIotaVar`) for the `l ≠ 0` faces and the finite-support carrier blocking the
negative-pairing case of the `l = 0` face.

**W2 ACCEPTED — `Definitions/MMFLM87SugawaraBracketClose.lean`** (294 lines, namespace `MM.FLM87SBC`, 6
guards; verified by the dispatcher: rebuild from a deleted `.olean` `--- OK ---`, token check at zero,
`FinalCheck.lean` acceptance build `--- OK ---`).  **Theorem 8.10.2 now has `hvir` discharged as well —
modulo one hypothesis that the intended lattices cannot satisfy, which is the batch's finding.**

**The finding, and it is structural.**  FLM (8.7.9) at `c = dim 𝔥` was **already a theorem**
(`MM.FLM87ZBF.freeBosonSugawaraBracketZero_of_orth`,
`Definitions/MMFLM87ZeroModeBracketFinal.lean:357`, batch 117), from
`hOrth : MM.FLM87ST.OrthonormalBasis bL form`; and `MM.FLM87SG.sugawaraCollapse_iff_orthonormal`
(`:555`) shows the alternative hypothesis `SugawaraCollapse` is **equivalent** to `hOrth`.  So nothing
new was needed at `c = dim 𝔥` — what is genuinely open is the bracket with `hOrth` **removed**.  And:

    not_orthonormalBasis_of_even : (∀ a, 2 ∣ form a a) → (i : κ) → ¬ OrthonormalBasis bL form

**`hOrth` is FALSE on every even lattice** — FLM's (8.7.3) asks for an orthonormal basis of the
`F`-vector space `𝔥 = L ⊗ F` (where scaling by `1/√(b_i,b_i)` is allowed), whereas the repository's
predicate demands it of the **`ℤ`-basis `bL`**, which is unsatisfiable as soon as some `(b_i,b_i)` is
even.  **So the landed producer covers only odd lattices, and not the Leech lattice of Theorem
8.10.2.**  This is the fifth "the target was landed under an unguessed name" case, and the first where
the landed form turns out to be *unusable for the intended object* rather than merely equivalent.

**Landed here.**  `freeBosonSugawaraBracketZero_dimH` (the bracket at `c = card κ`, routed through the
landed `MM.FLM87ZP.freeBosonSugawaraBracketZero_of_named` plus the Fock and zero-mode brackets);
**`virasoroOnStateData_aligned_dimH`** — `hvir` **with no bracket hypothesis**, at
`Yzero (baseAligned E)`, central charge `dim 𝔥` — the brief's assembled form;
**`theorem8102_of_generatorFieldsAligned_dimH`** — **Theorem 8.10.2 with `hvir` and `hL0`
discharged**; `rankStatement_card`; and
`not_freeBosonSugawaraBracketZero_of_centralCharge_ne` (the bracket at any other central charge is
**false**, so `dim 𝔥` is forced).

**The exact residual.**  `MM.FLM87SBC.SugawaraBracketAtDimH bL form hnd := FreeBosonSugawaraBracketZero
bL form hnd ↑(card κ)` **without `hOrth`**.  It is unreachable from the landed route because the bridge
`sugawaraModeZero = fockSugawaraVL + zeroPart` is **false** without `hOrth`
(`MM.FLM87ZDGC.not_generalSugawaraDecomposition`,
`Definitions/MMFLM87ZeroModeDecompGeneralCounter.lean:318`, the rank-one `⟨b,b⟩ = 2` counterexample at
`n = 1`), and the collapse behind it is equivalent to `hOrth` (batch 118).  Closing it requires a **new
Gram-weighted Sugawara bracket** (dual-basis pairing `[h'_i(m), h_j(n)] = m δ_ij δ_{m+n,0}`); the agent
flagged, as an explicitly unverified idea, that the bracket may use only the cross pairing and so might
be a transport of the orthonormal one along the Gram twist.  **That is the next task on the Virasoro
front.**

**Brief errors.**  The brief called (8.7.9) open — it is landed since batch 117, and **not** in
`MMFockVirasoro.lean`, which carries only the Heisenberg-level `L_commutator_L`; the `V_L`-level
transport is `MM.FLM87ST.fockSugawaraVL_bracket`.  The arbitrary-`c` target is false.  And the
prescribed decomposition silently requires `hOrth` — which is where the entire gap sits.  The brief's
route audit was otherwise right: the `MM.FLM87ZBC` named route is dead
(`:245`, `:257`) and the direct route is alive.

**W1 ACCEPTED — `Definitions/MMFLM810CreationIntertwine.lean`** (417 lines, namespace `MM.FLM810CI`, 12
guards; verified by the dispatcher: rebuild from a deleted `.olean` `--- OK ---`, token check at zero,
`FinalCheck.lean` acceptance build `--- OK ---`).  **The creation intertwining is now a theorem at
arbitrary `w`** — the induction step of the translation field.

**Landed.**  `Yzero_modeVL_eq_actionFamily_nat (base) (n ≥ 1) (b) (w)`:

    Yzero base (modeVL b (−(n:ℤ)) w) = actionFamily (factorFactorZero n hn b) (YzeroField base w)

at **arbitrary `w`** — together with the `ℕ+` form, the lattice-generator specialisation at an arbitrary
base, the `α_a(−1) ι(a)` case at an arbitrary base, and a recovery of the landed model-base theorem.
The supporting machinery is new and worth naming: `Yzero_trunc` — **(8.5.6) at an arbitrary vector**
(`Yzero base w` is lower truncated, from `Submodule.span_induction` over the landed
`MM.FLM85Un.spanningStatement`) — which is what makes the bundling `YzeroField` possible;
`actionFamilyZero` (linearity of `w ↦ actionFamily X (YzeroField base w)`); `actionPrescriptionTAZero_cons`.

**A type correction the brief needed.**  `FLM85TA.actionFamily` consumes a **`TruncatedField`**, not a bare
coefficient family, so the brief's literal shape `actionFold [factor n b] (Yzero w)` is **not typable**;
the file introduces `YzeroField base w` (whose `fam` is `Yzero base w` by `rfl`) as the object the
action consumes.  That bundling is exactly what `Yzero_trunc` buys.

**What does *not* follow, and this is the honest state of the translation field.**  `TranslationResidual`
is **not** proved: the intertwining is its **induction step** only.  The base case is landed as the named
`Prop` `BaseResidual`, with two machine-checked reductions — `baseResidual_iff_actionFamily_base` (the
base case is equivalent to the base-field identity `actionFamily (baseFactor a) (base a) = dField (base a)`,
i.e. to the still-open `MM.FLM84LT.generalStateDeriv`) and `translationResidualBase_iff_baseResidual` (the
`sugawaraModeZero` spelling).  The remaining `dFold` assembly needs two pieces the file names and does
**not** claim: (a) the factor recurrence `dFactor (factor n b) = n • factor (n+1) b` as bundled `Factor`s
(`MM.FLM810TRF.factorFamilyZero_deriv` is for the bare family; there is no scalar action on `Factor` and
no `actionFamily` congruence in `Factor.bound`), and (b) the derivation form of the state half at
`sugawaraModeZero`.  **So `htrans` is now: the induction step (proved) + the `dFold` assembly (two named
pieces) + the base case (equivalent to `generalStateDeriv`).**

**Brief errors.**  The landed case the brief called unique
(`MM.FLM810TR.Yzero_modeVL_eq_actionFamily`) is really `MM.FLM810TC.…`
(`Definitions/MMFLM810TranslationCorrected.lean:121`), and a **second** instance was already landed under
an unguessed name — `MM.FLM810TRS.Yzero_modeVL_iotaVL`
(`Definitions/MMFLM810TranslationDerivationState.lean:74`, the lattice-generator case `α_a(−n)ι(c)`,
`n ≥ 1`), so the brief's ladder would have reproved it.  That is the fifth unguessed-name case, and it is
a reminder that the grep rule must be applied to the **statement shape**, not to a guessed identifier.

**X1 ACCEPTED — `Definitions/MMFLM87GramSugawara.lean`** (300 lines, namespace `MM.FLM87GS`, 9 guards;
verified by the dispatcher: rebuild from a deleted `.olean` `--- OK ---`, token check at zero,
`FinalCheck.lean` acceptance build `--- OK ---`).  The task was dispatched to close the Virasoro residual
for the Leech case; it came back with a **refutation of the only proposed shortcut**, which is the result
that matters.

**The Gram-twist transport idea is FALSE, in both readings, and both are now theorems.**

* `form_eq_of_gramTwist` — if an `F`-linear equivalence `S` of `V_L` conjugates the `form`-modes to the
  `form'`-modes (the generator-level twist `h_i ↦ h'_i`, with **no** algebra-map hypothesis), then
  `form' = form` on all lattice pairs.  The mechanism is exact: conjugation preserves the central
  coefficient `m⟨a,b⟩` of the CCR (7.1.5) `MM.FLM85MC.modeVL_mul_modeVL_eq`, so the form cannot change.
* `not_gramTwist_rankOne_even` — no such equivalence between `MM.FLM87ZDGC`'s `formOne` (`⟨b,b⟩ = 1`) and
  `formTwo` (`⟨b,b⟩ = 2`) over `ℚ` (it would force `2 = 1`).
* `not_exists_rat_two_mul_sq_eq_one` + `not_orthonormalVector_rankOne_even` — `2c² = 1` has no rational
  solution, so the "extend `F` if necessary" reading of FLM (8.7.3) needs `√2` and therefore **is**
  (8.7.3) rather than a way around it; and independently, `sugawaraModeZero` is built from the (8.4.16)
  folds of the **`ℤ`-basis** fields, so an orthonormal `F`-basis of `𝔥` could not be fed to it without
  redoing (8.7.5) for that basis.

**The residual is unchanged and now precisely located.**  `MM.FLM87SBC.SugawaraBracketAtDimH bL form hnd`
(FLM (8.7.9) at `c = card κ`) is **not** a theorem, and the repository has **no route** to it: the only
landed producer demands exactly the refuted hypothesis (the measured first error is
`don't know how to synthesize placeholder for argument 'hOrth'`, goal `⊢ OrthonormalBasis bL form`), and
`MM.Heisenberg.L_commutator_L` is stated only for the **identity** Gram while `MM.FLM87ST.fockSugawaraVL`
carries **no form argument** at all (it is the orthonormal-form Sugawara in the `bL` coordinates).  So the
general case is **not** a transport: it needs a new **two-contraction** mode computation — the dual pair
`(b_i, u^i)` of FLM (8.7.2) — and the file names the smallest open case as
`RankOneEvenSugawaraBracket` (`SugawaraBracketAtDimH` at `MM.FLM87ZDGC.formTwo` over `ℚ`, with
`rankOneEvenSugawaraBracket_iff : … ↔ FreeBosonSugawaraBracketZero … ↑(card Unit)`, `Iff.rfl`, so the
named case is exactly the target and not a weakening).  Landed starting points for that computation are
cited in the file's §5: `MM.FLM810CV.dualCoeff`, `MM.FLM810ZM.dualHBasis_sum`,
`MM.FLM810ZMT.dualCoeff_sum_smul_hBasis`, the unconditional decomposition
`MM.FLM87ZDG.sugawaraModeZero_apply_eq_regular_add_zeroCorr` with `MM.FLM87SG.zeroCorrPart`, and the
Gram-weighted `[Z,Z]` bracket `MM.FLM87ZBK.zeroPart_comm`; **the missing piece is the bracket of the
regular inverse-Gram part.**

**Brief errors.**  `dualCoeff` is `MM.FLM810CV` (`Definitions/MMFLM810ConformalVector.lean:98`), not
`MM.FLM87SA`; the flagged shortcut was not merely unverified but **false**; and the brief's
"Fock-level first" is not the cheap step it looks like, for the reason above.  The brief's other
references checked out.

**Y2 ACCEPTED — `Definitions/MMFLM810DFoldAssembly.lean`** (355 lines, namespace `MM.FLM810DFA`, 9
guards; verified by the dispatcher: rebuild from a deleted `.olean` `--- OK ---`, token check at zero,
`FinalCheck.lean` acceptance build `--- OK ---`).  **`TranslationResidual` is now ASSEMBLED** — the
induction, the intertwining and the fold bookkeeping are no longer part of the translation gap.

**Piece (1) LANDED, and its true shape differs from the brief's.**

    factorSmul (n : ℕ) (X : Factor F L) : Factor F L            -- the MISSING scalar action on Factor
      (fam := (n:F) • X.fam, nil/vac inherited at the SAME bound, since the ^+ half of n•X is n•(X^+))
    dFactor_factorFactorZero : dFactor (factorFactorZero n hn b) = factorSmul n (factorFactorZero (n+1) … b)
    actionFamily_dFactor_factorFactorZero : actionFamily (dFactor (factor n b)) Y m
                                          = (n:F) • actionFamily (factor (n+1) b) Y m

The brief's literal `dFactor (factor n b) = n • factor (n+1) b` is **not type-correct** (there is no
`SMul` on `Factor` — that is why `factorSmul` had to be built), and the library's landed recurrence is in
the **chain-rule orientation** `X_{n+1} = (n:ℚ)⁻¹ • deriv X_n`, so the `F`-scalar form needs both the
`ℚ → F` bridge (not `rfl`: the `ℚ`-action on `V_L` is the tensor-product left action, proved here via
`nsmul`) and a cancellation of `n`.  `MM.FLM810CI` §5(a) is thereby closed; the bound issue is resolved by
`MM.FLM85AA.tailWindow_eq_of_kills` window comparison rather than by a congruence.

**Piece (2) NAMED, not discharged — and the reason is a finding.**  `SugawaraDerivation` is landed as a
named `Prop`, with `sugawaraDerivation_iff_lMinusOneCreationMode` showing it is **exactly** the landed
`MM.FLM810TG.LMinusOneCreationMode`, plus the consumed form `Yzero_sugawaraModeZero_modeVL` conditional on
it.  It could not be discharged here because **the only landed proof of the state half carries
`hOrth : OrthonormalBasis`** (`MM.FLM810SHC.lMinusOneCreationMode`, verified by the dispatcher at
`Definitions/MMFLM810StateHalfClose.lean:226`--`:228`) — and `hOrth` is false on even lattices.  So the
translation field and the Virasoro field share **one and the same** open input: the non-orthonormal
Sugawara computation.  A useful by-product: **the state half is `E`-independent** (the
`LOf`/`sugawaraModeZero` bridge carries no central extension).

**The assembly LANDED.**

    translationResidual_of_baseResidual : (hs : form symmetric) → BaseResidual (baseRepaired E)
                                        → SugawaraDerivation → MM.FLM810SHC.TranslationResidual
    translationResidual_iff_baseResidual : TranslationResidual ↔ BaseResidual
    translationOnStateData_of_baseResidual : MM.FLM810RR.TranslationOnStateData

**So `htrans` is now exactly two named inputs**: (i) `MM.FLM810CI.BaseResidual (baseRepaired E)`,
equivalent to the unproved `MM.FLM84LT.generalStateDeriv`; and (ii) `SugawaraDerivation` — i.e. the state
half **without** `hOrth`, which is discharged by *closing `MM.FLM87SBC.SugawaraBracketAtDimH`*
(batch 165's task Y1).  The two remaining fronts have therefore **merged into one**: the
two-contraction Sugawara computation, plus the single base-case identity.

**Y1 ACCEPTED — `Definitions/MMFLM87SugawaraTwoContraction.lean`** (509 lines, namespace `MM.FLM87TC`, 7
guards; verified by the dispatcher: rebuild from a deleted `.olean` `--- OK ---`, token check at zero,
`FinalCheck.lean` acceptance build `--- OK ---`).  The non-orthonormal Sugawara computation — now the
**single shared blocker** of the Virasoro and translation fields — has its `[L, mode]` half landed and its
remaining computation named to the lemma.

**LANDED: the Witt action at an arbitrary basis, no `OrthonormalBasis`.**

    sugawaraModeZero_mul_modeZ_sub : [sugawaraModeZero m, modeZ (bL l) n]
                                   = (((−n : ℤ)) : F) • modeZ (bL l) (m + n)

for an **arbitrary `ℤ`-basis** and an **arbitrary symmetric form** — i.e. the first half of FLM (8.7.9),
and the first landed object in the repository that does not carry `hOrth`
(`MM.FLM87ZBK.fockSugawaraVL_comm_modeVL` and `MM.FLM810SHC.correctedStateHalf_general` both do).
Supporting landings: the corrected (7.1.33) **CCR** `modeZ_mul_modeZ_sub` (including the lattice zero
mode), the one-contraction pair identity `normalOrderedHeisPairZero_mul_modeZ_sub`, the Gram collapses
`inverseGram_collapse_left`/`_right` (+`_smul`), and the two contraction coefficients
`gramContraction`, `gramContraction_transpose`, `normContraction`.

**The exact residual, named**: `MM.FLM87TC.RegularInverseGramBracket` — FLM (8.7.9) for
`MM.FLM87SA.sugawaraMode` (the regular inverse-Gram part, the first summand of the landed unconditional
decomposition `MM.FLM87ZDG.sugawaraModeZero_apply_eq_regular_add_zeroCorr`) at `c = card κ`.  **The one
computation that remains is the double commutator**
`[:b_i(m−p)b_j(p):, :b_k(n−q)b_l(q):]`, where both contractions occur at once.  A second, separate gap was
measured: the brackets mixing `sugawaraMode` with `zeroCorrPart` are **not even statable today**, because
`zeroCorrPart` is a plain function rather than an endomorphism and the landed decomposition is pointwise
— a linearity statement for it is missing from the repository.

**The contraction picture, corrected from the brief's.**  In the bracket the brief asked for first, the
**Gram contraction** `Σ_j (M⁻¹)_{ij}⟨b_j,b_l⟩ = δ_{il}` is the *only* contraction, and it enters **twice**
— at `k = −n` against the first factor of `:b_i(m−k)b_j(k):` (coefficient `M_{il}`) and at `k = m+n`
against the second (coefficient `M_{jl}`) — each half contributing `−n·b_l(m+n)/2`, so that the outer
`(1/2)` of (8.7.5) is exactly what makes the halves sum to `−n·b_l(m+n)` rather than `−2n·b_l(m+n)`.
The **trace** `Σ_{ij}(M⁻¹)_{ij}⟨b_i,b_j⟩ = card κ` does **not** occur in `[L,mode]` at all: it is the
contraction *inside* one normal-ordered pair, so it can only appear in the self-bracket's double
commutator — **and that is where the central charge `dim 𝔥` comes from**.

**More brief corrections of record.**  `MM.FLM87GS` is not a file name (the namespace lives in
`Definitions/MMFLM87GramSugawara.lean`) — `grep` beats a guessed path.  The transport obstruction is
stronger and more concrete than batch 164's `form_eq_of_gramTwist` alone: `modeZ`'s dependence on `form`
sits **only in the annihilation half** (creation for `m < 0`, `derivOf (hForm …)` for `m > 0`), so any
transport would have to scale creation by `1` and annihilation by `λ` — no conjugation can do that, so
the two-contraction computation is unavoidable for coefficient-level reasons as well.  And
`RankOneEvenSugawaraBracket` is **not** the easier first milestone: `κ = Unit` does not simplify the
double commutator, because the difficulty is the normal-ordering/double-contraction algebra, not the rank
— so "smallest case first" buys no shorter proof here.

**Z1 ACCEPTED — `Definitions/MMFLM87DoubleCommutator.lean`** (346 lines, namespace `MM.FLM87DC`, 3
guards; verified by the dispatcher: rebuild from a deleted `.olean` `--- OK ---`, token check at zero,
`FinalCheck.lean` acceptance build `--- OK ---`).  The double commutator is classified, and the task
**found a shorter route to (8.7.9)** that avoids it.

**Landed: the one-contraction layer for the double normal-ordered pair.**
`mul_mul_mode_sub` (the plain-product form, exact) and
`normalOrderedHeisPairZero_mul_normalOrderedHeisPairZero_sub` (the normal-ordered form with the scalar
`doubleContraction`), with the supporting `pairDefect`/(D) in both directions,
`normalOrderedHeisPairZero_eq_mul_sub`, `mode_mul_mode_eq_pairZero_add`, `commutator_mul_mul`,
`sub_smul_one_mul_smul_one`.

**The classification, and it corrects the brief's picture** (with `P_cd(N,r) = :b_c(r)b_d(N−r):`,
`M_cd = ⟨b_c,b_d⟩`, and `[AB,CD] = A[B,C]D + AC[B,D] + [A,C]DB + C[A,D]B`): exactly **four** terms, one per
index combination — `(j,k)` coefficient `(m−p)M_jk`, delta `m−p+q = 0`, leftover `b_i(p)b_l(n−q)` (cross);
`(j,l)` coefficient `(m−p)M_jl`, delta `m+n−p−q = 0`, leftover `b_i(p)b_k(q)` (**transpose-Gram**);
`(i,k)` coefficient `pM_ik`, delta `p+q = 0`, leftover `b_l(n−q)b_j(m−p)` (cross); `(i,l)` coefficient
`pM_il`, delta `p+n−q = 0`, leftover `b_k(q)b_j(m−p)` (**Gram**).  There is **no fifth "no contraction"
term** (the four-fold product cancels identically), and re-ordering the leftovers into normal-ordered
pairs adds **one scalar** `doubleContraction` = `c₁z₁+c₂z₂+c₃z₃+c₄z₄`, where each `z_r` is the defect
`Q−P` of that leftover — **and that is the only place two contractions meet**: it is supported on
`m+n = 0`, its index patterns are `M_il M_jk` and `M_ik M_jl`, and the inverse-Gram contraction of those
is `Σ_{ij}(M⁻¹)_{ij}M_{ij} = card κ` (`MM.FLM87TC.normContraction`).  The agent also verified the end
result independently in an exact Weyl-algebra model (156816 cases, no failures) and checked
`[L(m),L(n)] = (m−n)L(m+n) + (m³−m)·card κ/12·δ` on the Fock model.

**The route correction, which is the batch's most useful finding.**  The double commutator is **not** on
the shortest path.  The product rule `[L,XY] = [L,X]Y + X[L,Y]`, the landed Witt action
`MM.FLM87TC.sugawaraModeZero_mul_modeZ_sub`, and the defect (D) give, **with no pair–pair commutator at
all**,

    [L(m), P_kl(n,q)] = −q·P_kl(m+n, m+q) − (n−q)·P_kl(m+n, q) + δ(q)·1

with `δ` supported on `m+n = 0`, `δ(q) = −q(m+q)M_kl` for `q ≤ 0 < m+q` and `+q(m+q)M_kl` for
`m+q ≤ 0 < q` (exactly `|m|` values of `q`).  That is landed as the named `Prop`
`MM.FLM87DC.PairVersusSugawaraBracket`.  After it, only two steps remain for `SugawaraBracketAtDimH`:
(i) the `q`-sum reindexing with `MM.FLM87ZDG.sugawaraModeZero_apply_eq_sum_pairZero` (the two `P`-terms
give `(m−n)L(m+n)`), and (ii) the finite interval sum `Σ_q δ(q) = (m³−m)M_kl/6` together with the collapse
`Σ_{kl}(M⁻¹)_{kl}M_{kl} = card κ`, giving the central term `(m³−m)·card κ/12`.  **This bypasses the
double commutator entirely** — the classification above is now the record of a computation that does not
need to be assembled.

**Brief errors of record**: the brief's "the Gram enters twice at `k = −n` and `k = m+n` with coefficients
`M_il` and `M_jl`" is batch 165's *`[L,mode]`* bookkeeping, not the pair–pair commutator (where the
Gram-type pairings sit at `p+n−q = 0` and `m+n−p−q = 0`, with two further **cross** contractions `M_ik`,
`M_jk`); the "double contraction" is one delta coefficient times a normal-ordering defect, **not** a
product of two deltas; and `RegularInverseGramBracket` is **not** an intermediate for the corrected
`SugawaraBracketAtDimH`, because the mixed `zeroCorrPart` brackets are not formal — so the residual
targets the corrected case directly.  The file is 346 lines (the coefficient layer is in, the assembly
deliberately is not; nothing was weakened).

**A1 ACCEPTED — `Definitions/MMFLM87SugawaraShortRoute.lean`** (1094 lines, namespace `MM.FLM87SR`, 15
guards; verified by the dispatcher: rebuild from a deleted `.olean` `--- OK ---`, token check at zero,
`FinalCheck.lean` acceptance build `--- OK ---`).  **THIS IS THE CAMPAIGN'S LARGEST SINGLE RESULT: FLM
(8.7.9) is proved at an arbitrary basis with NO `OrthonormalBasis`, and with it the Virasoro field of
Theorem 8.10.2 is closed for the Leech lattice, and the state half of the translation field falls too.**

**Landed, in the order of the route.**

* `pairVersusSugawaraBracket_holds (bL) (form) (hnd) (hs) : MM.FLM87DC.PairVersusSugawaraBracket` —
  batch 166's residual is **a theorem**, from the product rule + the landed Witt action
  `MM.FLM87TC.sugawaraModeZero_mul_modeZ_sub` + the defect (D).  **The double commutator is not used at
  all** — batch 166's classification stands as the record of a computation that turned out to be
  unnecessary.  Helpers: `mul_mul_sub_mul_mul`, `pairDefect_add_reversePairDefect`, and a **new
  reverse-order** (D) `reverse_mode_mul_mode_eq_pairZero_sub`.
* **Summation (1)**: `finsum_comp_add` (finsum over `ℤ` is translation-invariant), `finsum_pairZero_shift`,
  `finsum_pairZero_two_terms : Σᶠ_q ((−q)•P(m+n,m+q)v − (n−q)•P(m+n,q)v) = (m−n) • Σᶠ_q P(m+n,q)v`.
* **Summation (2)**: `straddle`, `pairVersusSugawaraDefect_eq_straddle`, `antiQuad`,
  `sum_range_quadratic`, `finsum_straddle : Σᶠ_q straddle m q = (m³−m)/6`, and
  `finsum_pairVersusSugawaraDefect : Σᶠ_q δ(q) = (if m+n = 0 then (m³−m)·(6:F)⁻¹·M_kl else 0)`, with
  `δ` supported on exactly the `|m|` integers strictly between `0` and `−m` (uniform window
  `min (1−m) 1 ≤ q ≤ max 0 (−m)`), equal to `−q(m+q)M_kl` on `q ≤ 0 < m+q` and `+q(m+q)M_kl` on
  `m+q ≤ 0 < q`.
* **THE HEADLINE**: `sugawaraBracketAtDimH (bL) (form) (hnd) (hs) : MM.FLM87SBC.SugawaraBracketAtDimH`
  — i.e. `FreeBosonSugawaraBracketZero` at `c = card κ` **with no `hOrth`**.  The assembly is the
  pointwise expansion `sugawaraModeZero_apply_eq_sum_pairZero` + `pairVersusSugawaraBracket` + the two
  summations + `MM.FLM87TC.normContraction`, with the central charge `(m³−m)·card κ/12` obtained exactly
  as batch 166 predicted: the `q`-sum reindexes (`q ↦ q+m`) to `(m−n)·Σ_q P_kl(m+n,q)`, which the outer
  inverse-Gram collapse turns into `(m−n)L(m+n)`, and `Σ_{kl}(M⁻¹)_{kl}M_{kl} = card κ` supplies the
  central term.
* **Both payoffs reached**:
  `virasoroOnStateData_aligned_dimH_noOrth : VirasoroOnStateData … (Yzero (baseAligned E)) … (card κ : ℚ)`
  (the `hOrth`-free form of batch 163's `virasoroOnStateData_aligned_dimH`, via the landed
  `MM.FLM87SA.virasoroOnStateData_of_freeBosonBracketZero`), **and** the state half:
  `correctedStateHalf_basis`, `correctedStateHalf_noOrth`,
  `sugawaraDerivation : MM.FLM810DFA.SugawaraDerivation …`, and
  `lMinusOneCreationMode_noOrth : MM.FLM810TG.LMinusOneCreationMode …` — **no `hOrth` anywhere**, built
  from the `m = −1` Witt action plus `ℤ`-linearity of `modeVL`, then the landed
  `sugawaraDerivation_iff_lMinusOneCreationMode`.

**Consequences for Theorem 8.10.2, and this is the state to resume from.**

| input | status |
| --- | --- |
| `hL0` (8.10.12) | **discharged** (batch 157) |
| `hvir` (8.10.9) | **discharged for arbitrary lattices** — `sugawaraBracketAtDimH` has no `hOrth`, so batch 163's `hOrth`-carrying form is superseded; |
| state half of `htrans` (8.10.13) | **discharged, no `hOrth`** — `lMinusOneCreationMode_noOrth`; |
| `htrans`'s base case | **open**: `MM.FLM810CI.BaseResidual`, equivalent to `MM.FLM84LT.generalStateDeriv`; |
| `hjac` (8.10.8) | **open**: `JacobiOnStateTargets`, no producer for any field; the `n = 2` `ι` maps are landed (`MM.FLMIotaVar`) for its `l ≠ 0` faces, and the `l = 0` face at `⟨a,b⟩ ≤ −2` needs a carrier that can express a contraction series. |

**A new tracked item**: because `hOrth` is now gone from the Virasoro route, batch 163's
`MM.FLM87SBC.virasoroOnStateData_aligned_dimH` / `theorem8102_of_generatorFieldsAligned_dimH` (which
carry `hOrth`) should be **restated** against `MM.FLM87SR.virasoroOnStateData_aligned_dimH_noOrth`, so that
`theorem8102_of_generatorFieldsAligned` has one fewer input.  That restatement is mechanical and is the
first task of the next batch.

**Brief corrections of record**: `pairVersusSugawaraBracket` cost ~90 lines, not the estimated ~40 (it
needs the new reverse-order defect and a vector-level finish); summation (1) does **not** need
`sugawaraModeZero_apply_eq_sum_pairZero` (it is about `normalOrderedHeisPairZero` alone); the working
support window is `min (1−m) 1 ≤ q ≤ max 0 (−m)`; `MM.FLM810DFA` is a namespace (the file is
`Definitions/MMFLM810DFoldAssembly.lean`); the state half was **not** derivable from the landed material
(`MM.FLM810SHC.correctedStateHalf` carries `hOrth`), so its basis-label case had to be redone from the
Witt action; and in this pin `rw [← smul_sub]`, `rw [← sub_smul]` and `rw [← finsum_smul]` **fail even at
the `V_L` level** (`smul_add` fires at the `End` level, but `smul_sub` does not) — the workaround is a
forward `rw [← h, smul_sub]` or `simp only`.  The file blew the 300-line budget (1094) because all four
items had to live in the single permitted file.

**B1 ACCEPTED — `Definitions/MMFLM810TheoremNoOrth.lean`** (300 lines, namespace `MM.FLM810TNO`, 6
guards; verified by the dispatcher: rebuild from a deleted `.olean` `--- OK ---`, token check at zero,
`FinalCheck.lean` acceptance build `--- OK ---`).  **Theorem 8.10.2 now has exactly TWO open inputs.**

**The boundary, with the `hOrth` gone** (verbatim):

    theorem8102_of_generatorFieldsAligned_noOrth (hnd) (hs) (E) (S) (hpd) (hW) (hfin)
      (hjac : JacobiOnStateTargets bL form (vertexOpOfField (Yaligned bL form E)))
      (hbase : FLM810CI.BaseResidual bL form (baseRepaired bL form E)) :
      Theorem8102 form ((Fintype.card κ : ℕ) : ℚ)

**Exhaustive remaining inputs**: `E`, `S` (data); `hnd` (the non-degeneracy interface — a named
`def … : Prop`, **not** derivable from `hpd`); `hs` (the (7.1.1) symmetry — **not** implied by
`PositiveDefinite`, which constrains only the diagonal); `hpd`; `hW`, `hfin` (inputs of
`theorem8102_of_fields`); **`hjac` (8.10.8) — open, no producer for any field**; and **`hbase` (8.10.13
at its new frontier) — open, equivalent to the landed `MM.FLM84LT.generalStateDeriv`**.

**Discharged, i.e. no longer inputs**: (8.10.9) at `c = card κ`
(`MM.FLM87SR.virasoroOnStateData_aligned_dimH_noOrth`), (8.10.12)
(`MM.FLM810TA2.l0OnWeightGenerators_aligned`), the translation **state half**
(`MM.FLM87SR.sugawaraDerivation`), and `RankStatement` at `card κ` (so the boundary has **no `r`
parameter** — a fact the brief had missed).  No `OrthonormalBasis`, `SugawaraCollapse`,
`tailLeibnizResidual`, `IsHomogeneousOp` or `heisFam` occurs anywhere in the file.

**The residual, reduced to its sharpest form.**  Landed reductions: `baseResidual_iff_baseStateDeriv`
(the residual **is** the honest (8.4.23) at every vector), `generalStateDeriv_iff_baseResidual` (it is the
landed `MM.FLM84LT.generalStateDeriv`), and the sharpest of the three —

    def CreationModeStep (E) : Prop :=
      ∀ a m b n, 1 ≤ n → ∀ w, DerivDefect E a m (modeVL b (−n) w)
                                = modeVL b (−n) (DerivDefect E a m w)

with `baseStateDeriv_iff_creationModeStep` and `baseResidual_iff_creationModeStep` proving it **equivalent**
to the residual (the `ι(c)` case is the landed `MM.FLM84LT.generalStateDeriv_iotaVL`; the extension is
`MM.FLM85Un.spanningStatement`).  **So the last input of translation is: the (8.4.23) defect commutes with
every creation mode.**

**The missing lemma, named by the failure itself.**  Two attempts were made and reverted (no `sorry`
left).  The first stalled on the creation-list induction (`rw [ih]` → *"Did not find an occurrence of the
pattern `(DerivDefect …) (stateOfVL …)`"*).  The second, after unfolding the corrected fold, left exactly

    headOp X.fam Y m (α w) + tailOp X Y.fam m (α w) − deriv Y.fam m (α w)
      = α ((headOp X.fam Y m + tailOp X Y.fam m − deriv Y.fam m) w)

and `map_sub` failed on it (*"Did not find an occurrence of the pattern `?f (?a - ?b)`"*).  So the missing
input is **the commutator of a creation mode with the `headOp`/`tailOp` windows of the assembled
three-factor field** — no landed lemma computes `headOp`/`tailOp` at `modeVL b (−n) w`, and the landed
(8.4.23) (`MM.FLM85ZM.deriv_YCoeff_eq_opOpenBox_heisFamZero`) is the **model base**, i.e. the creation
half only (`MM.FLM810TC.deriv_modelBase_ne_flat_add_tail`).

**Brief errors of record**: the Virasoro/`L0` inputs live at `Yaligned E = Yzero (baseAligned E)` while
(8.10.13) is assembled at `baseRepaired E`, and the two are *family*-`rfl`-equal but **not**
definitionally equal as `TruncatedField`s, so the bridge `Yzero_baseRepaired_eq_Yaligned` had to be
landed; the object actually consumed is `translationOnStateData_of_baseResidual`, not the residual-level
form; and `c = card κ` silently discharges `hrank`.

**R1 ACCEPTED (reconnaissance) — `Probe/B169_R1_JACOBI.md` (152 lines).**  The true cost of `hjac` is now
measured, and two campaign assumptions are corrected.

**(1) `JacobiOnStateTargets` is EQUIVALENT to `MM.FLM810.JacobiAxiom` for *every* field** — not a weaker
state-restricted form.  `MM.FLM810JF.FaithfulJacobi Y.coeff u v`
(`Definitions/MMFLM810JacobiFaithful.lean:290`) is `JacobiOnStateTargets`'s body
(`Definitions/MMFLM810RemainingReduction.lean:166`--`:172`) with `w` ranging over **all** of `VL F L`
instead of over `Set.range (stateOfVL …)`; the backward direction is the landed
`jacobiAxiom_of_stateTargets` (`:180`, using the proved `MM.FLM85Un.spanningStatement`,
`MMFLM85Uniqueness.lean:600`), and the forward direction is a one-line definitional restriction (not
landed, and not needed).  **So the state-target restriction buys nothing: `hjac` *is* (8.10.8) at
`Yaligned`.**  The producer ledger is unchanged — six files mention it, all as definitions or hypotheses;
**no producer for any field**.

**(2) Both finiteness conjuncts of `hjac` are FREE at the boundary field.**  Truncation is landed
(`MM.FLM810TA2.componentTruncationAligned`, `Definitions/MMFLM810TheoremAligned.lean:121`) and the
window conjuncts follow from it (`crossWindowVectorwise_of_componentTruncation`,
`iterateWindowVectorwise_of_componentTruncation`, `Definitions/MMFLM810Axioms.lean:445`/`:452`), with the
assembly landed (`faithfulJacobi_of_crossWindowOn_stable`, `:464`).  **The entire cost of `hjac` is
therefore the component identity at a window past both vanishings** — and the window conjuncts cannot be
dropped, since at `N = 0` both sides are the empty sum
(`jacobiIdentity_zero_window`, `Definitions/MMFLM810JacobiNamedBridgeAux.lean:116`).

**(3) The `l = 0` face, verified.**  `MM.FLM86CI.commutator_YFam_eq_zero_of_even`
(`Definitions/MMFLM86CommIota.lean:186`) is unconditional at the even cocycle for `0 ≤ ⟨a,b⟩`, and
`commutator_rankOne_negTwo_ne_zero` (`:338`) is unconditional, with the carrier obstruction confirmed at
`not_locallyFinite2_contrFam_of_neg` (`Definitions/MMFLM86ProductExpansion.lean:351`) and
`ratioCoeff_ne_zero_of_neg` (`:236`).  Residue status: `0 ≤ ⟨a,b⟩` closes; `⟨a,b⟩ = −1` is closed form but
**conditional on `hY` *and* on `h851 = Corollary851` itself**
(`Definitions/MMFLM86IterateIota.lean:265`--`:296`); `⟨a,b⟩ ≤ −2` is unexpanded.  **The precise remaining
inequality** is the residue index `(i:ℤ) + form a b + 1 < 0`, i.e. `(i:ℤ) ≤ -form a b - 2`, where the
state `creationExpVL a (-(i:ℤ)-1-⟨a,b⟩) (anchorVL E a b)` is a positive-degree Fock descendant and **no
landed theorem evaluates `Y` at it** — the named gap `MM.FLM85IS.IterateProductStepOnCreated` =
`MM.FLM86SF.Display8610List` (8.6.10).  Only the top index `i = -⟨a,b⟩-1` is closed
(`iterateOf_lattice_boundary`, `MMFLM86IterateIota.lean:119`).

**(4) A structural surprise, and a correction to this campaign's own map.**  `MM.FLMIotaVar` (the `n = 2`
Hahn carrier landed in batch 156) is **isolated** — imported only by `FinalCheck.lean:386` — because the
§8.10 rationality layer uses a **different, already general-`n`** `ι`: `MM.FLM810.iota`
(`Definitions/MMFLM810Iota.lean:198`, with `iota_injective` at `:254`) and `toAllPerm`
(`Definitions/MMFLM810Theta.lean:121`).  Props 8.10.3/8.10.4/8.10.5/8.10.7 are all `def … : Prop` with a
TODO ledger (`Definitions/MMFLM810Rationality.lean:91`--`:297`) — stated, not proved — and the `ι`-image
existence half is recorded as not provable from repository data
(`Definitions/MMFLM810BridgeProduct.lean:37`--`:50`).  **`FLM-CH8-MAP.md`'s §8.10 rows (`:373`, `:434`,
`:437`--`:441`) are STALE** — they mark "missing" objects that are landed or at least stated; there is no
FLM-CH8 section in `ROADMAP.md`, and the live status is at `ROADMAP.md:4179`/`:3589`.

**(5) Verdict, and it is the honest one.**  `hjac` is **the open Jacobi identity at the honest field with a
currently-UNKNOWN route** — not "several more batches", and not "carrier plus rationality" in any form the
report could measure.  The largest fragment reachable today is the `l = 0` face at the lattice generators
for `0 ≤ ⟨a,b⟩` with the even cocycle, unconditional **up to one definitional link**: the (8.5.2) instance
`YFamOnGenerators … ((vertexOpOfField (Yaligned E)).coeff)`, which is **not landed**
(`Definitions/MMFLM85BaseFieldAligned.lean:46` records it as unproved) but follows from
`vertexOpOfField_coeff` (`Definitions/MMFLM810Fields.lean:180`) + `Yaligned_iotaVL`
(`Definitions/MMFLM85BaseFieldAligned.lean:297`) + `genFam_apply` (`:85`) + one `ring`.  **Recommended
single next batch item**: land that link, plus the `crossBracket`/`commutatorFamily` and
`crossIterate`/`iterateOf` dictionaries at `l = 0`, and then the `0 ≤ ⟨a,b⟩` corner — with the explicit
caveat that this does **not** make Theorem 8.10.2 unconditional.

**A note the agent flagged and the dispatcher confirms**: `Definitions/MMFLM810CreationCommute.lean`
appeared in `git status` during this reconnaissance and is **the sibling task's file, not R1's** — a
reminder that in a shared tree a `git add` must name its paths explicitly (which this campaign's commits
have done throughout).

**D1 ACCEPTED — `Definitions/MMFLM810JacobiZeroFace.lean`** (299 lines, namespace `MM.FLM810JZF`, 6
guards; verified by the dispatcher: rebuild from a deleted `.olean` `--- OK ---`, token check at zero,
`FinalCheck.lean` acceptance build `--- OK ---`).  The reachable fragment of Jacobi is landed, and the
file produced **a producer for `Corollary851` itself** — which the reconnaissance had recorded as having
none.

**Landed.**  `yFamOnGenerators_vertexOpOfField_Yaligned` — the (8.5.2) instance for the boundary field, the
definitional link `Definitions/MMFLM85BaseFieldAligned.lean:46` had recorded as unproved — with the
helper `ZAligned E := (vertexOpOfField (Yaligned E)).coeff` and `ZAligned_apply` (`rfl`).  The three `l = 0`
dictionaries as thin re-exports at the boundary field (`crossBracket_zero_Yaligned`,
`crossIterate_zero_Yaligned`, `crossBracket_zero_iff_corollary851_Yaligned`, the last being the landed
`corollary851_iff_jacobiIdentity_zero_face` with its `hvan` discharged by a new
`bracketTerm_Yaligned_eq_zero_of_window` from `stateVanishesBelow_lattice`).  And the corner itself,
**unconditional**: `commutatorFamily_Yaligned_eq_zero`, **`corollary851_Yaligned_lattice_of_nonneg`** (for
the even cocycle `C`, `(2:F) ≠ 0`, symmetric form, `0 ≤ ⟨a,b⟩` — at **every** window `N`, which is stronger
than the brief's fixed-window sketch), and `crossBracket_zero_eq_crossIterate_Yaligned_lattice_of_nonneg`
(the `l = 0` component identity, LHS zero by `commutator_YFam_eq_zero_of_even`, RHS zero by
`corollary851Residue_eq_zero_of_nonneg`).

**The caution in the brief was checked and cleared, machine-verifiably.**  `YFamOnGenerators` was refuted
at the **model base** (batch 157); the file lands
`Yaligned_ne_Yzero_modelBaseField (E) (weight) (i) (c) (hp : form (bL i) c = -1) : Yaligned E ≠
Yzero (modelBaseField E weight)` and shows the refutation's witness has no analogue here — `Yaligned E`
reads `ι(a)` as `genFam E a = reflectFam (YFam E a)`, `ZAligned`'s definition *is* the (8.5.15) reindex,
so at `n = 0` both sides of the link are `YFam E a 0`.  `modelBase` occurs only inside the two separation
lemmas, never in a statement.

**What is explicitly NOT claimed.**  `⟨a,b⟩ = −1` remains conditional on `hY ∧ h851` (and that pair is
refuted at rank one); `⟨a,b⟩ ≤ −2` needs the (8.6.10) gap
(`MM.FLM86SF.Display8610List` / `MM.FLM85IS.IterateProductStepOnCreated`); `l ≠ 0`, arbitrary `u, v` and
the Fock half are untouched; and **`hjac` is not discharged** — no `FaithfulJacobi`/`JacobiAxiom`/
`JacobiOnStateTargets` statement occurs in the file.  The corner is a real theorem, not a reduction of the
whole identity.

**Brief corrections**: the item-3 sketch fixed the window at 1 (landed is every `N ≥ 1`, plus a producer
for `Corollary851`); and the `⇔` dictionary additionally needed the (8.5.11) finiteness at the generators,
which the file discharges rather than assuming.  The agent also notes that `AGENTS.md`'s build section
still says `tools/build.sh` is "NOT usable here" — **it is stale**, the driver works on this box, and the
claim should be corrected in the next ledger pass.

**C1 ACCEPTED — `Definitions/MMFLM810CreationCommute.lean`** (454 lines, namespace `MM.FLM810CC`, 6
guards; verified by the dispatcher: rebuild from a deleted `.olean` `--- OK ---`, token check at zero,
`FinalCheck.lean` acceptance build `--- OK ---`).  It was dispatched to close the translation base case by
proving that the (8.4.23) defect commutes with creation modes; **it proved that claim false and found the
honest law**, which is a better result than the one ordered.

**The claim is false: the mode SHIFTS the field family, it does not commute with it.**  The classical law
is `[α_b(−n), Y(a,z)] = ⟨b,a⟩ z^{−n} Y(a,z)`, and the file lands its consequences window by window — the
three window commutators the brief asked for are all **false as asked**, and each has a landed exact
discrepancy in its place:

* `modeVL_headWindow_sub` (unconditional discrepancy) and `headWindow_modeVL_sub`
  (`= −⟨b,a⟩ • hW_{m+n}`);
* `modeVL_tailWindow_sub` (discrepancy = the field commutator plus the `^+`-half commutator) and
  `tailWindow_modeVL_sub` under the hypothesis `n ≤ M` (the window must contain the mode index):
  `tW(αw) − α tW(w) = −⟨b,a⟩•tW_{m+n} + (n⟨a,b⟩)•Y_{m+n+1}w`, with the pointwise input
  `modeVL_splitPlus_heisFamZero_sub` (at `k = 0` the (7.1.33) zero mode commutes; for `k ≥ 1` the
  discrepancy is `−n⟨b,a⟩δ_{k,n}`);
* `modeVL_deriv_sub` and `deriv_modeVL_sub` (`= −(m+1)•(⟨b,a⟩•Y_{m+n+1}w)`).

**The honest form of the translation target, and the residual it isolates.**

    window_carry : (hWα − α hW) + (tWα − α tW) − (dα − α d) = −⟨b,a⟩ • (hW_{m+n} + tW_{m+n} − d_{m+n})
    derivDefect_comp_modeVL_sub (hs) :
      DerivDefect E a m (α_b(−n) w) − α_b(−n) (DerivDefect E a m w) = −⟨b,a⟩ • DerivDefect E a (m+n) w
    creationMode_comm_iff_defect_shift (hs) :   -- with ⟨b,a⟩ ≠ 0
      (the fixed-mode commutation)  ↔  (∀ m w, DerivDefect E a (m+n) w = 0)

**So `CreationModeStep` is not an easier equivalent form of the base case — it is *exactly* (8.4.23) at
the shifted index.**  No window rule can close it, and `BaseResidual` / `generalStateDeriv` are unchanged.
The real missing lemma is now named:

    def CurrentFieldComm (E) : Prop :=
      ∀ a b n, 1 ≤ n → ∀ k,
        modeVL b (−n) * (baseRepaired E a).fam k − (baseRepaired E a).fam k * modeVL b (−n)
          = ((form b a : ℤ) : F) • (baseRepaired E a).fam (k+n)

with its pieces partly landed (creation modes commute with `E^-`,
`MM.FLM84CM.modeVL_comm_creationExpVL_int`; `a z^∂` acts on the lattice factor alone,
`MM.FLM84LF.midCoeff_mul_modeVL`) and two parts missing: the same commutator **with `E^+`** (a creation mode
acts on the annihilation exponential as a derivation) and pushing it through the two `convWith`s of `YFam`.

**The indexing trap, and it is the subtlest one this campaign has hit.**  `YFam` is the **reflected**
family (`baseRepaired_fam : (baseRepaired E a).fam k = YFam E a (−k−1)`), so **the shift reverses**: the
honest statement in the windows' indexing is `k ↦ k+n` with scalar `form b a`, while a `YFam`-indexed
`j+n` form is **false** (the correct `YFam` form is `j ↦ j−n`).  The agent's first statement of
`CurrentFieldComm` had the wrong indexing, and the error surfaced not as a type error but as a **`whnf`
timeout** in the assembly — cured by correcting the statement rather than by raising `maxHeartbeats`.
That is the third time in this campaign that a wrong statement has presented itself as a performance
problem, and it is now the standing rule: **a timeout is evidence about the statement first and the
machine second.**

**Brief errors**: the three window commutators are false (only the defect combination is covariant);
"defect commutes" is *equivalent* to (8.4.23) at the shifted index rather than easier than it; the
"suggested state half" (`lMinusOneCreationMode_noOrth`) is **not** the `deriv` half, since
`deriv Y m (αw) = α(deriv Y m w)` would need `Y_{m+1}` to commute with `α`, which is false.  API traps
recorded: there is no `LinearMap.map_zsmul` (use the generic `map_zsmul`) and no
`Module.End.sub_apply` (use `LinearMap.sub_apply`); plain `simp` does not close `−(c•w) = (−c)•w`;
`Finset.sum_congr` must be applied with `apply` (a `refine … fun i hi => ?_` leaves an
`AddCommMonoid ?m` stuck); and `smul_smul` merges rather than splits in this pin.

**J1 ACCEPTED — `Definitions/MMFLM810JacobiFull.lean`** (502 lines, namespace `MM.FLM810JF2`, 12
guards; verified by the dispatcher: rebuild from a deleted `.olean` `--- OK ---`, token check at zero,
`FinalCheck.lean` acceptance build `--- OK ---`).  It was dispatched, on the owner's instruction, to take
Jacobi **in full by the book**; it came back with a **refutation of the boundary's `hjac` as stated** and
the structural reason — which is worth more than the proof would have been.

**HEADLINE: `hjac` is FALSE at an admissible datum.**

    not_jacobiOnStateTargets_ZAligned_trivialCocycle :
      ¬ JacobiOnStateTargets (F := ℚ) basisUnitInt formR (vertexOpOfField (Yaligned … extR))
    not_jacobiAxiom_ZAligned_trivialCocycle :  -- the same, in the `JacobiAxiom` spelling

i.e. **verbatim the boundary's second open input**, at `(ℚ, ℤ, Unit, trivial cocycle, ⟨m,n⟩ = mn)`.  The
theorem's hypotheses do not exclude that datum: `hs` is `mul_comm` and `hpd` holds there (the agent had to
*prove* `positiveDefinite_formR` to say so), and `E` is unconstrained data.  **So "prove 8.10.3/8.10.4/
8.10.5 for `Yaligned E`" is impossible in general** — which is exactly why FLM states (8.6.1) and Theorem
8.6.1 under the cocycle condition `c(a,b) = (−1)^{⟨a,b⟩}`, the same hypothesis batch 156's task T2 had
already isolated from the other side.

**The structural form, and it is the acceptance criterion for any future `hjac` task.**

    cocycleCompatible_of_jacobiOnStateTargets :
      JacobiOnStateTargets … E → 0 ≤ ⟨a,b⟩ → (YFam E a m * YFam E b n) (stateOfVL D) ≠ 0
        → E.c b a * (-1)^⟨a,b⟩ = 1

**The boundary's own input *implies* the (7.2.22) cocycle compatibility it does not assume.**  So the repair
is to **fix the cocycle** — `CocycleCompatible … E`, or `E := C.toCentralExtension h2` for
`C : MM.FLMLattice.EvenCocycle` — before any further Jacobi work.

**A correction to this campaign's own brief, and it is the third time a brief called a `Prop` a theorem.**
`MM.FLM810Rat.Proposition8107` (`Definitions/MMFLM810Rationality.lean:218`) is a **`def … : Prop`** — its
docstring says *"Stated, not proved"* and its TODO ledger is at `:295` — **not** the landed implication this
batch's brief asserted.  So FLM's §8.10 route has **four** inputs (8.10.3, 8.10.4, 8.10.5 **and** the 8.10.7
assembly), not three, and none of them is landed.

**Also landed, for the corrected route**: `RationalityLayer S Y` (the three propositions at one field, as a
single `Prop`); `jacobiAxiom_of_rationalityLayer` (the reduction with 8.10.7 carried as a hypothesis);
`CocycleCompatible` with `cocycleCompatible_evenCocycle`; the unconditional `0 ≤ ⟨a,b⟩` corner **at a
compatible cocycle** (`corollary851_ZAligned_lattice_of_compatible`); and the exact residual
`JacobiResidualAligned S` with `jacobiAxiom_aligned_evenCocycle_of_residual`.  A **new producer for
`Corollary851`** is landed too — `corollary851_of_faithfulJacobi_lattice`, which the reconnaissance had
recorded as having **none** anywhere in the repository; the mechanism is that producer plus the landed
`not_YFamOnGenerators_and_corollary851_rankOne`.

**`Definitions/MMFLM810BridgeProduct.lean:37`--`:50` still holds verbatim**: the `ι`-image existence half is
**not provable from repository data** (`prodCoef` comes from an arbitrary `VertexOp`; no bridge from the
component layer to `ι₁₂`'s image is landed).  What it would need is the `Res_{z₀}` argument producing the
`f` of shape (8.10.51) — i.e. (8.10.8) itself, or an independent rationality input.

**Brief errors**: `Proposition8107` is not a theorem; "the whole of Jacobi reduces to 8.10.3/8.10.4/8.10.5"
is false at arbitrary `E` (refuted above); the named blocker (`BridgeProduct`) is real but not the
operative one at the boundary as stated; the brief missed the then-uncommitted sibling file
`MMFLM810JacobiZeroFace.lean` (the agent re-proved its three small dictionary lemmas locally so that its
file depends only on committed modules); and `MM.FLM810ZMU.Yaligned` names a file that does not exist
(`Yaligned` is `MM.FLM85BA.Yaligned`, `Definitions/MMFLM85BaseFieldAligned.lean:290`, and `Yzero` lives in
`MMFLM810ZeroModeL0Uncond.lean`).

**C2 ACCEPTED — `Definitions/MMFLM810CurrentFieldComm.lean`** (476 lines, namespace `MM.FLM810CFC`, 9
guards; verified by the dispatcher: rebuild from a deleted `.olean` `--- OK ---`, token check at zero,
`FinalCheck.lean` acceptance build `--- OK ---`).  **THE TRANSLATION FIELD OF THEOREM 8.10.2 IS CLOSED.**
The batch estimate made before dispatch (the dispatcher's own answer to the owner's question) was
"two batches, maybe one" — it took **one**.

**The E⁺ commutator, which C1 had named as missing, and the correction.**  The dispatcher predicted it was
already landed as `MM.FLM86TN.modeVL_comm_annihExp_pos`; **that prediction was wrong** — that lemma is the
*positive-index* (annihilation) mode commuting with `E⁺`, not the creation-mode derivation.  What landed is
the real thing:

    modeVL_mul_annihExp_sub (hs) (a b) {n} (hn) (q) :
      modeVL b (−↑n) * annihExp a q − annihExp a q * modeVL b (−↑n)
        = ((form b a : ℤ) : F) • annihExp a (q + ↑n)

with family form `modeVL_annihFam_add` (`i ↦ i−n`).  The engine was the landed
`MM.FLM84CT.expField_comm_of_delta` (the abstract derivation property), cited rather than reproved.

**The convolution step**: `convWith_modeVL_add … : modeVL b (−↑n) ((A⊛B) m v) = (A⊛B) m (modeVL b (−↑n) v)
+ c • (A⊛B) (m−↑n) v` — the (8.5.15) convolution differentiated, with `B` applied first (so the shift has
the same direction), reindexed by the landed `MM.FLM85CP.sum_image_shift_eq` over
`T = supp(B·v) ∪ supp(B·αv) ∪ (supp(B·v)+n)`.

**The headline and the whole chain, from `hs` (7.1.1) alone**:

    currentFieldComm (hs) (E) : MM.FLM810CC.CurrentFieldComm E
    baseStateDeriv_of_currentFieldComm : BaseStateDeriv E
    creationModeStep_of_currentFieldComm : CreationModeStep E
    baseResidual_of_currentFieldComm : MM.FLM810CI.BaseResidual (baseRepaired E)
    generalStateDeriv_of_currentFieldComm (hs) : MM.FLM84LT.generalStateDeriv   -- the long-open Prop
    translationResidual_of_currentFieldComm (hnd) (hs) (E) : MM.FLM810SHC.TranslationResidual
    translationOnStateData_of_currentFieldComm
    theorem8102_of_hs : MM.FLM810TNO.theorem8102_of_generatorFieldsAligned_noOrth with `hbase` DISCHARGED

**So Theorem 8.10.2 now has exactly ONE open input: `hjac` (8.10.8)** — and that one is *refuted* in the
boundary's current form (batch 171's `not_jacobiOnStateTargets_ZAligned_trivialCocycle`), so it must first
be re-scoped to a compatible cocycle (`CocycleCompatible` / `MM.FLMLattice.EvenCocycle`) before any proof
attempt.  The remaining inputs of `theorem8102_of_hs` are data and interface hypotheses only: `hnd`, `hs`,
`E`, `S`, `hpd`, `hW`, `hfin`, plus `hjac`.

**Where the brief was incomplete, and it matters for future chains**: `derivDefect_comp_modeVL_sub` is a
*shift* law, so it does **not** by itself give `CreationModeStep`; what closes it is an induction over the
(8.5.4) creation list with the stronger hypothesis `∀ m', D(m') = 0` — base
`MM.FLM84LT.generalStateDeriv_iotaVL`, step the shift law at `m'` and `m'+n`, spanning
`MM.FLM85Un.spanningStatement`.  No `⟨b,a⟩ ≠ 0` case split is needed (`c = 0` is harmless).  The landed
`baseStateDeriv_iff_creationModeStep` cannot be reused because its IH is at fixed `m`.

**A new API trap, and it cost the batch's budget**: `Module.End F (VL F L)` carries **two
non-syntactically-equal `SMul ℚ` instances** (`LinearMap.instSMul` vs `Algebra.toSMul` of
`Module.End.instAlgebra`), so `rw [MM.FLM84CT.commutator_smul]` **fails** on the goal produced by
`MM.FLM87.annihilationExponent_apply`; it must be applied as `refine (…).trans ?_` (unification, not
matching).  Also measured: `simp only []` errors *"simp made no progress"* when there is no beta-redex;
`sub_self` does not fire on `0 − 0` at type `Module.End` (use `sub_zero`); `Finset.sum_add_distrib` must
follow the pointwise `show` rather than share the `map_sum` `rw`; and `set` for the Finset witnesses had to
become `let`.  The indexing warning did hold (`fam k = YFam (−k−1)`, `j ↦ j−n`) and was load-bearing.

### Batch 173 — the §8.10 substitution σ is CONSTRUCTED (and `SubstForm`'s reformulation is REFUTED); (8.10.73)/(8.10.46)/(8.10.72) land; the ι-bridge is measured; and **the book's Jacobi axiom is §8.8 Theorem 8.8.9, whose proof repeats Theorem 8.6.1**

Owner's directive for this session (2026-09-20): *complete Theorem 8.10.2 following the reference book*.
The dispatcher's own earlier "Route B" — hand-computing the component identity
`crossBracket = crossIterate` face by face — is **withdrawn**: it is not the book's route.  Two fresh
agents, one batch, disjoint new files; both verified here (rebuild from a deleted `.olean`,
`--- OK ---`; `Probe/forbidden_tokens.py` at zero; the `#guard_msgs` guards inside the files; and the
`FinalCheck.lean` acceptance build — see the last paragraph).

**The book's proof of the Jacobi axiom, read from the page images this session, and it corrects the
campaign's own map.**

* Printed 245: Theorem 8.10.2 is the **summary of Chapter 8** ("We have already established a number of
  axiomatic properties of vertex operator algebras ... summarized as follows").  So its (8.10.8) is what
  Chapter 8 established, namely **Theorem 8.8.9**.
* Printed 231, Theorem 8.8.9's proof, verbatim: "**We repeat the proof of Theorem 8.6.1 except for the
  following changes:** ... [Y(A,z₁) x_n Y(B,z₂)] is the coefficient of z₂^{−n−l} in (8.6.24) and the
  Theorem follows as in the earlier proof."  **Jacobi is a repeat of Theorem 8.6.1's proof**, `x_n`
  reducing the pole order (Remark 8.8.10).
* Printed 233, verbatim: "Taking Res_{z₀} leads to **an interesting alternate form of Theorem 8.6.1**:
  [Y(u,z₁),Y(v,z₂)] = Y((Y(u,z₁−z₂) − Y(u,−z₂+z₁))v, z₂). **(8.8.39)**".  So (8.8.39) — the input
  §8.10's Proposition 8.10.3 uses ("the commutator formula ... in the form (8.8.39), easily implies") —
  **is Theorem 8.6.1**, proved independently in §8.6.
* **Correction of record**: `Definitions/MMFLM810Rationality.lean:263`--`:269` says
  `ProductRationality` "needs the `Res_{z_0}` of the Jacobi identity in the form (8.8.39) — i.e. a proof
  of the Jacobi identity, which is what §8.10 is a route *to*".  That is a **circular reading**: the
  `Res_{z_0}` face is Theorem 8.6.1 and does not presuppose Jacobi.  It also explains A2's measurement
  below that the bridge is false for an arbitrary `VertexOp`: the bridge must be stated under Theorem
  8.6.1's hypothesis.
* **So the book has one engine and two continuations, not two routes**: (§8.8) repeat Theorem 8.6.1's
  proof; (§8.10) from the same Theorem 8.6.1, Props 8.10.3 → 8.10.4 → 8.10.5 → Prop 8.10.7.  Both need
  **Theorem 8.6.1 at the repaired honest field, for all states** — the single crux.

**A1 ACCEPTED — `Definitions/MMFLM810RatSubst.lean`** (375 lines against a 300-line target — reported;
namespace `MM.FLM810RSub`, 7 guards).  **The substitution σ of (8.10.66)/(8.10.67) exists**: `polySubst`
on `MvPolynomial (Fin 2) F` (`X 0 ↦ X 0+X 1`, `X 1 ↦ X 1`) and its descent
`locSubst : Loc F 2 →+* Loc F 2` by `IsLocalization.lift`, with `@[simp]` generator equations, the
uniqueness `eq_locSubst_of_fixes_constants`, and `rationalForm_locSubst` (shape (8.10.51) ↦ shape
(8.10.63): the roles of `r` and `t` interchange, the linear form becomes `z_0+z_2`).  The side
condition needed the `SSub`-generator argument (`polySubst_linForm` + `substCoefs_ne` +
`IsLocalization.map_units`), not an `F`-algebra structure.

**HEADLINE: `SubstForm` AS STATED IS TOO STRONG — `SubstForm f h ↔ h = locSubst f` is false, and so are
`Proposition8104b`/`Proposition8105` over a general field.**  `SubstForm`
(`MMFLM810RatCoef.lean:268`) quantifies over **all** ring homs σ with the two generator equations; such a
σ need not fix the constants.  For any field endomorphism φ, the φ-semilinear substitution (the lift of
`eval₂Hom` with constant map `C` followed by φ) satisfies both equations and sends `locC a` to
`locC (φ a)`; the side condition survives because φ is injective.  At `F = ℚ(x)`, `φ(x) = x²` (or
`𝔽_p(t)` and Frobenius) it differs from `locSubst`, so `SubstForm (locC x) (locSubst (locC x))`
fails.  Landed instead: `SubstFormAlg` (the constants-fixing form),
`substFormAlg_iff : SubstFormAlg f h ↔ h = locSubst f`, the unconditional half
`substForm_imp : SubstForm f h → h = locSubst f`, and `substForm_iff_iff_unique` — the brief's
equivalence for all `f,h` is **equivalent to** the uniqueness `∀σ, IsSubstHom σ → σ = locSubst`, which
is exactly what fails.  **The repair for the §8.10 statements is to restate
`Proposition8104b`/`Proposition8105` with `SubstFormAlg`**; for `F = ℚ` the two coincide (`ℚ` has no
nontrivial endomorphism), so the moonshine target is unaffected, but the general-`F` statements are
false as they stand.

**A2 ACCEPTED — `Definitions/MMFLM810ThetaKill.lean`** (250 lines, namespace `MM.FLM810TK`, 5 guards).
(8.10.73) `thetaVanishesOnZOnePoles` is landed verbatim, with the reusable
`isLaurentPolynomialMV_algebraMap_mul_laurentMonomial` and the general forms
`theta2_algebraMap_mul_laurentMonomial` (every `p·z^k`) and `theta2_algebraMap_mul_zUnit_zpow`
(`p·z_1^j`).  (8.10.46) `theta2MulMonomialStatement` **was already a theorem**
(`MMFLM810ThetaZero.lean:262`; the brief was stale) — the new content is the Laurent-polynomial
multiplier form `theta2_mul_of_isLaurentPolynomialMV`.  (8.10.72) is landed as the **existence** form
`recoverySplit_exists : ∃ Wsing Wreg, RecoverySplit Y W Wsing Wreg n`, with explicit threshold
families; **measured: the stated `RecoverySplit` needs no finiteness hypothesis** (threshold
bookkeeping), the finiteness (8.10.5) is named `IterateVanishesAbove`, and the genuinely missing datum
is `IteratePartialSumHomState` — `HomState` is homogeneous, so a finite sum of states of differing
weights has no `HomState`.

**The ι-bridge ledger, measured.**  (a) A bridge **is** landed but **vacuous**:
`MM.FLM810JB.iotaBridge_of_jacobi8108` (`MMFLM810JacobiToBridge.lean:242`) with
`jacobi8108_iff_zeroProduct` (`:130`, char 0 — the landed hypothesis `Jacobi8108` is *equivalent to*
every component product vanishing), so `prodCoef = 0` (`:201`) and `f = 0`.  (b) For an **arbitrary**
`VertexOp` the bridge is **false**: `MM.FLM810RatModel.not_productRationality`
(`MMFLM810RationalityModel.lean:231`), model computation `:203`, obstruction
`hahnSeries_coeff_bounded_above` `:66`.  (c) `ι`'s image is characterised only as a subring
(`realizableAt_iff_mem_range_iota`, `MMFLM810BridgeClosure.lean:90`, closures `:115`--`:241`); **no
support/cone characterisation**, and the only computations of `toAllPerm (ι ·)` are
`iota_laurentMonomial` (`MMFLM810ThetaZero.lean:112`) and `toAllPerm_towerMono`
(`MMFLM810Tower.lean:188`) — there is **no formula for `ι (mk' g D)`**.  (d) `MM.FLMIotaVar.lean` is a
**separate carrier** (`FLMLoc.LocRing`/`AllSeriesN` vs `Loc`/`Tower`), with no lemma linking it to
`MM.FLM810.iota`.  (e) `prodCoef` has exactly **one** landed computation — the counterexample (b); the
lattice `twoVarField` computations (`MMFLM84TwoVarProduct.lean:359`/`:375`/`:385`,
`MMFLM86ProductExpansion.lean:571`) are a **different object**, and there is no
`prodCoef ↔ twoVarField` dictionary.  (f) The only class reachable from landed material is the
square-zero one, landed here as `iotaBridge_of_zeroProduct` (`:216`), characteristic-free.  **All of
this is consistent with the book**: the bridge is a consequence of Theorem 8.6.1, so it cannot hold for
an arbitrary `VertexOp` and its statement must carry that hypothesis.

**Also**: `FinalCheck.lean` **did not import the §8.10 layer at all** (a grep for
`MMFLM810Rationality`/`MMFLM810RatCoef` was empty); the two new modules are now imported
(`FinalCheck.lean:411`, `:412`), so this layer enters the acceptance gate for the first time.
`Probe/` is gitignored — `Probe/briefs/B173-A1-subst.md`, `Probe/briefs/B173-A2-thetakill-bridge.md`
and `Probe/B173_A2_BRIDGE.md` are working files, not deliverables; their content is in this entry and
in `NOTES.md` §70.

**Batch 174 (next), on the book's route**: measure and land **Theorem 8.6.1's display at the repaired
honest field `Yaligned (C.toCentralExtension h2)` for all states** — the single input both of the book's
continuations need.  Specifically: (i) what the batch-152 "one landed `Prop`" of the display is at the
repaired field, and whether it is still the gap; (ii) the (8.4.25) two-point product at the honest field
versus the creation-half `Yzero` form of `MMFLM86ExpProduct`; (iii) the `x_n`/pole-order machinery of
(8.6.17)--(8.6.24) that Theorem 8.8.9's proof repeats.  **Batch 174 (next), on the book's route**: measure and land **Theorem 8.6.1's display at the repaired
honest field `Yaligned (C.toCentralExtension h2)` for all states** — the single input both of the book's
continuations need.  Specifically: (i) what the batch-152 "one landed `Prop`" of the display is at the
repaired field, and whether it is still the gap; (ii) the (8.4.25) two-point product at the honest field
versus the creation-half `Yzero` form of `MMFLM86ExpProduct`; (iii) the `x_n`/pole-order machinery of
(8.6.17)--(8.6.24) that Theorem 8.8.9's proof repeats.  Dispatcher estimate: 4--8 batches for Theorem
8.6.1, 2--4 more for Jacobi via §8.8, 1--2 for the assembly.

---

### Batch 174 — (8.6.3) at the boundary field IS Corollary 8.5.1 at the (8.6.2) states; the honest-field (8.4.25) engine was ALREADY LANDED and is now reindexed to `Yaligned`; and the old-field obstruction does not bite

Two fresh agents, disjoint new files, one batch; both verified here (rebuild from a deleted `.olean`,
`--- OK ---`; `Probe/forbidden_tokens.py` at zero; the in-file `#guard_msgs`; the `FinalCheck.lean`
acceptance build after wiring both in).

**B174-A ACCEPTED — `Definitions/MMFLM810Theorem861Aligned.lean`** (300 lines, namespace
`MM.FLM810T861A`, 5 guards).  The **source-faithful** (8.6.3) at the boundary field, and its exact
residual.  `Source861Aligned (C) (h2) (u' v') (a b) K := Theorem861SourceStatement form
(C.toCentralExtension h2) (ZAligned E) u' v' a b K` (`ZAligned E = (vertexOpOfField (Yaligned E)).coeff`),
`Residual861Aligned := Corollary851 (ZAligned E) (stateOf u' a) (stateOf v' b) K`.

* **HEADLINE: `source861Aligned_iff_residual` — at `1 ≤ K`, the source-faithful (8.6.3) at the boundary
  field IS `Corollary851` (8.5.13) at the (8.6.2) states, both directions.**  So the whole of `hjac` is
  now one named `Prop`, and it is the display (8.5.13), not a §8.6 chain.  (`source861Aligned_iff_core`
  removes (8.6.1) entirely; `source861Aligned_of_residual` / `residual_of_source861Aligned` are the two
  directions; `display861Aligned_iff_residue`, `residue861Aligned_iff_residual` the landed readings.)
* **Discharged at this field**: (8.6.1)'s cocycle half is a *theorem* (`hypothesis861_of_evenCocycle`);
  integrality is free by type; there is **no** base/compatibility hypothesis (`Yaligned E = Yzero
  (baseAligned E)` is a definition); the grading factor is redundant here
  (`sourceResidue861_eq_unshifted`).  **Not** part of the cost: `componentTruncationAligned` — (8.6.3) has
  no finiteness conjunct — and `FaithfulJacobi`, which only *implies* (8.6.3) vectorwise
  (`source861Aligned_vectorwise`; the uniform window is not derivable from `hjac`).
* **Proved corner**: `source861Aligned_iotaVL_of_nonneg` — `u' = v' = 1`, `0 ≤ ⟨a,b⟩`, every `K ≥ 1`,
  unconditional (from `corollary851_Yaligned_lattice_of_nonneg`).
* **The old obstructions do not bite.**  `not_l0OnWeightGenerators_modelBase` and
  `not_translationOnStateData_modelBase` are stated with `hY` about `actionPrescriptionTA …
  modelBaseField`; `Yaligned E` has base `baseAligned E` and `Yaligned_ne_Yzero_modelBaseField`
  separates the fields, so that `hY` is unsatisfiable there — sharper, both refuted predicates are
  **theorems** at the aligned field.  Batch 152's "one landed `Prop`" was `JacobiOnStateTargets`, not the
  display's residual.
* **§8.6's direct route (8.6.4)--(8.6.27), re-read at the repaired field.**  The repair changes exactly one
  item: the (8.5.2) link, refuted at `modelBase` and now the theorem
  `yFamOnGenerators_vertexOpOfField_Yaligned`.  Proved source-faithful: (8.6.4), (8.6.7), (8.6.11),
  (8.6.13), (8.6.14) per monomial, (8.6.22).  Absent: (8.6.5) (multi-variable series of states; `k = 1`
  only), (8.6.23)/(8.6.24) (scalar bookkeeping only), (8.6.25)--(8.6.27).  Stated not proved: (8.6.6),
  (8.6.8)--(8.6.10) — needing the `k`-fold normal-ordered product and the **shifted field
  `Y(a,z+w)`** — (8.6.12)/(8.6.15)/(8.6.17), and (8.6.20)/(8.6.21) whose commutator side *is* (8.6.3)'s
  extraction.  Legacy proxies: (8.6.16), (8.6.18), (8.6.19).  **So FLM's §8.6 route is blocked by three
  missing objects — the multi-variable series of states, the shifted field `Y(a,z+w)`, and the
  `k`-fold normal-ordered product — and this batch bypasses it via Remark 8.8.11**, leaving (8.5.13) at
  the (8.6.2) states as the residual.  Remark 8.3.9's spanning is still not landed.

**B174-B ACCEPTED — `Definitions/MMFLM86ExpProductAligned.lean`** (300 lines, namespace `MM.FLM86EPA`,
7 guards).  The honest-field (8.4.25) two-point product at the lattice states.

* **The P1 deficit is closed definitionally, not by new mathematics**: `Yaligned E (ι(a)) m = YFam E a
  (−m−1)` (`Yaligned_iotaVL_apply`), and `normOrdered2 = creationPair2 ⊛ midBlock2 (ab z₁^∂z₂^∂) ⊛
  annihPair2` — so the grading shift and the annihilation half that B153-P1's `Yzero (modelBaseField …)`
  lacked are simply there; the fix is the reindexing `−(−m−1)−1 = m`.
* Landed at `Yaligned E`: `Yaligned_mul_Yaligned`; `Yaligned_mul_eq_sum_ratioCoeff` (`0 ≤ ⟨a,b⟩`: the
  convolution with the contraction series `ratioCoeff`), `Yaligned_mul_iotaVL_eq_sum` (at `ι(c)`),
  `Yaligned_mul_eq_threshold` (sign-unconditional per-state threshold, from `MM.FLM86RX.threshold_holds`),
  `Yaligned_mul_eq_normOrdered2`; `Yaligned_iotaVL_apply_iotaVL` ((8.5.11)); `commutatorFamily_Yaligned`
  and `commutatorFamily_Yaligned_eq_smul` (`0 ≤ ⟨a,b⟩`: `(1 − c(b,a)(−1)^{⟨a,b⟩}) • product`) and
  `commutatorFamily_Yaligned_eq_zero` (`CocycleCompatible`).
* **The honest-field engine was ALREADY LANDED** — `MMFLM86YFamProduct` (B154-Q1) and `MMFLM86CommIota`
  (B156-T2) — and this file reindexes it; nothing was reproved.  The brief's premise "this task redoes the
  P1 engine" was stale.
* **New named residual `ResidueNegativeAligned (E) := ∀ a b, form a b < 0 → Corollary851 (ZAligned E)
  (ι a) (ι b) ((−(form a b)).toNat)`**, with `residueNegativeAligned_iff_carrier` both directions.
* Remains: (i) `⟨a,b⟩ < 0` has only the per-state threshold form, the operator-level carrier form being
  **refuted** (`not_prodExpTwoVarEnd_rankOne`, `not_locallyFinite2_contrFam_of_neg`); (ii) the outer
  evaluation `YFam E a m (creationExpVL b j (anchorVL E b c))` at Fock descendants is evaluated nowhere
  (B154-Q1 residual (i)); (iii) the Fock half for general `u, v`; (iv) `ResidueNegativeAligned`.

**Where Theorem 8.10.2 now stands.**  Everything left of `hjac` is **`Corollary851 (ZAligned E)
(stateOf u' a) (stateOf v' b) K` for all Fock `u', v' ∈ M(1)`, all `a, b ∈ L`, all windows `K`** (A's
equivalence), plus the boundary restatement at the even cocycle — the old `theorem8102_of_hs` carries `E`
free and is refuted there (`not_jacobiOnStateTargets_ZAligned_trivialCocycle`).  **Batch 175** attacks the
two open halves on disjoint files: **(A)** the Fock half at `0 ≤ ⟨a,b⟩` (arbitrary `u'`, extending
`corollary851_Yaligned_lattice_of_nonneg`; the creation-list induction of batch 172's C2 is the model),
**(B)** the negative-pairing lattice case `ResidueNegativeAligned` (a carrier question: the finite-support
**Where Theorem 8.10.2 now stands.**  Everything left of `hjac` is **`Corollary851 (ZAligned E)
(stateOf u' a) (stateOf v' b) K` for all Fock `u', v' ∈ M(1)`, all `a, b ∈ L`, all windows `K`** (A's
equivalence), plus the boundary restatement at the even cocycle — the old `theorem8102_of_hs` carries `E`
free and is refuted there (`not_jacobiOnStateTargets_ZAligned_trivialCocycle`).  **Batch 175** attacks the
two open halves on disjoint files: **(A)** the Fock half at `0 ≤ ⟨a,b⟩` (arbitrary `u'`, extending
`corollary851_Yaligned_lattice_of_nonneg`; the creation-list induction of batch 172's C2 is the model),
**(B)** the negative-pairing lattice case `ResidueNegativeAligned` (a carrier question: the finite-support
carrier cannot express `contrFam N` for `N < 0`, measured in `not_locallyFinite2_contrFam_of_neg`).

---

### Batch 175 — the Fock half reduces to the Fock MONOMIAL BASIS (and the book's (8.5.13) is the lattice case only); the negative pairing splits into `= −1` and `≤ −2`; and the carrier is NOT the wall — `NegativeChamberExpansion` is

Two fresh agents, disjoint new files, one batch; both verified here (rebuild from a deleted `.olean`,
`--- OK ---`; `Probe/forbidden_tokens.py` at zero; the in-file `#guard_msgs`; the `FinalCheck.lean`
acceptance build after wiring both in).

**B175-A ACCEPTED — `Definitions/MMFLM85CorollaryFock.lean`** (347 lines against a 300-line target,
reported; namespace `MM.FLM85Fock`, 3 guards).  The Fock half of (8.5.13) at the boundary field.

* **The book's quantification, read off the page images**: p. 195 Theorem 8.4.2 is for `a, b ∈ L̂`; p. 200
  Corollary 8.5.1, verbatim, "In the notation of Theorem 8.4.2, `[Y(a,z₁),Y(b,z₂)] = Res_{z₀}
  z₂^{−1}Y(Y(a,z₀)ι(b),z₂)e^{−z₀(∂/∂z₁)}δ(z₁/z₂)` (8.5.13)" — **it has no quantification of its own, so
  (8.5.13) is the lattice states only**.  General `u, v ∈ V_L` is Theorem 8.6.1.  **So
  `MM.FLM85Co.Corollary851` at arbitrary `a b : V_L` is strictly stronger than the book's (8.5.13)** — it
  is Theorem 8.6.1's content, which is exactly why the Fock half cannot be read off Corollary 8.5.1.
* Landed: both sides are `F`-linear in each state (`commutatorFamily_add_left`/`_smul_left`/`_add_right`/
  `_smul_right`, `corollary851Residue_*`, `corollary851_*`); the spanning reductions
  `corollary851_of_mem_span`, `corollary851_of_spanning`; `corollary851_iff_spanningStates` (with the
  landed `spanningStatement`); and **the exact reduction of the Fock half**
  `corollary851_stateOf_basis` / `residual861Aligned_fockBasis_iff`: the Fock half is *equivalent* to
  `Corollary851` at the **Fock monomial basis** `stateOf (fockMonomialBasis bL d) a`,
  `d : ModeBasisIndex κ →₀ ℕ`.  `corollary851_vl_iff_residual861Aligned` shows the whole-`V_L` statement
  is equivalent to the Fock-half residual, so working at (8.6.2) loses nothing.
* **Not landed, and the reason**: no law computes `Corollary851` at `b_i(−n)·x` from `x`; FLM (8.5.5)
  coefficientwise at `u = a(−n)x` is nowhere landed; and the created-state evaluation
  `Y ((Y (stateOf u' a)) k (stateOf v' b)) j` — which the residue itself contains — is computed nowhere.
  The missing engine already exists as unproved Props: `MM.FLM86SF.Display8610One`/`Display8610List`
  ((8.6.10), the shifted field), `MM.FLM85IS.IterateProductStepAt`/`IterateProductStepOnCreated`, and
  `MM.FLM85Co.ProductExpansion` = `MM.FLM85Ext.iterateStatement` ((8.5.12)).  B174-B computes a product
  at `ι(c)`, not the field of a created state.  **The mixed case `u' = 1`, `v'` arbitrary is not free
  either** — do not dispatch it as if it were.

**B175-B ACCEPTED — `Definitions/MMFLM86ResidueNegative.lean`** (297 lines, namespace `MM.FLM86RN`,
7 guards).  The negative-pairing case at the lattice pair, split and measured.

* **The book's own production of the negative-pairing terms** is Corollary 8.4.4 (8.4.39) (p. 196), the
  coefficient of `z_0^{−⟨a,b⟩−1}` in (8.4.38): `z_2^{−1}Σ_{k+l=−⟨a,b⟩−1} 8p_k(…)Y(ab,z_2)8
  (−1)^l(1/l!)(∂/∂z₁)^lδ(z₁/z₂)` — finite, exactly `N = −⟨a,b⟩` terms — with the verbatim case list
  (8.4.41) `≥ 0 ⇒ 0`, (8.4.42) `= −1`, (8.4.43) `= −2` (two terms), `N` terms at `−N`.
* Landed at `ZAligned E` with the (8.5.2) conjunct discharged: `residueNegative_canonical` (the exact
  (8.5.26) sum), `residueNegative_summand_shape` (the `i`-th state is the created state
  `creationExpVL a ((N:ℤ)−1−i) (anchorVL E a b)`), `residueNegative_lower_pos` (positive degree for
  `i+1 < N`), `residueNegative_top_summand` (the `i = N−1` summand closed as `ε(a,b)·YFam E (a+b) (…)`),
  `residueNegative_split`; `residue_negOne` and `negOne_iff_commutator` (the `⟨a,b⟩ = −1` case is
  *equivalent* to the honest commutator (8.4.42) at `YFam`); and
  `residueNegativeAligned_iff_negOne_and_below : ResidueNegativeAligned ↔ NegOneResidualAligned ∧
  NegTwoAndBelowResidualAligned`.
* **Correction of record — the campaign's recorded obstruction was too strong**: `negContraction_tail`
  proves `∃K, ∀k ≥ K, normOrdered2 E a b (P+k−⟨a,b⟩) (Q−k) v = 0` for **every vector** (from the landed
  `normOrdered2_locallyFinite2OneSided`), so **the finite-support carrier is not the wall**.  What the
  negative chamber needs is the (8.4.26)/(8.4.27) **two-chamber identity**, now named
  `NegativeChamberExpansion (E)` (a `def … : Prop`, consumers recorded), whose finiteness half is already
  a theorem and whose scalar split is landed (`MM.FLM87.expansionA_sub_expansionB_of_neg`).
* **Correction**: the `≤ −2` gap is **not** (8.6.10)/`Display8610List` — that display needs a product
  `mul : CoefFamily → CoefFamily → CoefFamily` the carrier does not have (which is why `mul` is a parameter
  there) — but the two-point (8.5.12) `:Y(a,z₂+w)Y(b,z₂): w^{⟨a,b⟩}` at negative `w`-powers.  And
  `YFamOnGenerators` being a theorem does **not** make `⟨a,b⟩ = −1` unconditional: the remaining conjunct
  is `h851` itself, i.e. (8.4.42).

**Where Theorem 8.10.2 now stands, and batch 176.**  Everything left of `hjac` is `Corollary851
(ZAligned E) u v K` for all `u, v ∈ V_L` (equivalently at the Fock monomial basis, B175-A), plus the
boundary restatement at the even cocycle.  Both halves converge on **one missing engine: the field at a
created state** — FLM (8.5.5) coefficientwise at `Yaligned E`, equivalently the shifted field (8.6.10)
`Y(Y(a,w)ι(1),z) = Y(a,z+w)`, whose named Props exist unproved.  **Batch 176**: (A) the (8.5.5)
coefficientwise normal-ordered formula for `Y(u,z)` at `Yaligned E` (the Fock path's engine; landed
starting points `Yzero_spec`/`Yzero_unique`, `generalVertexOperator_existsUnique`); (B)
`NegativeChamberExpansion` (the negative path's engine).  Dispatcher estimate: 2--4 batches for (A),
1--3 for (B).

---

### Batch 176 — `NegativeChamberExpansion` is a THEOREM in 11 lines (the wall moves to the Θ-collapse); (8.5.5) coefficientwise and the vacuum field of (8.6.10) are landed; and **`BaseResidual` is already a theorem at that base, so the `j = 1` level of (8.6.10) is unconditional**

Two fresh agents, disjoint new files, one batch; all four files verified here (rebuild from deleted
`.olean`s, `--- OK ---` for each; `Probe/forbidden_tokens.py` at zero; the in-file `#guard_msgs`; the
`FinalCheck.lean` acceptance build after wiring them in).

**B176-B ACCEPTED — `Definitions/MMFLM86NegativeChamber.lean`** (299 lines, namespace `MM.FLM86NC`,
7 guards).

* **(8.4.26)/(8.4.27), verbatim off the page images** (printed 192--193 = PDF 246--247):
  `[Y(a,z1),Y(b,z2)] = 1Y(a,z1)Y(b,z2)1 z1^{⟨a,b⟩}(1 − z2/z1)^{⟨a,b⟩} − (−z2/z1)^{⟨a,b⟩}(1 − z1/z2)^{⟨a,b⟩}1
  = −1Y(a,z1)Y(b,z2)1 (−z2)^{⟨a,b⟩} Θ_{z1/z2}((1 − z1/z2)^{⟨a,b⟩})` (8.4.26), and
  `Θ((1 − z)^{⟨a,b⟩}) = 0` for `⟨a,b⟩ ≥ 0`, `= (1/N!)δ^{(N)}(z)` for `⟨a,b⟩ < 0`, `N = −⟨a,b⟩ − 1`
  (8.4.27)/(8.4.28).  (The dispatcher re-checked the folio: PDF 246 → printed 192, 249 → 195, 254 → 200,
  259 → 205, so **the offset is −54**; B176-B's "−55" is wrong, and `MMFLM86SpineGap.lean`'s −54 stands.)
* **`negativeChamberExpansion : NegativeChamberExpansion E` IS A THEOREM** — batch 175's named `def`,
  verbatim, witness `Finset.range (max T0 K + 1)`, **11 lines**.  The reason it was cheap: the equality
  was *already landed* for **every** pairing as `MM.FLM86RX.threshold_holds`
  (`MMFLM86Reindex8612.lean:880`), and the finiteness was already `MM.FLM86RN.negContraction_tail`.
  `positiveChamberExpansion` is the re-indexing `P = −m−1, Q = −n−1` of the landed
  `MM.FLM86YP.YFam_mul_apply_eq_sum_ratioCoeff`.
* **The wall moves to the collapse.**  `NegOneResidualAligned` does **not** follow from the chamber
  expansion: the landed equivalence chain is
  `negOneResidualAligned_iff_chamberCollapse` and `negOneChamberCollapse_iff_blockCollapse`, with
  `NegOneBlockCollapse` the **pure-block telescoping residual** (the (8.4.27)
  `Θ((1−z)^{−1}) = δ(z)` step read on the antidiagonal `P+Q+1`), and
  `negTwoAndBelow_iff_chamberResidue : NegTwoAndBelowResidualAligned ↔ NegTwoChamberResidueIdentity`
  (whose right-hand side keeps the `ZAligned`-at-created-states summands — the same gap as B176-A's).
  Engine landed: `negChamberCommutator_eq_sum_ratioCoeff_of_window`.

**B176-A ACCEPTED — `Definitions/MMFLM85PrescriptionCoeff.lean` (244 lines) and
`MMFLM85PrescriptionCoeffLevel.lean` (247 lines)** (namespace `MM.FLM85PC`, 8 + 8 guards).

* **(8.5.5), verbatim off printed p. 198 (PDF 252)**: `Y(v,z) = Y_Z(v,z) = ⟨(1/(n₁−1)!)(d/dz)^{n₁−1}α₁(z) …
  (1/(n_k−1)!)(d/dz)^{n_k−1}α_k(z) Y(a,z)⟩` for `v = α₁(−n₁)…α_k(−n_k)·ι(a)` (8.5.4), "the normal ordered
  product being defined in an obvious way extending the rules already given"; (8.6.10) on printed p. 207.
* **(8.5.5) coefficientwise** is landed at `ZAligned E`: `ZAligned_spec` (general (8.5.4) state),
  `ZAligned_spec_cons` (the list with one more factor, unfolded as the (8.4.16) `headOp` + `tailOp`),
  `ZAligned_creationStep` (arbitrary vector `w ∈ V_L`).  **Honest note: this was already landed at an
  arbitrary base** (`MM.FLM810CI.Yzero_modeVL_eq_actionFamily`, `…_nat`,
  `actionPrescriptionTAZero_cons`, `MMFLM810CreationIntertwine.lean:205`--`:249`); these are the
  `ZAligned`/(8.5.15)-indexed transports — do not re-prove them.
* **`vacuumFieldAt_ZAligned : VacuumFieldAt (ZAligned E) vacuumVL` is a THEOREM** (from
  `Yaligned_vacuum`).  Consequence: at the honest field `Display8610List`/`Display8610One` follow from
  the **single** input `IterateProductStep (ZAligned E) conv`
  (`display8610List_ZAligned_of_step`, `display8610One_ZAligned_of_step`), which is *equivalent* to the
  landed `MM.FLM85IS.CreatedStepResidual` (`createdStep_ZAligned_iff`), and at `k = 1` is the landed
  (8.5.12) `MM.FLM85Ext.iterateStatement` (`display8610One_ZAligned_iff_iterateStatement`).  **The two
  named gaps of batches 119/121 are one at the honest field.**
* **The levels of (8.6.10) at `ZAligned`**: `j = 0` is a **theorem**
  (`display8610One_zero_level_ZAligned`); `j = 1` is **equivalent** to
  `MM.FLM810CI.BaseResidual (baseAligned E)` (`display8610One_one_level_iff_baseResidual`, both
  directions) and is proved at the lattice generators (`display8610One_one_level_iotaVL`);
  `j ≥ 2` untouched.
* **The uniqueness route does NOT work for (8.6.10)** — `Yzero_unique` compares linear maps
  `V_L → (End V_L){z}`, while `Display8610One` is about the field at one state; neither side of (8.6.10)
  is a state-linear prescription-satisfying field.  What spanning/uniqueness does give is (8.5.5) at an
  arbitrary vector, which is the landed `MM.FLM810CI` content above.

**THE DISPATCHER'S FINDING, and it is a free theorem.**
`MM.FLM810TNO.baseRepaired_eq_baseAligned` (`MMFLM810TheoremNoOrth.lean:97`) proves
`baseRepaired E = baseAligned E`, and batch 172 landed
`MM.FLM810CFC.baseResidual_of_currentFieldComm : BaseResidual (baseRepaired E)`
(`MMFLM810CurrentFieldComm.lean:392`).  **So `BaseResidual (baseAligned E)` is already a theorem**, and
B176-A's equivalence therefore makes the `j = 1` level of (8.6.10) at `ZAligned E` **unconditional** —
neither B176-A nor B176-B noticed this (B176-A calls `BaseResidual` "the open
`MM.FLM810CI.BaseResidual`").  Batch 177-A lands that corollary first and then attacks `j ≥ 2`.

**Where Theorem 8.10.2 stands, and batch 177.**  `hjac` = `Corollary851 (ZAligned E) u v K` for all
`u, v ∈ V_L`; the two named residuals are now `MM.FLM85IS.CreatedStepResidual` (the one-step (8.4.25) at
created states — B176-A's, whose `j ≤ 1` levels are unconditional and whose `j ≥ 2` levels are open) and
`MM.FLM86NC.NegOneBlockCollapse` plus `NegTwoChamberResidueIdentity` (B176-B's).  **Batch 177**: (A) land
the unconditional `j = 1` slice from `baseResidual_of_currentFieldComm` +
`baseRepaired_eq_baseAligned`, then attack `j ≥ 2` of the one-step; (B) prove `NegOneBlockCollapse` by
telescoping `normOrdered2` along the antidiagonal via its three-factor form
(`MMFLM86TwoVarNorm.lean:528`) and `MM.FLM86TN.normOrdered2_swap`, with the landed scalar shadow
**Batch 177**: (A) land
the unconditional `j = 1` slice from `baseResidual_of_currentFieldComm` +
`baseRepaired_eq_baseAligned`, then attack `j ≥ 2` of the one-step; (B) prove `NegOneBlockCollapse` by
telescoping `normOrdered2` along the antidiagonal via its three-factor form
(`MMFLM86TwoVarNorm.lean:528`) and `MM.FLM86TN.normOrdered2_swap`, with the landed scalar shadow
`MM.FLM87.expansionA_sub_expansionB_of_neg` (`MMFLMY87.lean:910`).

---

### Batch 177 — the `k = 1` case of (8.6.10) is a FREE theorem from (8.5.23)+(8.10.13) (the tower collapses to level 1); and `NegOneBlockCollapse` is REFUTED as stated — it needs the source cocycle, and the wall is the `c`-free antidiagonal collapse

Four new files, two fresh agents, one batch; all verified here (rebuild from deleted `.olean`s,
`--- OK ---` for each, `Probe/forbidden_tokens.py` at zero, the in-file `#guard_msgs`, and the
`FinalCheck.lean` acceptance build after wiring them in).

**B177-A ACCEPTED — `Definitions/MMFLM85LevelShift.lean` (190 lines) and
`Definitions/MMFLM85PrescriptionCoeffLevel2.lean` (241 lines)** (namespaces `MM.FLM85LS` / `MM.FLM85PC2`;
15 guards, 14 at `[propext, Classical.choice, Quot.sound]` and `creationCoeff_lminusOne` at the strictly
smaller `[propext, Quot.sound]`, recorded truthfully).

* **Item 1, the free theorem**: `baseResidual_baseAligned_of_symm (hs) (E) : BaseResidual (baseAligned E)`
  (from `baseRepaired_eq_baseAligned` + batch 172's `baseResidual_of_currentFieldComm`), hence
  `display8610One_one_level_ZAligned` (the `j = 1` slice, at every lattice generator and every `n`), the
  family/all-`a` forms, `one_level_ZAligned_iff_baseResidual` (both directions) and
  `display8610List_one_level_ZAligned`.
* **HEADLINE: the `k = 1` case of (8.6.10) is no longer a gap.**  The `L(-1)`/`deriv` induction works,
  but with the **base-shifted** recurrence:
  `creationCoeff_lminusOne : state_j(L(-1)v) = (j+1)·state_{j+1}(v)` (a pure reindexing of (8.10.13)),
  `levelAt_succ_iff_levelAt_lminusOne : LevelAt Y vac v (j+1) ↔ LevelAt Y vac (L(-1)v) j` (**both**
  directions), `levelAt_succ_iterate_iff_levelAt_one`, and the creation property (8.5.23) at level 0.
  Hence `display8610One_of_creation_translation : CreationProperty Y vac → TranslationAxiom Y om →
  Display8610One Y.coeff a vac` — **(8.5.23) + (8.10.13) ⟹ (8.6.10) at `k = 1`, with no Jacobi, no
  (8.4.25), no base residual and no uniqueness argument**; `Yzero_unique` was measured *not* to apply.
  The other recurrence `L(-1)·state_j = (j+1)state_{j+1}` is neither landed nor needed.
* At the boundary field (only `hnd`, `hs`): `display8610One_ZAligned`, `levelAt_ZAligned`,
  `iterateStatement_ZAligned` ((8.5.12) at `b = ι(1)`) and `iterateProductStepAt_vacuum_ZAligned` —
  **the vacuum half of `IterateProductStep` is a theorem**.
* **New exact residual**: `def NonvacuumCreatedStep (E) : Prop := ∀ l : List (VL F L × ℕ), l ≠ [] →
  IterateProductStepAt (ZAligned E) conv (iteratedCreation (ZAligned E) vacuumVL l)`, with
  `createdStepResidual_ZAligned_iff_nonvacuumStep` and `display8610List_ZAligned_iff_nonvacuumStep`
  (both directions).  So §8.6's Fock residual is now the one-step (8.4.25) at **non-vacuum** created
  states (equivalently `k ≥ 2` of (8.6.10)); `IterateProductStep` at an arbitrary `v` stays open.

**B177-B ACCEPTED — `Definitions/MMFLM86NegOneCollapse.lean` (258 lines) and
`MMFLM86NegOneCollapseDiag.lean` (248 lines)** (namespace `MM.FLM86NOC`, 8 guards, all exactly the
three-axiom set).

* **HEADLINE: batch 176's `NegOneBlockCollapse` IS REFUTED AS STATED** — it is missing the source's
  (8.6.1) cocycle hypothesis.  The verbatim obstruction:
  `negOneBlockCollapse_eps_add_smul_eq_zero (h : NegOneBlockCollapse E) (hp : form a b = −1)
  (hq : form b a = −1) : (E.eps a b + E.eps b a) • YFam E (a+b) (m+n) v = 0` — the residual's instances
  at `(a,b,m,n)` and `(b,a,n,m)` are the *same two window sums* read as `S−T = ε(a,b)Y` and
  `T−S = ε(b,a)Y`.  Hence `(1 + E.c b a) • YFam E (a+b) N v = 0`, and at `a = b` (where `c a a = 1`)
  `YFam E (2a) ≡ 0`.  `not_negOneBlockCollapse_rankOne` (with
  `not_negOneChamberCollapse_rankOne`, `not_negOneResidualAligned_rankOne`) exhibits the failure at the
  landed rank-one datum `formR m n = mn`, `E = extR`; the general obstruction applies verbatim to an
  even form (e.g. hyperbolic `ℤ²` with the trivial cocycle).  This is the campaign's sixth
  "the named residual is false as landed" case.
* **No pointwise telescoping exists**: at `⟨a,b⟩ = −1` the two chamber sums are the two **complementary
  half-lines** `p ≥ −m` and `p ≤ −m−1` of the *same* antidiagonal, in the two families related by
  `c(b,a)`.  The corrected residual is `AntidiagonalCollapse` (**`c`-free**: the two half-sums of the
  single family `normOrdered2 E a b` add up to `ε(a,b)·YFam E (a+b)(m+n)`), and the main equivalence is
  `negOneBlockCollapse_iff_antidiagonalCollapse (hs) (E) (hc : NegOneSourceCocycle E)` — **both
  directions**.  `NegOneSourceCocycle` is exactly the source's (8.6.1) at `⟨a,b⟩ = −1`, and it **holds at
  the boundary field**: `CocycleCompatible` gives `c(b,a)·(−1)^{−1} = 1`, i.e. `c(b,a) = −1`.  What is
  left is the assembly identity `(:YY:_{ab})(z,z) = ε(a,b)·Y(a+b,z)` (creation exponentials add,
  annihilation adds, the middle block gives `ε(a,b)(a+b)z^{(a+b)(0)}`), stated not proved.
* **`≤ −2`**: untouched beyond its statement; its summands are `ZAligned` at the (8.5.11) created states,
  i.e. exactly `MM.FLM85IS.IterateProductStepOnCreated`.  **The `−1` case is the first step of that
  induction and the `≤ −2` case its iterate** — so B177-A's and B177-B's residuals meet at the same
  created-state evaluation.

**Where Theorem 8.10.2 stands at the end of batch 177.**  `hjac` at the boundary field is
`Corollary851 (ZAligned E) u v K` for all `u, v ∈ V_L`; its two named residuals are now
**(1)** `NonvacuumCreatedStep E` (the one-step (8.4.25) at non-vacuum created states; equivalently
`k ≥ 2` of (8.6.10)) and **(2)** `AntidiagonalCollapse` under `NegOneSourceCocycle` (whose remaining
assembly is `(:YY:_{ab})(z,z) = ε(a,b)Y(a+b,z)`), plus the `≤ −2` iterate (3) which meets (1).
**Batch 178**: (A) `NonvacuumCreatedStep` — measure whether the `(8.5.23)+(8.10.13)` structure extends
to the non-vacuum states or whether the genuine normal-ordering step is irreducible; (B)
`AntidiagonalCollapse`'s assembly identity.

**Brief errors this batch**: B177-B's task was a refutation, not a proof (the brief's requested
telescoping cannot exist); two measured API traps — `rw`/`simp` cannot rewrite under a `Finset.sum`
binder whose pattern mentions the bound variable (use `apply Finset.sum_congr rfl; intro k _`), and
`neg_one_smul` does not match `(-1 : F) • x` here (use `exact neg_one_smul F x`); and the `LMinusOneCreationMode`
machinery the brief pointed at is not used by B177-A.

**Brief errors this batch (176)**: `Definitions/MMFLM86LS.lean` does not exist (`normOrdered2_locallyFinite2OneSided`
is `MMFLM86LimitStep.lean:320`); `Definitions/MMFLM86PE*.lean` matches no file (it is
`MMFLM86ProductExpansion.lean`); `MM.FLM87.expansionA_sub_expansionB_of_neg` is in `MMFLMY87.lean`;
the display-page range in B176-A's brief was one leaf early; and `YFam_mul_eq_convWith2_contrFam` is
conditional on `0 ≤ ⟨a,b⟩` so it is not the negative-pairing route.

**Brief errors this batch**: `Definitions/MMFLM86CI.lean` does not exist (the namespace `MM.FLM86CI` is in
`MMFLM86CommIota.lean`; the `−1` lemmas are in `MMFLM86IterateIota.lean`); `Display8612`/
`OperatorDisplay` do not exist; the recorded "carrier cannot express `contrFam N`" is too strong;
`tools/build.sh` **is** usable on this box; and (8.6.3) carries `((z₁/z₂)^a δ(z₁/z₂))` with
`e^{−z₀(∂/∂z₂)}`, not the `e^{−z₀(∂/∂z₁)}δ` of (8.5.13).

**Brief errors this batch**: `YFam` is in `Definitions/MMFLM84LatticeField.lean` (not `MMFLM84LF.lean`);
the import targets `MMFLM810ZMU/TB`, `MMFLM85Co`, `MMFLM810JZF/JF2` live in
`MMFLM810ZeroModeL0Uncond.lean`, `MMFLM810TranslationBase.lean`, `MMFLM85Corollary.lean`,
`MMFLM810JacobiZeroFace.lean`, `MMFLM810JacobiFull.lean`; `twoVarField_iotaVL`,
`vacuumVL_right_eq_YCoeff`, `vacuumVL_zero` do not exist (`YFam_iotaVL`, `YFam_vacuumVL_eq_YCoeff`,
`YFam_vacuumVL` are the names); `tools/build.sh` **is** usable in this environment (its hardcoded
`FLT` exists) — the "not usable here" of `AGENTS.md` is stale for this box; and (B174-A) the brief's
inventory list `Theorem861`/`Theorem861.of_cutoff`/`theorem861Residue_*` points at legacy scalar proxies
(`MMFLM86SpineGap.lean:52`--`:61`), none of which states (8.6.3).

---

### Batch 178 — `NonvacuumCreatedStep` is REFUTED (the `conv` carrier gates on two-sided finiteness, so the whole `conv`-form §8.6 chain is false); and `AssemblyDiagonal` is the single named residual of the `⟨a,b⟩ = −1` case, whose hard step is `expField` multiplicativity

Three new files, two fresh agents, one batch; all verified here (rebuild from deleted `.olean`s,
`--- OK ---` for each, `Probe/forbidden_tokens.py` at zero, the in-file `#guard_msgs`, and the
`FinalCheck.lean` acceptance build after wiring them in).

**B178-A ACCEPTED — `Definitions/MMFLM85NonvacuumStep.lean` (233 lines) and
`MMFLM85NonvacuumRefute.lean` (226 lines)** (namespaces `MM.FLM85NV` / `MM.FLM85NVR`; all guards exactly
the three-axiom set).

* **Reduction, hypothesis-free**: `nonvacuumCreatedStep_iff_singletonCreatedStep` — the created-state
  step at *any* state is the step at a **one-factor** created state (FLM (8.5.23) does it in one line; no
  induction on the list is needed) — and `nonvacuumCreatedStep_iff_iterateProductStep`; with `hnd`, `hs`,
  `nonvacuumCreatedStep_iff_display8610Two` and `display8610List_iff_display8610Two` (so `k = 2` implies
  every `k`).  The shortest case is `Display8610Two E`, i.e. (8.6.10) at `k = 2`.
* **HEADLINE: the short case is FALSE, and with it the whole `conv`-form §8.6 chain.**  For every
  `a ≠ 0`: `ZAligned E (ι(a)) j ι(1) ≠ 0` for all `j ≤ −1` (`ZAligned_iotaVL_vacuumVL_ne_zero`), so
  `¬ LocallyFiniteCoef (ZAligned E (ι(a)))` (`not_locallyFiniteCoef_ZAligned_iotaVL`).  `conv` is a
  `dite` on `LocallyFiniteCoef` of its **second** argument, hence `conv A (YFam E a) = 0`; the step at
  the created state `ι(a)` at `x = ι(1)`, `j = 0` demands `YFam E a = conv vacuumField (YFam E a) = 0`,
  and the left side is nonzero on the vacuum.  Landed: `not_iterateProductStepAt_ZAligned_iotaVL`,
  `not_nonvacuumCreatedStep_of_ne_zero`, `not_createdStepResidual_ZAligned`,
  `not_iterateProductStep_ZAligned`, `not_display8610List_ZAligned`, `not_display8610Two_of_ne_zero`.
* **This is a carrier defect, not FLM's mathematics**: the defect class is batch 133's (`MMFLM86IS`
  refuted the two-sided `LocallyFinite2`), and the repair must consume the **one-sided** (8.10.5)
  (`MM.FLM84LF.YFam_eq_zero_of_gt`) and/or the joint `8…8` object of (8.6.25).  The missing FLM object is
  **(8.6.6)'s third family** — the ordinary product vs the normal-ordered one, i.e. the OPE contraction
  `(z₁−z₂)^{⟨ā,b̄⟩}`, `MM.FLM86KF.Display866Family3` — **not** (8.6.5)/(8.6.9)/`kFoldNormalOrder`, which
  are landed.
* **Consequences for the ledger**: `NonvacuumCreatedStep`, `CreatedStepResidual`, `IterateProductStep`
  and `Display8610List` are all false at `ZAligned`/`conv`, so **no Jacobi-type input (hjac included)
  can prove them**; B177-A's `display8610List_ZAligned_iff_nonvacuumStep` is a valid equivalence between
  two false statements.  `hjac`/`JacobiAxiom` itself is untouched, and the `k = 1` theorem and the vacuum
  step of batch 177 stand (they do not apply `conv` to a non-locally-finite family).
* **A repo trap worth keeping**: `Definitions/MMFLM85StepLinear.lean` is imported by **nothing** yet holds
  `iterateProductStepAt_iff_apply`, which `MMFLM85IS`'s header cites as landed; without the explicit
  import the statement silently elaborates with `sorryAx` (the guards catch it).

**B178-B ACCEPTED — `Definitions/MMFLM86AntidiagonalAssembly.lean`** (331 lines against a 300-line target,
reported; namespace `MM.FLM86AA`, 3 guards plus an all-declarations check).

* **`AssemblyDiagonal (E)` is landed** (pairing-free, FLM (8.4.20) at `z₁ = z₂ = z`):
  `∀ a b m n v, ∃ s : Finset ℤ, (∀ p ∉ s, normOrdered2 E a b p (−m−n−1−p) v = 0) ∧
  diagSum (normOrdered2 E a b) s (−m−n−1) v = ε(a,b) • YFam E (a+b) (m+n) v`, i.e.
  `(:YY:_{ab})(z,z) = ε(a,b)·Y(a+b,z)` in carrier form; `NegOneAssemblyDiagonal` is its
  `⟨a,b⟩ = −1` instance.
* **Both directions**: `assemblyDiagonal_iff_collapse : NegOneAssemblyDiagonal E ↔ AntidiagonalCollapse E`
  — the reindexing of the `ℤ`-window into the two `ℕ`-half-windows, with no product, commutator or
  residue used.  Consumer map: `AssemblyDiagonal → AntidiagonalCollapse → (NegOneSourceCocycle) →
  NegOneBlockCollapse → NegOneResidualAligned`, with `negOneSourceCocycle_of_cocycleCompatible`
  (`CocycleCompatible` + `⟨a,b⟩ = −1` ⇒ `c(b,a) = −1`) and
  **`negOneResidualAligned_evenCocycle_of_assemblyDiagonal : AssemblyDiagonal (C.toCentralExtension h2) →
  NegOneResidualAligned (C.toCentralExtension h2)`** — the `−1` case now has exactly **one** named missing
  input, no cocycle or window hypothesis.
* **The exact residual is `AssemblyDiagonal`**, whose proof is the factor diagonalisation in four steps
  documented in the file's header.  **The hard step is `expField` multiplicativity**
  (`E⁻_a(z)E⁻_b(z) = E⁻_{a+b}(z)`), which is landed **nowhere** — the landed
  `MM.FLM86EP.creationExpVL_mul_comm` is *commutation*, not exponent addition.  Finiteness is not the
  wall (`MM.FLM86LS.normOrdered2_locallyFinite2OneSided` already gives the `∃ s`).
* The `≤ −2` case is untouched (`NegTwoChamberResidueIdentity`).

**Where Theorem 8.10.2 stands at the end of batch 178.**  `hjac` = `Corollary851 (ZAligned E) u v K` for
all `u, v ∈ V_L` (untouched by the refutation — it is a statement about `commutatorFamily` and the
residue, not about the `conv`-gated product).  The live residuals are **(1)** the `⟨a,b⟩ = −1` case's
`AssemblyDiagonal` (hard step: `expField` multiplicativity), **(2)** the `≤ −2` iterate, and **(3)** the
Fock half, whose route through the `conv`-form (8.6.10) is now **dead** and must be rebuilt on a one-sided
product (the (8.6.6) third-family object).  **Batch 179**: (A) `expField` multiplicativity then the four
steps of `AssemblyDiagonal`; (B) the one-sided carrier repair — define the product with one-sided
finiteness and restate (8.6.10) on it, measuring how much of batch 177's `k = 1` theorem survives.

**Brief errors this batch**: `creationExpVL_mul` does not exist (`creationExpVL_mul_comm` is commutation);
the prescribed list induction was unnecessary; the suggested per-state product tools are a dead end for the
created-state step; and the `≤ 300`-line cap forced a split.

---

### Batch 179 — the one-sided product repair is landed (the previously refuted vacuum step is now a theorem); `expField` multiplicativity is a theorem (via a NEW Leibniz rule) and 2 of `AssemblyDiagonal`'s 4 steps are done

Four new files, two fresh agents, one batch; all verified here (rebuild from deleted `.olean`s,
`--- OK ---` for each, `Probe/forbidden_tokens.py` at zero, the in-file `#guard_msgs`, and the
`FinalCheck.lean` acceptance build after wiring them in).

**B179-A ACCEPTED — `Definitions/MMFLM86ExpFieldMul.lean` (300 lines) and `MMFLM86ExpFieldMulLat.lean`
(248 lines)** (namespace `MM.FLM86EFM`, 5 guards, all exactly the three-axiom set).

* **`expField` multiplicativity IS LANDED**: `flatMul_expField_of_commute (h : ∀ i j, Om i * Om' j =
  Om' j * Om i) : flatMul (expField Om) (expField Om') = expField (Om + Om')` over an arbitrary
  `Ring A`/`Algebra ℚ A`.  **The real missing engine was a Leibniz rule**,
  `deriv_flatMul : deriv (flatMul X Y) = flatMul (deriv X) Y + flatMul X (deriv Y)`, landed nowhere
  before (`MMFLM84Diag`/`MMFLM84Subst` record it as open); with `expField_unique`, `flatMul_assoc` and
  `flatMul_swap` the multiplicativity follows.  Exponent linearity (`exponentField_add`,
  `creationExponent_add`, `annihilationExponent_add`) is landed too.
* **Steps 1 and 3 of `AssemblyDiagonal` are landed**, in the antidiagonal forms the carrier consumes:
  `flatMul_creationExpVL` (`E⁻_aE⁻_b = E⁻_{a+b}`), `sum_creationPair2_antidiagonal`,
  `flatMul_annihExp` and `sum_annihPair2_antidiagonal` — the `E⁺` half on the **mirrored** families
  `i ↦ annihExp a (−i)` (a separate statement with a separate (7.1.5) input
  `MM.FLM86TN.modeVL_comm_of_pos`), not "the same proof".
* **Steps 2 and 4 are NOT proved, so `AssemblyDiagonal` is still a `Prop`.**  Step 2 is landed as the
  exact named residual `MiddleBlockDiagonal (E)` (the middle-block antidiagonal collapse
  `Σ_{p} midBlock2 a b p (N−p) = ε(a,b)·(a+b)z^N`, a `def … : Prop` clearly marked not proved, with its
  route recorded); its proof did not compile in three rounds (`iota` defeq and induction-motive
  normalisation), and per the budget rule the agent stopped rather than weaken it.  Step 4 (nested
  `convWith2` → `convWith`) is documented but is not a single clean identity (the inner index enters
  through the total degree).

**B179-B ACCEPTED — `Definitions/MMFLM85OneSidedProduct.lean` (281 lines) and `MMFLM85OneSidedChain.lean`
(238 lines)** (namespace `MM.FLM85OSP`, 7 guards, all exactly the three-axiom set).

* **The defect, verbatim**: `conv A B` is a `dite` on `LocallyFiniteCoef B` (a **two-sided** per-state
  window), while the honest field satisfies only the upper half (FLM (8.10.5),
  `MM.FLM84LF.YFam_eq_zero_of_gt`), so `conv A (YFam E a) = 0` — the batch-178 refutation.
* **The repair**: `PairFinite A B := ∀ m w, ∃ s, ∀ j ∉ s, A (m−1−j) (B j w) = 0` and
  `convOneSided` (total).  `convOneSided_eq_conv` proves **agreement wherever `conv` was defined**, so
  nothing landed is lost; `convOneSided_vacuumField(_left)` are **unconditional** where the landed
  `conv_vacuumField_left` needed `LocallyFiniteCoef`; `convOneSided_assoc` holds on the locally-finite
  class; `pairFinite_of_opAbove_of_above`, `pairFinite_singletonCoef_left`,
  `pairFinite_vacuumField_left/_right`.
* **The batch-178 refuted instance is now a THEOREM**:
  `vacuumShiftStep_ZAligned_convOneSided : ZAligned E ((ZAligned E vacuumVL) (−1) v) =
  convOneSided (shiftedField (ZAligned E) vacuumVL 0) (ZAligned E v)` for **every** `v`; batch 177's
  `display8610One_ZAligned`, `levelAt_ZAligned`, `iterateStatement_ZAligned` contain no `conv` and
  transport verbatim; `display8610List_convOneSided` gives (8.6.10) at every `k`.
* **New shortest residual**: `Display8610TwoOneSided` / `SingletonCreatedStepOneSided`, with
  `createdStepOneSided_iff_twoOneSided`.
* **Measurement that matters**: **two honest fields still do not form a `PairFinite` pair** (recorded, not
  asserted, as `PairFiniteHonestFields`: at `m = 2⟨a,a⟩−1` the summand has `j`-independent weight, so the
  sum is infinite).  So the general two-factor display is **not** a carrier problem — it is the OPE gap
  `MM.FLM86KF.Display866Family3` ((8.6.6)'s third family), exactly as batch 178 predicted.

**Where Theorem 8.10.2 stands at the end of batch 179.**  `hjac` = `Corollary851 (ZAligned E) u v K`
for all `u, v ∈ V_L`.  Live residuals: **(1)** `AssemblyDiagonal` — now only its steps 2
(`MiddleBlockDiagonal`) and 4; **(2)** the `≤ −2` iterate; **(3)** the Fock half, whose repaired
one-sided chain reduces to `Display8610TwoOneSided`, i.e. the OPE third family
`Display866Family3`.  **Batch 180**: (A) `MiddleBlockDiagonal` + step 4 (with the route batch 179-A
recorded: collapse on `F{L}` via `zPowH_iota` and `form a c + form b c = N`, then the tensor-factor
transfer); (B) the OPE third family `Display866Family3` — connect the general two-factor product to the
landed two-variable `normOrdered2`/`ratioCoeff` product (`MM.FLM86EPA.Yaligned_mul_eq_sum_ratioCoeff`)
and restate the `k = 2` display at the two-variable level.

---

### Batch 180 — `MiddleBlockDiagonal` (step 2 of `AssemblyDiagonal`) is LANDED; step 4 becomes the single residual `AssemblyDiagonalResidual` (pure index regrouping) with the reduction proved; and the OPE third family is corrected and bridged

Three new files, two fresh agents, one batch; all verified here (rebuild from deleted `.olean`s,
`--- OK ---` for each, `Probe/forbidden_tokens.py` at zero, the in-file `#guard_msgs`, and the
`FinalCheck.lean` acceptance build after wiring them in).

**B180-A ACCEPTED — `Definitions/MMFLM86MiddleBlockDiag.lean` (294 lines) and
`MMFLM86MiddleBlockDiagStep4.lean` (295 lines)** (namespace `MM.FLM86MBD`; 7 guards, all exactly the
three-axiom set).

* **Step 2 IS LANDED**: `middleBlockDiagonal (E) : MM.FLM86EFM.MiddleBlockDiagonal (F := F) form E` —
  verbatim batch 179-A's def (the middle block on the antidiagonal collapses to
  `ε(a,b)·(a+b)z^N`).  Route confirmed: on `F{L}` the window is the image of the support under `form a`,
  `zPowH_iota` gives the pair condition and `form a c + form b c = form (a+b) c` turns it into the single
  eigenvalue (`midGrad_iota_diag`); then `TensorProduct.induction_on` + `zPowHVL = TensorProduct.map id`
  (`tensor_map_id_id`) + `actVL_mul_actVL`.  **The earlier failures were caused by the wrong tool**:
  `Finsupp.induction_linear` (add/single) with `Finset.sum_subset` works where extensionality did not.
* **Step 4 is reduced to ONE named `Prop`**: `diagSum_convWith2` (the total-degree reindexing of a
  `convWith2` diagonal sum, unconditional) and `diagSum_convWith2_nested` (the same for
  `(X ⊛ Y) ⊛ Z`, i.e. `normOrdered2`'s shape) and `sum_creationPair2_window` (step 1 in window form) are
  landed, and `def AssemblyDiagonalResidual (E) : Prop` states the remaining regrouping:
  `∑_{q∈w}∑_{q'∈w' q} creationExpVL (a+b) (N−q.1−q.2−q'.1−q'.2) (midBlock2 q'.1 q'.2 (annihPair2 q.1 q.2 v))
  = ε(a,b) • YFam (a+b) (−N−1) v`.  **`assemblyDiagonal_of_residual (hs) (E) : AssemblyDiagonalResidual E →
  AssemblyDiagonal E` is PROVED.**  Recipe for the next task, measured: take a **uniform** middle window
  `w' q = ⋃_{q∈w}(canonical window)` (with per-`q` windows the `K'`-ranges differ and the fibers cannot be
  combined), then group by `J = q.1+q.2` (step 3), group by `K' = q'.1+q'.2` (step 2, now a theorem), and
  read `(J,K')` as `YFam`'s `(j,i) = (−J−1,−K'−1)` via `YFam_eq_sum_three_factor`.
* So the `⟨a,b⟩ = −1` case at `ZAligned (C.toCentralExtension h2)` is unconditional **modulo
  `MM.FLM86MBD.AssemblyDiagonalResidual` alone**; the landed `assemblyDiagonal_iff_collapse` chain is
  unchanged (it consumes `AssemblyDiagonal`, which is still a `Prop`).

**B180-B ACCEPTED — `Definitions/MMFLM86OPEThird.lean`** (427 lines against a 300-line target, reported;
namespace `MM.FLM86OPE` — `MM.FLM86OT` is taken by `MMFLM86OrdTwo.lean`; 11 guards, all exactly the
three-axiom set).

* **(8.6.6) verbatim off printed p. 206** — `A = 8Y(a₁,w₁)…Y(a_k,w_k)8 ι(1) = Y(a₁,w₁)…Y(a_k,w_k)ι(1)
  ∏_{i<j}(w_i−w_j)^{−⟨ā_i,ā_j⟩}`, with the sentence above it: "…understood as the binomial expansion in
  non-negative integral powers of `w_j`".  (8.4.25) and (8.5.12) are quoted too.
* **Correction of record**: the repository's `MM.FLM86KF.Display866Family3` **is not** the landed
  two-variable product — it reads the printed *series* factor as a *scalar* `pairingFactor … w n` (true
  only at null exponents), and its type `Fin k → ℕ → VL` reads the multidegree only in its first
  coordinate, so it can never carry a `k`-variable coefficient family.  Landed instead: the corrected
  multi-index form `Display866Family3TwoIdx` with its proved null-pair instance
  `display866Family3_twoIdx_honest_nullPair`, and **`Display866Family3Two` + `display866Family3Two`
  (the printed third family at `k = 2`, unconditional for `0 ≤ ⟨a,b⟩`)**.
* **The bridge the display consumes is landed**: `shiftedField_ZAligned_apply_eq_sum_ratioCoeff` — (8.4.25)
  at the **shifted** honest fields, i.e. the `w₁^j w₂^l`-coefficient of `Y(a,z+w₁)Y(b,y,z+w₂)` equals
  `(binom …)·Σ_i ratioCoeff ⟨a,b⟩ i • normOrdered2 E a b (…+i−⟨a,b⟩) (…−i) v` — the display's right-hand
  side evaluated by the landed `normOrdered2`+`ratioCoeff` expansion.  Auxiliaries:
  `ZAligned_mul_apply_eq_sum_ratioCoeff` (batch 174 re-indexed), `ZAligned_mul_eq_normOrdered2`,
  `convOneSidedWith_eq_sum_choose`.
* The display is restated as `Display8610TwoGenerators` (its lattice-generator slice, with
  `display8610TwoGenerators_of_display` from `Display8610TwoOneSided`), the datum
  `ShiftedPairFiniteGenerators` is **recorded, not asserted**, and
  `display8610TwoGenerators_iff_ratioForm` is proved **both directions** given that datum.
* **The Fock half is not closed**: the field-of-the-created-state equation itself, the datum (or a
  carrier-free replacement), and the full display at arbitrary `x y : V_L` remain open.  The brief's
  "not a carrier problem, it is the OPE gap" was half right: the `convOneSided` form carries
  `PairFinite` (and returns 0 without it), the OPE is what evaluates the product where it holds.

**Where Theorem 8.10.2 stands at the end of batch 180.**  `hjac` = `Corollary851 (ZAligned E) u v K` for
all `u, v ∈ V_L`.  Live residuals: **(1)** `MM.FLM86MBD.AssemblyDiagonalResidual` — a pure double-sum
regrouping whose recipe is now fully measured, and which alone gives the `⟨a,b⟩ = −1` case at the even
cocycle; **(2)** the `≤ −2` iterate; **(3)** the Fock half's created-state equation (the two-factor
display), with the OPE bridge now landed.  **Batch 181**: (A) prove `AssemblyDiagonalResidual` by the
recorded regrouping (uniform middle window, group by `J` and `K'`, read via `YFam_eq_sum_three_factor`);
(B) the two-factor created-state display at the lattice-generator slice, using B180-B's bridge and the
landed (8.5.5) prescription.

---

### Batch 181 — MILESTONE: `AssemblyDiagonal` is a THEOREM (the `⟨a,b⟩ = −1` case of `hjac` is unconditional at the boundary field); and the two-factor display is refuted wherever the carrier datum fails, with `PairFinite` itself now UNDECIDED

Four new files, two fresh agents, one batch; all verified here (rebuild from deleted `.olean`s,
`--- OK ---` for each, `Probe/forbidden_tokens.py` at zero, the in-file `#guard_msgs`, and the
`FinalCheck.lean` acceptance build after wiring them in).

**B181-A ACCEPTED — `Definitions/MMFLM86AssemblyResidualFibers.lean` (219 lines),
`MMFLM86AssemblyResidualAux.lean` (168 lines), `MMFLM86AssemblyResidual.lean` (264 lines)** (namespace
`MM.FLM86AR`, 7 guards, all exactly the three-axiom set).

* **`assemblyDiagonalResidual : AssemblyDiagonalResidual (F := F) bL form E` is proved VERBATIM**
  (batch 180-A's `def`, no weakening), with the new fiber lemmas `sum_filter_fiber_eq`,
  `sum_annihPair2_fiber` (step 3 in fiber form over **any** window), `sum_midBlock2_fiber` (step 2 fiber
  form) and the regrouping `sum_sum_window_regroup`.
* **`assemblyDiagonal (hs) (E) : AssemblyDiagonal (F := F) bL form E` — `AssemblyDiagonal` IS A THEOREM**,
  and composing with the landed `MM.FLM86AA.negOneResidualAligned_evenCocycle_of_assemblyDiagonal` gives
  **`MM.FLM86AR.negOneResidualAligned_evenCocycle`**: `NegOneResidualAligned` at
  `ZAligned (C.toCentralExtension h2)` for **every symmetric form**, no residual hypothesis.  **The
  `⟨a,b⟩ = −1` case of `hjac` is closed** (with the `0 ≤ ⟨a,b⟩` corner closed in batch 170 D1).
* **Method correction of record**: the middle window must be **one common window for all `q`** — the agent
  used the rectangle `WP ×ˢ WQ` of the projections of the per-`q` windows, **not** the brief's
  `Finset.biUnion`-of-canonical-windows (with a `q`-dependent `w' q` the `K'`-image differs per `q` and
  the fibers cannot be combined).  Also: the landed `sum_annihPair2_antidiagonal`/`middleBlockDiagonal`
  are canonical-window statements, so pinning them to the residual's arbitrary window needed the two new
  fiber lemmas; `YFam_eq_sum_three_factor` lives in `MM.FLM84TV`; and the annihilation/middle total
  degrees are `≤ 0`/any integer respectively (the brief's ranges were sign slips).

**B181-B ACCEPTED — `Definitions/MMFLM86TwoFactorDisplay.lean`** (469 lines against a 300-line target,
reported; namespace `MM.FLM86TFD`, 12 guards).

* **State dictionary and the `j = l = 0` slice are fully evaluated**: `zAligned_iotaVL_family`,
  `creationCoeff_iotaVL` (the created state is the (8.5.11) coherent state),
  `iteratedCreation_two_zero : the display's LHS state = creationExpVL a (−⟨a,b⟩) (anchorVL E a b)`,
  `iteratedCreation_two_zero_eq_anchor` at `⟨a,b⟩ = 0`, and `createdState_zero_zero_ne_zero` — the LHS
  at component `−1` on the vacuum is `ε(a,b)·ι(a+b) ≠ 0`.
* **HEADLINE, and it refutes the display on the failing datum**: `convOneSided A B = 0` definitionally
  when `¬ PairFinite A B` (`convOneSided_eq_zero`), hence `not_display8610TwoGenerators` and
  **`display8610TwoGenerators_false`**: at **any** null pair, if `PairFinite (ZAligned E (ι a))
  (ZAligned E (ι b))` fails then `Display8610TwoGenerators E` is **false**; and
  `not_display8610TwoOneSided` shows the repaired `convOneSided` carrier is **not neutral** — the display
  inherits its 0-on-failure behaviour.  So the generator slice is exactly **[carrier datum AND the (8.4.25)
  ratio form]**, not merely the LHS evaluation (`display8610TwoGenerators_iff_ratioForm` is the exact
  two-direction statement).
* **The batch-179-B reason for expecting the datum to fail is NOT a proof**: the `j`-independent-weight
  summand shows only that all summands lie in one weight space, which is infinite-dimensional and
  compatible with either behaviour.  **`PairFinite (ZAligned E (ι a)) (ZAligned E (ι b))` is UNDECIDED in
  the repository** (no proof, no refutation).  **This is the fork for the Fock half**: if it holds (on the
  needed slices), the Fock half is exactly the created-state evaluation — the (8.5.5) prescription of the
  coherent state `YFam E a (−j−1)(creationExpVL b l (ι b))`, which is **not** a (8.5.4) state for `l > 0`;
  if it fails, `Display8610TwoGenerators`/`Display8610TwoOneSided` are false and the honest replacement is
  the OPE product `Display866Family3Two` (landed for `0 ≤ ⟨a,b⟩`).
* The span reduction cannot be transported through the `dite` carrier (linearity holds with the datum
  fixed, but `convOneSided` need not be additive), so the brief's span route is blocked by the carrier,
  not by `spanningStatement`.

**Where Theorem 8.10.2 stands at the end of batch 181.**  `hjac` = `Corollary851 (ZAligned E) u v K`;
of its lattice-pair cases, `0 ≤ ⟨a,b⟩` (batch 170) and `⟨a,b⟩ = −1` (this batch) are **closed**, and the
live residuals are **(1)** the `⟨a,b⟩ ≤ −2` case (`NegTwoAndBelowResidualAligned` /
`NegTwoChamberResidueIdentity`), **(2)** the Fock half, now reduced to **deciding `PairFinite`** on the
shifted honest pairs, and **(3)** the final boundary restatement at the even cocycle.
**Batch 182**: (A) decide `PairFinite (shiftedField (ZAligned E) (ι a) j) (shiftedField (ZAligned E) (ι b) l)`
— prove it or refute it, and either way land the resulting Fock-half statement; (B) the `≤ −2` case.

**Brief errors this batch**: the `⋃_q` middle-window recipe; `YFam_eq_sum_three_factor`'s file; the two
degree-range slips; "apply the free `k = 1` theorem twice" cannot work (it needs the creation property at
the same vacuum, and the non-vacuum one-step is the missing input); "the RHS evaluation is landed" only
given the undecided datum; and route (b) cannot start for `l > 0` because the created state is not a
(8.5.4) state.

**Brief errors this batch**: step 4 did not go through (it is now one `Prop`); `Finsupp` extensionality is
not the step-2 route (`induction_linear` is); the consumer chain cannot close until `AssemblyDiagonal` is
proved; `MM.FLM86KF/EPA/TN.lean` are not file names (those namespaces live in `MMFLM86KFold.lean`,
`MMFLM86ExpProductAligned.lean`, `MMFLM86TwoVarNorm.lean`); and `Display866Family3` is a scalar
mis-reading of a series factor.

**Brief errors this batch**: the missing engine was the Leibniz rule, not `expField_unique`; the
multiplicativity needs the coefficient-level commutation hypothesis consumed in `flatMul_swap`; the
`E⁺` half is a separate statement; the folio "`−54`" vs "`−55`" is a 0-based/1-based convention
difference, not an error; and the general one-sided product gate must be on the **pair**, not on the
second argument alone.

---

### Batch 182 — `PairFinite` is PROVED on the honest pairs (the Fock fork closes in favour of the datum, and the batch-179/181 "infinite sum" recording is REFUTED); and the `⟨a,b⟩ = −2` residue is reduced to `NegTwoBlockCollapse`

Five new files, two fresh agents, one batch; all verified here (rebuild from deleted `.olean`s,
`--- OK ---` for each, `Probe/forbidden_tokens.py` at zero, the in-file `#guard_msgs`, and the
`FinalCheck.lean` acceptance build after wiring them in).  **The dispatcher also corrected five
now-false docstrings in existing files** (see the last paragraph).

**B182-A ACCEPTED — `Definitions/MMFLM85PairFiniteDecide.lean`** (340 lines against a 300-line target,
reported; namespace `MM.FLM85PFD`, 8 guards, all exactly the three-axiom set).

* **HEADLINE: `PairFinite` is PROVED, not refuted.**  Key identity
  `ZAligned_convSummand_eq_sum_ratioCoeff`: for symmetric `form` and `0 ≤ ⟨a,b⟩`,
  `(ZAligned E (ι a) (m−1−j)) ((ZAligned E (ι b) j) w) = Σ_{k ≤ ⟨a,b⟩} ratioCoeff ⟨a,b⟩ k •
  normOrdered2 a b (j−m+k−⟨a,b⟩) (−j−1−k) w` — (8.4.25) along the diagonal `(m−1−j, j)`, the two
  `normOrdered2` indices moving in **opposite** directions in `j`.  The per-state (8.10.5) bounds
  (`normOrdered2_locallyFinite2OneSided`) then confine the support to `Finset.Icc (K1+m+1) (−K2−2)`
  (shifted: `Icc (K1+m−j+1) (l−K2−2)`).  **The `j`-independence of the summand weight is compatible
  with finite support, and the support is finite** — the batch-179-B/180-B/181-B argument was not a proof
  and its conclusion was false.
* Landed: `pairFinite_ZAligned_iotaVL`, `pairFinite_yFam`, `pairFinite_ZAligned_iotaVL_self`,
  `pairFinite_shiftedField_ZAligned_iotaVL`, **`shiftedPairFiniteGenerators_of_nonneg`** (batch 180-B's
  recorded datum is now a theorem under `hs` and `∀ a b, 0 ≤ form a b`) and
  `pairFiniteHonestFields_of_nonneg`; and **`display8610TwoGenerators_iff_ratioForm_of_nonneg`** — the
  display's generator slice **is** the landed (8.4.25) ratio form, the datum discharged.
* **The batch-181-B "refutations" are vacuous**: `not_not_pairFinite_ZAligned_iotaVL` shows
  `not_display8610TwoGenerators` / `display8610TwoGenerators_false` / `not_display8610TwoOneSided` have an
  **unsatisfiable** hypothesis wherever `0 ≤ ⟨a,b⟩`.  `Display8610TwoGenerators`/`Display8610TwoOneSided`
  are neither proved nor refuted.
* **The exact Fock residual is now the created-state evaluation**:
  `display8610TwoGenerators_iff_createdState` — the generator slice is equivalent to the (8.5.5)
  evaluation of `YFam E a (−j−1) (creationExpVL b l (ι b))` for `l > 0`.  Open: `PairFinite` for
  `form a b < 0` (no finite (8.4.25) expansion there) and that evaluation.
* **A misreading of the source, corrected**: the docstrings' citation of "FLM Remark 8.5.2" for the
  infinite-sum claim was wrong — that Remark (printed p. 200) says the *created-state* identity (8.5.14)
  is not rigorous, while **Corollary 8.5.4 (printed p. 201) records "the sum being finite"**.

**B182-B ACCEPTED — `Definitions/MMFLM86NegTwoResidue.lean`** (467 lines, namespace `MM.FLM86NTR`,
12 guards, all exactly the three-axiom set).

* **At `⟨a,b⟩ = −2` nothing is left unreached on the evaluation side**: `coherentField_one` (the (8.5.5)
  one-step at `creationExpVL a 1 (anchorVL E a b)`, with the `^+` tail), `ZAligned_anchorVL`
  (`ZAligned E (anchorVL E a b) = ε(a,b) • YFam E (a+b)`) and `residueSum_negTwo_two_terms` — the whole
  `N = 2` residue equals two closed terms (degrees 0 and 1).
* **The exact residual is the commutator side**, named `NegTwoBlockCollapse` (the (8.4.26) chamber
  difference at `⟨a,b⟩ = −2` on some window equals the two closed terms; `normOrdered2`+`ratioCoeff`+
  `actionFamily` only — no `ZAligned`, no residue, no commutator), with
  `negTwoExactResidualAligned_iff_blockCollapse` **both directions**;
  `negTwoAndBelow_iff_exact_and_below` splits `≤ −2` into `= −2` and `≤ −3` (both ways).
* **Composition with the closed `−1` chain**: `residueNegativeAligned_evenCocycle_iff_negTwoAndBelow`
  — at `C.toCentralExtension h2` the negative-pairing residue is **exactly** `NegTwoBlockCollapse` +
  `NegThreeAndBelowResidualAligned`; `residueNegativeAligned_evenCocycle_of_blockCollapse` is the
  consumer form.
* This is the `N = 2` analogue of the closed `−1` chain (`AssemblyDiagonal` was the `z_0^0` line;
  `NegTwoBlockCollapse` is the `z_0^1` line).  `≤ −3` has no closed form and is left named, not faked.
* **Source-language finding**: the residue's states are the coefficients of the **negative** powers
  `z_0^{−N}…z_0^{−1}` of `Y(a,z_0)ι(b)` (the pole part, `latticeState_eq_exp`), while the landed
  display language is `ℕ`-indexed and *defines* the negative-power coefficients to be `0`
  (`creationCoeffZ_of_neg`) — so the landed (8.6.10) cannot even state the evaluations the residue needs.
  Also: the landed flat coefficients `MM.FLM8512.rhsFam`/`MM.FLM84T.coeff40Family` use
  `opOpenBox`/`openBoxCoeff`, which **drop the `^+` tail** of (8.4.16)
  (`MM.FLM85Ac.actionFamily_eq_opOpenBox_add_tail`); a residual stated with them would be the wrong
  statement, so the honest `actionFamily` form is used.

**The dispatcher's corrections of record (batch 182).**  The five false docstring paragraphs in
`MMFLM85OneSidedProduct.lean`, `MMFLM85OneSidedChain.lean` (`PairFiniteHonestFields`),
`MMFLM86OPEThird.lean` (two) and `MMFLM86TwoFactorDisplay.lean` (two) are rewritten in place as
"CORRECTION OF RECORD (batch 182)" blocks and the four modules rebuilt (`--- OK ---`, tokens at zero).
The corresponding paragraphs of this log and of `NOTES.md` §§75--78 are superseded by this entry and by
§79.

**Where Theorem 8.10.2 stands at the end of batch 182.**  `hjac` = `Corollary851 (ZAligned E) u v K`;
the lattice-pair cases `0 ≤ ⟨a,b⟩` and `⟨a,b⟩ = −1` are **closed**, and the live residuals are
**(1)** `NegTwoBlockCollapse` (`⟨a,b⟩ = −2`), **(2)** `NegThreeAndBelowResidualAligned` (`≤ −3`,
needing a closed form for the iterated (8.5.5) at degree `≥ 2`), **(3)** the Fock half's created-state
evaluation of `YFam E a (−j−1)(creationExpVL b l (ι b))`, and **(4)** the boundary restatement.
**Batch 183**: (A) `NegTwoBlockCollapse` — the batch-181 `AssemblyDiagonal` machinery one order higher;
(B) the coherent-state field `ZAligned E (creationExpVL a k (anchorVL E a b))` for general `k` by the
landed `ZAligned_creationStep` — the object shared by the `≤ −3` case and the Fock display.

**Brief errors this batch**: the `MMFLM86KF/EPA/TN.lean` file-name trap again;
`MM.FLM85PC.iterateStatement_ZAligned` does not exist (it is `MM.FLM85PC2.iterateStatement_ZAligned`, and
only at `b = ι(1)`); the general `MM.FLM85Ext.iterateStatement` is a `def … : Prop` **stated, not
proved**; the mode label of `creationCoeff` equals its anchor label; and the landed `ℕ`-indexed display
zeroes the negative-power coefficients.

---

### Batch 183 — the `−2` line needs the source cocycle: batch 182-B's `NegTwoBlockCollapse` is REFUTED as stated and the corrected residual is the WEIGHTED antidiagonal `NegTwoDiagonalWeight`; and the coherent-state field is a RECURSION, with the `≤ −3` case expanded to the honest (8.5.5) form

Four new files, two fresh agents, one batch; all verified here (rebuild from deleted `.olean`s,
`--- OK ---` for each, `Probe/forbidden_tokens.py` at zero, the in-file `#guard_msgs`, and the
`FinalCheck.lean` acceptance build after wiring them in).

**B183-A ACCEPTED — `Definitions/MMFLM86NegTwoDiagonal.lean` (209 lines),
`MMFLM86NegTwoBlockCollapse.lean` (503 lines against a 300-line target, measured deviation reported),
`MMFLM86NegTwoBlockCollapseAux.lean` (204 lines)** (namespaces `MM.FLM86NTD`/`MM.FLM86NTB`; guards all
exactly the three-axiom set).

* **HEADLINE: batch 182-B's `NegTwoBlockCollapse` is REFUTED as stated** (it carries no cocycle
  hypothesis).  The proved obstruction: it forces `(E.eps a b − E.eps b a) • YFam E (a+b) (N−1) v = 0` for
  all `N, v`, i.e. by (7.1.26) `c(a,b) = 1` or `YFam E (a+b) ≡ 0` — the exact analogue of batch 177-B's
  refutation of `NegOneBlockCollapse`.  At `⟨a,b⟩ = −2` the source's own (8.4.34) condition
  `c(a,b) = (−1)^{⟨a,b⟩}` **is** `c(a,b) = 1`, so that hypothesis is the source's, not an extra one.
* **The corrected `c`-free residual is `NegTwoDiagonalWeight (E)`** — the `z_0^1` line of
  (8.4.36)/(8.4.40): a **weighted** antidiagonal sum (`diagSumWeighted (fun p ↦ ((p+m:ℤ):F))`) equals the
  two closed terms of batch 182-B's `residueSum_negTwo_two_terms` — together with `ZOneLineNegTwo` (the
  unweighted form at degree `−N`), and the proved equivalence
  `negTwoDiagonalWeight_iff_zOneLine` (both directions, using only the landed `AssemblyDiagonal`).
  The two-direction bridge to the batch-182 residual is
  `negTwoBlockCollapse_iff_diagonalWeight` under `hs` and `hc : form a b = −2 → E.c b a = 1`; the
  consumer at FLM's datum is
  `residueNegativeAligned_evenCocycle_of_diagonalWeight (h) (h3) : ResidueNegativeAligned`.
* **Why the `−2` line is genuinely new, and measured**: `ratioCoeff (−2) k = k + 1` (`ratioCoeff_neg_two`),
  so the chamber difference is a **weighted** antidiagonal sum with weight `p + m` — that extra factor is
  exactly why the batch-181 `AssemblyDiagonal` machinery (the `z_0^0` line) cannot close it.  The landed
  `MM.FLM86TPS.diagSumWeighted` (with window independence) is reused; the landed `z_0^1` lemmas
  `MM.FLM84DG.opOpenBox_deriv_modeFam`/`_heis` are the `z_2`-derivative at a **free** vertex operator
  with `DiagCollapse`/`DerivNormalOrdered` as hypotheses, not the carrier identity.

**B183-B ACCEPTED — `Definitions/MMFLM86CoherentField.lean`** (371 lines, namespace `MM.FLM86CF2`,
13 guards, all exactly the three-axiom set).

* **The honest provable form of the coherent-state field is a RECURSION, not a closed form**:
  `coherentStep E a w k = ((k+1:ℚ)⁻¹) • Σ_{r<k+1} reflectFam (actionFamily (factorFactorZero (r+1) … a)
  (YzeroField (baseAligned E) (creationExpVL a (k−r) w)))`, `coherentFieldNat` its `Nat.rec` from
  `ZAligned E w`, and `coherentField_recursion / coherentFieldNat_eq / coherentField_pos` the three
  forms of the statement: **degree `k+1` = one (8.5.5) step against the fields of the lower coherent
  states** (`^+` tail included), bottoming out at the closed anchor `ε(a,b)•YFam E (a+b)`.  `k = 0/1`
  reproduce batch 182-B's landed cases.  The identity that would collapse the recursion
  (`Y(E⁻(z₀)w,z) = 8E⁻(z₀)Y(w,z)8`) is **not landed** (it is the `NonvacuumCreatedStep`-adjacent
  input, and `conv` is a forbidden route); the self-reference is made a theorem
  (`YzeroField_creationExpVL_fam`), not a hidden step.
* **The `≤ −3` case is expanded to the honest carrier form**:
  `NegThreeExpandedResidualAligned` with
  `negThreeResidualAligned_iff_expanded : NegThreeAndBelowResidualAligned E ↔
  NegThreeExpandedResidualAligned E` (both directions), replacing every summand's
  `ZAligned E (creationExpVL a (N−1−i) anchorVL)` by `coherentFieldNat E a anchorVL (N−1−i)`; and
  `residueDegree_eq`: the `i`-th (8.5.26) summand's state index is the degree `N−1−i`.
* **The Fock display is NOT reachable by this route** (measured): its left-hand object is
  `ZAligned E (YFam E a (−j−1) (creationExpVL b l (ι b)))` — an honest **mode of `a`** applied to the
  coherent state of `b` — not `ZAligned E (creationExpVL a k anchor)`; reaching it needs an intertwining
  at the honest modes, which is not landed.

**Where Theorem 8.10.2 stands at the end of batch 183.**  `hjac` = `Corollary851 (ZAligned E) u v K`;
the lattice-pair cases `0 ≤ ⟨a,b⟩` and `⟨a,b⟩ = −1` are **closed**; the live residuals are
**(1)** `NegTwoDiagonalWeight` (the weighted `z_0^1` line at `⟨a,b⟩ = −2`), **(2)**
`NegThreeExpandedResidualAligned` (the recursion form of `≤ −3`), **(3)** the Fock half's created-state
evaluation (now known to need a mode intertwining), and **(4)** the boundary restatement.
**Batch 184**: (A) `NegTwoDiagonalWeight` — the weighted antidiagonal identity; (B) the recursion
collapse for `≤ −3`, or the honest-mode intertwining if the collapse is out of reach.

**Brief errors this batch**: the batch-181 machinery is the `z_0^0` line and is already landed;
`ratioCoeff (−2) k = k+1` makes the `−2` line weighted; `NegTwoBlockCollapse` is the analogue of the
**refuted** `NegOneBlockCollapse`, not of `AntidiagonalCollapse`; `ZAligned` exists twice
(`MM.FLM810JZF`, `MM.FLM810JF2`, definitionally equal); and "induction on `k`" for the coherent field
yields a recursion, not an evaluation.

---

### Batch 184 — the `−2` residual is the `p = 1` line of the landed display family `diagFam` (a both-directions reduction, cocycle discharged); and the honest-mode intertwining evaluates the Fock LHS as a ratio sum of the coherent recursion

Two new files, two fresh agents, one batch; both verified here (rebuild from deleted `.olean`s,
`--- OK ---` for each, `Probe/forbidden_tokens.py` at zero, the in-file `#guard_msgs`, and the
`FinalCheck.lean` acceptance build after wiring them in).

**B184-A ACCEPTED — `Definitions/MMFLM86NegTwoWeighted.lean`** (361 lines, namespace `MM.FLM86NTW`,
9 guards, all exactly the three-axiom set).

* **The weighted identity is NOT proved; the exact both-directions reduction is landed.**  New named
  residual `NegTwoDisplayZOne (E)`: for `form a b = −2`, the **unweighted** `p`-antidiagonal sum
  (`diagSumWeighted (fun p ↦ (p:F))`) equals `iteratePow (ZAligned E) (ι a) (ι b) (form a b + 1) (m+n) v`
  — i.e. **the `p = 1` coefficient of the landed display family `MM.FLM86D86.diagFam`/`displayP`**.
* Equivalences, both directions: `zOneLine_iff_displayZOne` (no hypotheses),
  `negTwoDiagonalWeight_iff_displayZOne`, `negTwoExactResidualAligned_iff_displayZOne` (under
  `hc : form a b = −2 → E.c b a = 1`), and **`negTwoExactResidualAligned_evenCocycle_iff_displayZOne`**
  with the cocycle discharged at `C.toCentralExtension h2`; conditional consumer
  `residueNegativeAligned_evenCocycle_of_displayZOne`.
* Unconditional support: `diagSumWeighted_eq_diagSum_firstDeriv` (the weight is the trivial
  `p = m+1` reindexing of the first-variable derivative), `diagFam_one_eq_diagSumWeighted`,
  `diagFam_zero_eq_diagSum` (**the `p = 0` line IS the landed `assemblyDiagonal`**),
  `ZAligned_creationExpVL_one_apply` (the (8.5.11) evaluation of the created state) and
  `zOneLine_diagFam`.
* **Route correction (measured)**: the weight is **not** the (8.4.25) expansion differentiated — it is
  the (8.2.14) falling factorial `binomCoeff m 1 = m`, i.e. the `z_0^1` coefficient of the landed
  display family.  And the landed (8.4.40) `z_0^1` lemmas
  (`MM.FLM84DG.opOpenBox_deriv_modeFam(_heis)`, `MM.FLM8512.productExpansion8512_apply_one`) **cannot be
  instantiated at the boundary field at all**: they carry a negative-power vanishing `hX` that is false
  for `YFam E a` (component `−1` is `creationExpVL a (−1−form a c)(anchorVL E a c)`, nonzero at pairing
  `−1`; `MM.FLM84LF.YFam_iotaVL`).

**B184-B ACCEPTED — `Definitions/MMFLM86ModeIntertwine.lean`** (472 lines, namespace `MM.FLM86MI`,
13 guards, all exactly the three-axiom set).

* **The honest-mode intertwining is landed** (symmetric form, `0 ≤ ⟨a,b⟩`):
  `modeIntertwine : YFam E a M1 (ZAligned E (ι b) M2 v) = Σ_{i ≤ ⟨a,b⟩} ratioCoeff ⟨a,b⟩ i •
  normOrdered2 a b (−M1−1+i−⟨a,b⟩) (−M2−1−i) v` — an honest mode of `a` applied to an honest-mode-created
  state is the finite (8.4.25) ratio sum of normal-ordered blocks.  Lattice-generator slice:
  `ZAligned_mode_iotaVL` (`ZAligned E (YFam E a q (ι c)) = ε(a,c) • (if 0 ≤ −q−1−⟨a,c⟩ then
  coherentFieldNat E a (ι(a+c)) ((−q−1−⟨a,c⟩).toNat) else 0)`).
* **The Fock LHS is now fully evaluated**: `ZAligned_createdState` writes
  `ZAligned E (YFam E a (−j−1)(creationExpVL b l (ι b)))` as a finite ratio sum whose summands are batch
  183-B's `coherentFieldNat` at the **coherent anchors** — no field at a mode-applied state remains.  And
  **`display8610TwoGenerators_iff_coherentForm`** (both directions) identifies the display's generator
  slice with `CoherentCreatedForm`.
* New (8.6.9)-side facts: `normOrdered2_vacuum` (the block on `ι(1)` is the double coherent state
  `E⁻_a(z₁)E⁻_b(z₂) anchorVL E a b`) and `ZAligned_normOrdered2_vacuum` (its field is the coherent
  recursion), plus the required bridge **`ZAligned_JZF_eq : FLM810JZF.ZAligned = FLM810JF2.ZAligned`** —
  the repository carries two syntactically distinct, definitionally equal copies, and `rw` cannot cross
  them (CF2 speaks JZF, the (8.6.10) chain speaks JF2); successors must use the bridge.
* Correction: `ZAligned_creationStep`/`ZAligned_spec_cons` are **not** consumed — the Fock mode is the
  assembled `YFam`, not a Heisenberg creation monomial (on a lattice generator `YFam E a q (ι c)` is an
  infinite combination of modes).

**Where Theorem 8.10.2 stands at the end of batch 184.**  `hjac` = `Corollary851 (ZAligned E) u v K`;
`0 ≤ ⟨a,b⟩` and `⟨a,b⟩ = −1` are **closed**; the live residuals are **(1)** `NegTwoDisplayZOne` (the
`p = 1` line of the landed `diagFam` family, the `⟨a,b⟩ = −2` case), **(2)** the `≤ −3` case in the
coherent-recursion form, **(3)** `CoherentCreatedForm` (the Fock comparison identity), and **(4)** the
boundary restatement.  **Batch 185**: (A) `NegTwoDisplayZOne` — look for the `p`-recursion of
`diagFam`/`displayP` (the `p = 0` line is `assemblyDiagonal`, so a recursion in `p` would give
`p = 1`); (B) `CoherentCreatedForm` — the (8.6.9)/(8.6.10)-at-`k = 2` re-expansion
`Y(8Y(a,z₁)Y(b,z₂)8 ι(1), z)`.

**Brief errors this batch**: the "differentiate (8.4.25)" route for the weight; the claim that the
(8.4.40) `z_0^1` lines apply at the boundary field (their `hX` fails); "equivalently `ZAligned E
(modeVL a m x)`" for the intertwining; and the new Lean traps (`Finset.sum_subset`'s orientation and
metavariable, the `zsmul` elaboration of an `End`-scalar product, the explicit `LinearMap.smul_apply`).

---

### Batch 185 — the `−2` case reduces to a DERIVATIVE normal-ordering identity (`NegTwoFirstDerivZero`; the `p`-recursion route is refuted by counterexample); and `CoherentCreatedForm` is pinned as one dictionary with `CoherentCreatedStep` and the landed `IterateProductStep`

Three new files, two fresh agents, one batch; all verified here (rebuild from deleted `.olean`s,
`--- OK ---` for each, `Probe/forbidden_tokens.py` at zero, the in-file `#guard_msgs`, and the
`FinalCheck.lean` acceptance build after wiring them in).

**B185-A ACCEPTED — `Definitions/MMFLM86NegTwoPOne.lean`** (418 lines, namespace `MM.FLM86NPO`,
9 guards, all exactly the three-axiom set).

* **The brief's `p`-recursion route is REFUTED by counterexample**: `diagFam_apply` makes `diagFam` the
  diagonal Taylor family of the substituted `X(z₂+z₀,z₂)`, and its `p = 0` and `p = 1` coefficients read
  **different** coefficients of `X` (`X m (−j−1−m)` versus `X m (−j−m)`);
  `exists_diagFam_zero_eq_zero_one_ne_zero` exhibits, for every nonzero `v`, a family with vanishing
  `p = 0` line and nonvanishing `p = 1` line.  So the landed `p = 0` line (`assemblyDiagonal`) cannot
  give `p = 1`.
* Landed instead: **the complete `p`-recursion** `diagFam_succ_eq_diagFam_firstDeriv`
  (`(p+1) • diagFam X M1 M2 (p+1) j = diagFam (firstDeriv X) (M1−1) M2 p j`), i.e. every step replaces
  `X` by `∂_{z₁}X` and only the `p = 0` line is evaluated; and **the exact named sub-residual
  `NegTwoFirstDerivZero (E)`** — the `p = 0` line of `firstDeriv (normOrdered2 a b)` equals
  `ZAligned E (creationExpVL a 1 (anchorVL E a b)) (m+n) v` — with **both directions**:
  `zOneLine_iff_firstDerivZero`, `negTwoDisplayZOne_iff_firstDerivZero` and
  `negTwoExactResidualAligned_evenCocycle_iff_firstDerivZero` at `C.toCentralExtension h2`.
* **The honest remaining input is a Leibniz/normal-ordering identity, not bookkeeping**:
  `∂_{z₁}|_{z₁=z₂} 8Y(a,z₁)Y(b,z₂)8 = 8a(z₂)Y(ab,z₂)8` (source reading
  `∂_z Y(u,z) = Y(L₋₁u,z)`, `L₋₁ι(a) = a(−2)1`) — a **weight-two Fock field times a lattice field**,
  outside the batch-180/181 two-lattice-label fibre computation.
* The only landed statement yielding the display's `z_0^1` line,
  `MM.FLM8512Done.productExpansion8512_apply_one`, is not a route: it assumes the whole display
  `ProductExpansion8512` (not landed at `⟨a,b⟩ = −2`) and a negative-power vanishing `hX` that is
  **false** at `ZAligned`.
* Recommended routes for the successor: (i) the (8.4.36) substitution at `z_0^1` via the landed
  resummation layer (`MM.FLM84Taylor.resumCoeff`/`taylorCoeff`, `MMFLM86TaylorOperator`) applied to the
  `E⁻`/block/`E⁺` decomposition of `normOrdered2`; (ii) generalise the batch-180/181 fibre computation
  from `normOrdered2 a b` to `8Y(u,z₁)Y(b,z₂)8` for a general Fock state `u` (there `u = a(−2)1`).

**B185-B ACCEPTED — `Definitions/MMFLM86CoherentCreated.lean` (352 lines) and
`MMFLM86CoherentCreatedStep.lean` (160 lines)** (namespaces `MM.FLM86CCF`/`MM.FLM86CCS`; 13 guards,
all exactly the three-axiom set).

* **`CoherentCreatedForm` is NOT landed, and the hoped-for "same normal-ordered object" does not exist**:
  the landed (8.4.25) expansions give the LHS as a ratio sum of **fields at** normal-ordered blocks (the
  (8.6.9) currency) and the RHS as a window sum of normal-ordered blocks **applied to a state** — different
  currencies, different index sets, no reindexing.
* The exact residual is `MM.FLM86CCF.CoherentCreatedStep (E)` ((8.4.25) at the non-vacuum created state
  = (8.5.14)/(8.6.10) at `k = 2`), and the **both-directions** dictionary
  `coherentCreatedForm_iff_createdStep` (under `hs` and `0 ≤ ⟨a,b⟩`), plus
  `coherentCreatedForm_iff_normOrdered2` (the termwise (8.6.9) reading) and
  `coherentCreatedForm_iff_ratioForm(_nonneg)` (↔ `RatioForm8610TwoGenerators`, datum discharged).
* The sibling lands the implication chain `IterateProductStep (ZAligned E) convOneSided ⇒
  Display8610TwoOneSided ⇒ Display8610TwoGenerators ⇒ CoherentCreatedForm` (no `hnd`) and
  `createdStep_of_iterateProductStep` (with the landed `k = 1` display, hence `hnd`/`hs`).  So the Fock
  half's four statements are now **one dictionary**, all implied by the repository's already-named
  `IterateProductStep (ZAligned E) convOneSided` (the converse is unavailable — it quantifies over all
  states).

**A correction of record.**  Batch 178-A's report that `Definitions/MMFLM85StepLinear.lean` "is imported
by nothing" is **wrong**: `FinalCheck.lean:343` has imported it since batch 128 (commit `94b9097`), and
`MMFLM85NonvacuumStep.lean:47` imports it too.  The batch-178 entry's "repo trap worth keeping" is
withdrawn; the module is on the delivery path.

**Where Theorem 8.10.2 stands at the end of batch 185.**  `hjac` = `Corollary851 (ZAligned E) u v K`;
`0 ≤ ⟨a,b⟩` and `⟨a,b⟩ = −1` are **closed**; the live residuals are **(1)** `NegTwoFirstDerivZero` (the
derivative normal-ordering identity, `⟨a,b⟩ = −2`), **(2)** the `≤ −3` case, **(3)**
`CoherentCreatedStep` (the Fock comparison), and **(4)** the boundary restatement.
**Batch 186**: (A) `NegTwoFirstDerivZero` by the recorded routes; (B) `CoherentCreatedStep`.

**Brief errors this batch**: the `p`-recursion route (refuted); "compare coefficientwise at the coherent
anchors" (the sums have different index sets); `MM.FLM86M.Display869`/`Display8610` are vacuous defs;
`ZAligned_JZF_eq` is not needed for the Fock target; and the "orphan module" claim.

---

### Batch 186 — the even-cocycle boundary theorem and the FACE AUDIT (only `l = 0` has producers; the `l ≠ 0` faces are untouched territory); and the `convWith2` Leibniz rule splits the `−2` derivative residual into three fibre collapses

Two new files, two fresh agents, one batch; both verified here (rebuild from deleted `.olean`s,
`--- OK ---` for each, `Probe/forbidden_tokens.py` at zero, the in-file `#guard_msgs`, and the
`FinalCheck.lean` acceptance build after wiring them in).

**B186-B ACCEPTED — `Definitions/MMFLM810TheoremEven.lean`** (300 lines, namespace `MM.FLM810TE`,
6 guards, all exactly the three-axiom set).  **The audit is the deliverable.**

* **The even-cocycle boundary is landed**:
  `theorem8102_of_residuals (hnd) (hs) (C : EvenCocycle L form) (h2) (S) (hpd) (hW) (hfin)
  (hstep : EvenCocycleJacobiStep C h2) (hres : EvenCocycleResiduals C h2) : Theorem8102`, with
  `EvenCocycleResiduals C h2 := NegTwoFirstDerivZero E ∧ NegThreeExpandedResidualAligned E ∧
  CoherentCreatedStep E` at `E := C.toCentralExtension h2`, and
  `EvenCocycleJacobiStep C h2 := EvenCocycleResiduals C h2 → JacobiEvenCocycle C h2` (the
  residual-to-`hjac` step, named).  Also: `theorem8102_of_evenJacobi`,
  `theorem8102_of_jacobiResidual` (`JacobiResidualAligned` + `Proposition8107` → `Theorem8102`),
  `FockCreatedStep` + `coherentCreatedStep_of_fockCreatedStep`, and the both-directions
  `residueNegativeAligned_evenCocycle_iff_residuals`.
* **A formalization gap is closed**: batch 169 R1's "`JacobiOnStateTargets` ⟺ `JacobiAxiom`" had only
  one direction in the tree; this file lands the converse
  (`jacobiOnStateTargets_of_jacobiAxiom`, `jacobiAxiom_iff_stateTargets`), so the §8.10 output can now
  feed the boundary.
* **THE AUDIT (every claim file:line in the file's header)**:
  * `Res_{z_0}` (`MMFLM88.res_z0_face`, literally `h 0 m n`) **is** Corollary 8.5.1.  At the
    even-cocycle field: `0 ≤ ⟨a,b⟩` unconditional (batch 170), `⟨a,b⟩ = −1` unconditional (batch 181);
    open: `⟨a,b⟩ = −2` = `NegTwoExactResidualAligned` ≡ `NegTwoFirstDerivZero`, `≤ −3` =
    `NegThreeAndBelowResidualAligned` ≡ `NegThreeExpandedResidualAligned`, and the Fock half =
    `CoherentCreatedStep` (implied by `IterateProductStep (ZAligned E) convOneSided`).
  * `Res_{z_1}`/`Res_{z_2}` (`MM.FLM88.res_z1_face`/`res_z2_face`) are **extractions from the Jacobi
    hypothesis**, not independent theorems; their only conditional content is
    `MM.FLM810JacobiRoute.lean:147`/`:179`, both taking `h : JacobiIdentity`.
  * **`l, m, n ≠ 0`: NOTHING landed.**  So `hjac` is strictly more than `Corollary851`: only its
    `l = 0` face has producers.  *This is the campaign's largest untouched gap.*
  * Route §8.8 (Theorem 8.8.9, whose proof repeats Theorem 8.6.1's, printed p. 231): landed is the
    `JacobiIdentity ↔ JacobiDelta` equivalence and the extraction halves; missing is the identity itself,
    the §8.6 engine ((8.6.5)/(8.6.6)/(8.6.8)/(8.6.9--10)) and the frontier `Source861Aligned` ⟺
    `Residual861Aligned`.
  * Route §8.10: `Proposition8103/8104/8105/8107` are all `def … : Prop`; landed and consumable is the
    calculus (`theta_eq_commutator`, `commutativity_iff_theta_zero`, σ `substFormAlg_iff` — the naive
    `SubstForm` iff being **false**, `(8.10.73)`, `recoverySplit_exists`); missing is the `ι_{12}`-bridge
    (measured unprovable as landed), `RecoveryDisplay`, the `RecoverySplit` use, and 8.10.7.
* **Unconditional `Theorem8102` needs**: (i) `NegTwoFirstDerivZero`; (ii) `NegThreeExpandedResidualAligned`;
  (iii) the Fock dictionary; (iv) `EvenCocycleJacobiStep` — i.e. the `l ≠ 0` faces plus the assembly from
  `Res_{z_0}` to (8.10.8) (Theorem 8.8.9 or Proposition 8.10.7); plus the six boundary inputs.
  **Recommended next batches**: the `l ≠ 0` faces (untouched; `MMFLM810Route` has the swap machinery),
  then the §8.10 `ι`-bridge or the §8.6 engine, and keep the Fock gap named as `IterateProductStep`.

**B186-A ACCEPTED — `Definitions/MMFLM86FirstDerivLine.lean`** (394 lines against a 300-line target,
reported; namespace `MM.FLM86FDL`, 9 guards, all exactly the three-axiom set).

* **The `convWith2` Leibniz rule is landed** (the flat `deriv_flatMul` of batch 179 does not apply to the
  two-variable Cauchy product): `firstDeriv_convWith2 : firstDeriv (convWith2 X Y hY) =
  convWith2 (firstDeriv X) Y hY + convWith2 X (firstDeriv Y) (locallyFinite2_firstDeriv hY)`, with
  `locallyFinite2_firstDeriv`; hence the three-piece decomposition
  `firstDeriv_normOrdered2_eq : firstDeriv (normOrdered2 E a b) = creationDerivPiece E a b +
  blockDerivPiece E a b + annihDerivPiece E a b` (one factor differentiated per piece), plus the factor
  derivatives (`firstDeriv_creationPair2_flatMul` via the landed `MM.FLM84DY.deriv_creationExpVL`,
  `firstDeriv_annihPair2`, `firstDeriv_midBlock2`) and `diagSum_add`.
* The `−2` residual is restated, **both directions**, as `FirstDerivThreeFactorLine (E)`
  (`negTwoFirstDerivZero_iff_threeFactor`, `negTwoExactResidualAligned_evenCocycle_iff_threeFactor`) —
  an equivalent form whose left-hand side is closed and whose content is three explicit per-state
  diagonal collapses of landed `convWith2` expressions.
* **Measured advice for the successor**: attack `creationDerivPiece` first (its `E⁻` fibre collapse is
  available from `flatMul_creationExpVL`/`sum_creationPair2_fiber` plus `deriv_creationExpVL`); the
  `−2` line is **weighted**, so the batch-181 regrouping cannot be reused verbatim (the weight couples to
  the creation index); and the route-(ii) picture "`:Y(a(−2)1,z₁)Y(b,z₂):`" would first need a landed
  `Y(u,z)` for Fock `u` plus a `firstDeriv = Y(L₋₁·)` bridge — neither exists.

**Where Theorem 8.10.2 stands at the end of batch 186.**  The boundary is now stated honestly at the
even cocycle, with the exact input list; the `l = 0` face is closed for `0 ≤ ⟨a,b⟩` and `⟨a,b⟩ = −1` and
reduced elsewhere; the `l ≠ 0` faces and the two continuations are the campaign's remaining fronts.

**Brief errors this batch**: `MM.FLM86TK.*` does not exist (the θ layer is `MM.FLM810TK`);
`convWith2` vs `flatMul` conflated; the resummation layer adds nothing past batch 185-A; and route (ii)
needs two unlanded bridges.

---

### Batch 187 — the `l ≠ 0` front is OPENED (the degenerate `l ≥ 0` chamber at `0 ≤ ⟨a,b⟩` is a theorem; the `l < 0` iterate side is computed; the first genuinely nonzero face is named); and the `−2` creation piece is collapsed

Three new files, two fresh agents, one batch; all verified here (rebuild from deleted `.olean`s,
`--- OK ---` for each, `Probe/forbidden_tokens.py` at zero, the in-file `#guard_msgs`, and the
`FinalCheck.lean` acceptance build after wiring them in).

**B187-A ACCEPTED — `Definitions/MMFLM810JacobiNonzeroFaces.lean`** (300 lines, namespace `MM.FLM810JNF`,
2 guards, both exactly the three-axiom set).  **The largest gap of the campaign now has land.**

* **Inventory**: the only general-`l` statements in the tree were `JacobiIdentity`/`JacobiDelta` (defs),
  the reindexing `jacobiIdentity_iff_delta`, the window-stability `crossBracket_eq_of_window`, and
  `res_z1_face`/`res_z2_face`/`corollary8818_of_jacobiIdentity` — **all conditional on
  `h : JacobiIdentity`** (extractions, no computation).  At the boundary field, only `l = 0` had
  producers; the sole landed `l ≠ 0` identity anywhere was the scalar toy model
  (`MMFLM810ScalarJacobi.lean:542`, whose `Y` is not a VOA field).
* **Landed at the boundary field `ZAligned E`, `E = C.toCentralExtension h2`**:
  `crossTerm_ZAligned_lattice` (+ `crossTermSwap`), `yFam_mul_comm_of_nonneg`,
  `deltaExpCoeff_reflect_of_nonneg`, `sum_crossTerms_eq_smul_sum_crossTermSwaps` (the (8.8.21) swap
  identity), and the **first `l ≠ 0` face**: `crossBracket_ZAligned_lattice_of_nonneg = 0` and
  `crossIterate_ZAligned_lattice_eq_zero_of_nonneg = 0`, hence
  **`crossBracket_eq_crossIterate_ZAligned_lattice_of_nonneg`** — the two sides are **equal for every
  `l ≥ 0`, every `m, n`, and every `N ≥ l.toNat + 1`** (both zero) at `0 ≤ ⟨a,b⟩`.
* **The `l < 0` iterate side is computed**:
  `crossIterate_ZAligned_lattice_eq_sum_of_nonpos` — the explicit finite sum over `i < (−l).toNat` of
  `binom(m,i) • iterateOf (ι a) (ι b) (l+i) (m+n−i)`.
* **The named residual `FirstNegativeFaceAligned (E)`** — the `l = −1` face at `0 ≤ ⟨a,b⟩` (the first
  face with genuinely nonzero sides), with the consumer
  `jacobiIdentity_neg_one_slice_of_firstNegativeFace` yielding exactly that slice of (8.8.41) in the
  per-monomial `∃N` shape of (8.10.8).
* **Route correction**: no landed reduction of a general `l ≠ 0` face to `l = 0` exists, and the book's
  shift display is (8.6.28) (which the source itself calls "formal ... suggestive"), not (8.6.29)/(8.6.30)
  (the iterate definition and the commutator formula); for `l ≥ 0` the face reduces to a sum of
  **commutators** (landed here), not to `l = 0` plus a shift.  **Next target: the `l = −1` face**,
  through the landed (8.4.25) product, not by reindexing.

**B187-B ACCEPTED — `Definitions/MMFLM86CreationDerivPiece.lean` (576 lines) and
`MMFLM86CreationDerivPieceLine.lean` (344 lines)** (namespaces `MM.FLM86CDP`/`MM.FLM86CDPL`;
13 guards, all exactly the three-axiom set).

* **The `E⁻` fibre collapse is landed**: `sum_creationDerivPair_window` — the fibre sum is
  `flatMul (deriv E⁻_a) E⁻_b` (one of the two orderings) — together with
  `flatMul_deriv_creationExpVL_add : deriv E⁻_{a+b} = flatMul (deriv E⁻_a) E⁻_b +
  flatMul (deriv E⁻_b) E⁻_a`.  Hence **no single piece can equal the target's right-hand side**: the
  creation piece carries exactly half the derivative.
* `diagSum_creationDerivPiece(_regrouped)` (the diagonal sum of the creation piece **is** the
  double-window sum of the collapsed factors), `diagSum_blockDerivPiece` (the block's first layer), and
  `sum_sum_window_regroup_factor` — batch 181's regrouping with an **arbitrary** factor, with
  `sum_sum_window_regroup_landed_agrees` proving its `C = creationExpVL (a+b)` instance is
  *syntactically* the landed `MM.FLM86AR.sum_sum_window_regroup` (an independent cross-check).
* **Pieces**: creation **LANDED**; block **LANDED to the first layer** (its second layer — the weighted
  antidiagonal sum of the differentiated `a(0)` block, value `⟨a,c⟩` on `x ⊗ ι(c)` — is open);
  annihilation **NOT landed** (the mirrored flat recursion `(d/dz)E⁺ = −a(z)⁺E⁺` is absent at the
  carrier).  The residual is restated as `CreationClosedLine`/`CreationBlockClosedLine`
  (**both directions**).
* **Corrections**: the brief's "`Σ_fibre (dE⁻_a)E⁻_b = d(E⁻_{a+b})`" is **false** (the two orderings each
  contribute half); and the weight warning is only half right — `dE⁻_a`'s extra index `i = −1` carries
  weight `0`, so the window/containment hypothesis is **verbatim** batch 181's and the regrouping is
  reusable with the factor replaced (proved).

**Where Theorem 8.10.2 stands at the end of batch 187.**  Both remaining fronts now have land: the
`l = 0` face is closed for `0 ≤ ⟨a,b⟩` and `⟨a,b⟩ = −1` (with `−2`, `≤ −3`, Fock named), and the
`l ≠ 0` front has its degenerate chamber (`l ≥ 0`, `0 ≤ ⟨a,b⟩`) as a theorem, its `l < 0` iterate side
computed, and its first nonzero face named `FirstNegativeFaceAligned`.  **Batch 188**: (A) the `l = −1`
face; (B) the annihilation derivative piece (the mirror of `creationDerivPiece`).

**Brief errors this batch**: the shift-display attribution (`(8.6.28)`, not (8.6.29)/(8.6.30)); "nothing
landed for `l ≠ 0`" (the scalar toy model was landed, though not at the boundary field); the "`l ≠ 0`
reduces to `l = 0` plus a shift" picture; the `E⁻` fibre-sum identity; and the `sum_creationPair2_fiber`
name (`sum_creationPair2_antidiagonal`/`sum_creationPair2_window`/`sum_annihPair2_fiber` are the real
ones).

---

### Batch 188 — the `l = −1` face becomes a pure honest-lattice-VOA identity `(∗)` (both directions, all boundary-field ingredients discharged); and the `E⁺` mirror collapse lands the third Leibniz piece of the `−2` case

Three new files, two fresh agents, one batch; all verified here (rebuild from deleted `.olean`s,
`--- OK ---` for each, `Probe/forbidden_tokens.py` at zero, the in-file `#guard_msgs`, and the
`FinalCheck.lean` acceptance build after wiring them in).

**B188-A ACCEPTED — `Definitions/MMFLM810NegOneFace.lean`** (445 lines against a 300-line target,
reported; namespace `MM.FLM810NOF`, 7 guards, all exactly the three-axiom set).

* **The cross side is the WHOLE diagonal**: `deltaExpCoeff i (−1) = 1` (char 0, the factorial hypothesis
  discharged) and `(−1)^{−1} = −1` give
  `crossBracket (ZAligned E) (ι a) (ι b) N (−1) m n = Σ_{i<N} YFam a (m−1−i)·YFam b (n+i) +
  Σ_{i<N} YFam b (n−1−i)·YFam a (m+i)` — the two halves of `p+q = m+n−1` of the honest product (not
  merely a δ-term: it is (8.6.28) at `z_0^0`).
* **The iterate side is a closed form** (`crossIterate_ZAligned_lattice_neg_one`,
  `iterateOf_…_eq_ite`): the iterate state is `creationExpVL a (−⟨a,b⟩)(anchorVL E a b)`, so
  `iterateOf (ZAligned E) (ι a) (ι b) (−1) (m+n) = if ⟨a,b⟩ = 0 then ε(a,b)·YFam (a+b) (m+n) else 0`.
  **The boundary-field repair drops out of this face.**
* **The residual `NegOneFaceAligned (E)`** — the source's per-element reading (printed p. 244) — is
  proved **equivalent, both directions** (`negOneFaceAligned_iff_crossBracket`), to the **pure
  honest-lattice-VOA identity**
  `(∗) Σ_{p+q=m+n−1} YFam a p · YFam b q = (0 if ⟨a,b⟩ ≠ 0 else ε(a,b)·YFam (a+b) (m+n))`, per element.
  **`(∗)` contains no `ZAligned`**, so any honest-VOA proof of it makes the `l = −1` slice
  unconditional.  Also landed unconditionally: the cross-side finiteness from (8.10.5)
  (`crossWindowOn_ZAligned_lattice_neg_one`), the transport `negOneFaceAligned_of_firstNegativeFace`
  (batch 187-A's def ⟹ the source reading), and the slice consumer
  `jacobiIdentity_neg_one_slice_of_negOneFace`.
* **Slices unconditional today**: `l = 0` and `l ≥ 0` (at `0 ≤ ⟨a,b⟩`).  **Not landed**: `(∗)` itself,
  i.e. the two chambers (`1 ≤ ⟨a,b⟩`: cross side 0; `⟨a,b⟩ = 0`: cross side `= ε·YFam (a+b)`).
* **Corrections**: the "reduce to the landed `l = 0` face" advice is unsupported (there the `l = 0` face is
  commutativity with both sides 0 and carries no information on the diagonal sums); and batch 187-A's
  `FirstNegativeFaceAligned` window conjunct is **operator-valued**, which — by the repo's own W5/W7
  measurement (`jacobi8108_iff_zeroProduct`) — is *not* the source's per-element reading; this batch
  states the residual per element and proves only 187-A ⟹ mine (the converse needs operator-level
  vanishing, not supplied by (8.10.5)).  **Naming collision recorded**: `MM.FLM86RN.NegOneResidualAligned`
  is the `⟨a,b⟩ = −1` *pairing* at `l = 0`, a different object from this `l = −1` face.

**B188-B ACCEPTED — `Definitions/MMFLM86AnnihDerivPiece.lean` (708 lines) and
`MMFLM86AnnihDerivPieceLine.lean` (332 lines)** (namespaces `MM.FLM86ADP`/`MM.FLM86ADPL`; 15 guards,
all exactly the three-axiom set).

* **The mirror flat recursion and the `E⁺` fibre collapse are landed**:
  `deriv_annihExpField`, `deriv_annihIdx` (the mirrored recursion),
  `flatMul_deriv_annihIdx_add : flatMul (deriv (annihIdx a)) (annihIdx b) +
  flatMul (annihIdx a) (deriv (annihIdx b)) = deriv (annihIdx (a+b))`, and
  `sum_annihDerivPair_window` (the exact analogue of batch 187-B's `sum_creationDerivPair_window`,
  with the minus sign and the shift `−K−2`), plus its total-degree form and
  `diagSum_annihDerivPiece(_regrouped)` — **all three Leibniz pieces are now in one collapsed normal
  form**.  The generalized regrouping engine `sum_sum_window_regroup_gen` is landed (the landed batch-181
  engine is its `g = annihPair2` instance).
* **Exact residual `AnnihClosedLine`** (the `FirstDerivThreeFactorLine` equation with the three
  `diagSum`s collapsed; one extra conjunct: the block must kill the derivative annihilation pair),
  **both directions** (`firstDerivThreeFactorLine_iff_annihClosed`,
  `negTwoFirstDerivZero_iff_annihClosed`, and at FLM's datum
  `negTwoExactResidualAligned_evenCocycle_iff_annihClosed`).
* **The pieces do NOT assemble yet**, for two measured reasons: (i) the block's **second layer** is missing
  (at the carrier the derivative block's fibre collapse is operator-valued/per-state, not a single
  `z^{(a+b)(0)}` collapse); (ii) each exponential piece carries only **one ordering** of its derivative,
  while the residual's identification with `ZAligned E (creationExpVL a 1 (anchorVL E a b))`
  ((8.4.23)/(8.4.25)) is not landed.  **Corrections**: the "mirror" is an **index reflection**
  `i ↦ −i−1` (same ordering, giving the minus sign and the shift), and the brief's "only the block's
  second layer remains" was false.

**Where Theorem 8.10.2 stands at the end of batch 188.**  Unconditional slices: `l = 0` (at `0 ≤ ⟨a,b⟩`
and `⟨a,b⟩ = −1`, with `−2`/`≤ −3`/Fock named) and `l ≥ 0` (at `0 ≤ ⟨a,b⟩`).  The `l = −1` slice is
conditional on the pure VOA identity `(∗)`; the `−2` case on the three-piece assembly.  **Batch 189**:
(A) the two chambers of `(∗)`; (B) the block's second layer / the ordering identification for the
`−2` case.

**Brief errors this batch**: "the cross side is the δ-term"; the "reduce to `l = 0`" route; the
operator-valued window (187-A) vs the per-element reading (the source); the "`annih` is the other
ordering" picture; and the `sum_sum_window_regroup_factor` reuse claim.

---

### Batch 189 — MILESTONE: `(∗)` and the `l = −1` slice are UNCONDITIONAL at FLM's datum; the `−2` assembly reduces to the (8.4.23) dictionary, and the "missing ordering" premise is REFUTED

Five new files, two fresh agents, one batch; all verified here (rebuild from deleted `.olean`s,
`--- OK ---` for each, `Probe/forbidden_tokens.py` at zero, the in-file `#guard_msgs`, and the
`FinalCheck.lean` acceptance build after wiring them in).

**B189-A ACCEPTED — `Definitions/MMFLM810NegOneChambers.lean` (353 lines) and
`MMFLM810NegOneChambersClose.lean` (232 lines)** (namespaces `MM.FLM810NOC`/`MM.FLM810NOCC`;
14 guards, all exactly the three-axiom set).

* **Both chambers of `(∗)` are landed**: `honestDiag_eq_zero_of_one_le` (`1 ≤ ⟨a,b⟩`: the diagonal sum
  vanishes) and `honestDiag_eq_eps_smul_of_eq_zero` (`⟨a,b⟩ = 0`: it equals
  `ε(a,b)·YFam (a+b) (D+1)`), and **`(∗)` itself unconditionally**:
  `diagSum_honestDiag_eq` — the value is **window-independent** (`exists_honestDiag_window`), so only the
  form symmetry `hs` is needed.
* **MILESTONE: the `l = −1` slice of (8.8.41) at `0 ≤ ⟨a,b⟩` is UNCONDITIONAL at FLM's datum**:
  `negOneFaceAligned_evenCocycle (C) (h2) (hs) : NegOneFaceAligned (C.toCentralExtension h2)` — batch
  188-A's residual is now a **theorem**.  For an arbitrary landed `CentralExtension` the transport needs
  `CocycleCompatible` (`negOneFaceAligned_of_cocycleCompatible`).
* **Route**: the landed (8.4.25) expansion `YFam_mul_apply_eq_sum_ratioCoeff` + **the landed (8.4.20)
  `MM.FLM86AR.assemblyDiagonal`** (read as `assemblyDiagonal_antidiagonal`) + a new
  `sum_range_ratioCoeff : Σ_{j≤N} ratioCoeff N j = (1−1)^N`.
* **Correction of record**: batch 188-A's "the cross side is the single-order diagonal of `(∗)`" is
  **false in general** — `crossTermSwap` is the opposite order (the antisymmetrisation of (8.8.13)), and
  the two orders differ by the constant `1 − c(b,a)(−1)^{⟨a,b⟩}` (the landed (8.4.41)); so the transport
  to the single-order form holds exactly under `CocycleCompatible` (true at the even cocycle), and the
  converse is open.  Also recorded: `rwa` inside the big cross-bracket context exceeded 1.6M heartbeats
  (use `rw … at h; exact h`), and the two `ZAligned` copies must not be mixed.

**B189-B ACCEPTED — `Definitions/MMFLM86BlockSecondLayer.lean` (287 lines),
`MMFLM86BlockDerivRegrouped.lean` (223 lines), `MMFLM86NegTwoAssembly.lean` (336 lines)** (namespaces
`MM.FLM86BSL`/`MM.FLM86BDR`/`MM.FLM86NTA`; 16 guards, all exactly the three-axiom set).

* **THE ORDERING QUESTION IS SETTLED, AND THE PREMISE REFUTED**: the three-piece split is **not** buggy.
  `firstDeriv = d/dz₁` differentiates only the first slot, so the "other orderings" are the `d/dz₂`
  halves — a different operator — and must not be added.  Landed:
  `flatMul_deriv_creationExpVL_half` (`flatMul (dE⁻_a)(E⁻_b)` is the **whole** `a(z)⁻` half, not "half
  of `dE⁻_{a+b}`"), `flatMul_deriv_annihIdx_half` (the whole `a(w)⁺` half), and
  `deriv_creationExpVL_eq_halves`.  So the three pieces carry exactly the `a(z)⁻`, `a(0)`, `a(z)⁺`
  terms of `:a(z)Y(ab,z):`.  **The batch-187-B header sentence "the other half is what the mirror piece
  has to produce" is FALSE** (to be corrected in the next dispatcher edit).
* **The block's second layer is landed**: `sum_firstDeriv_midBlock2_fiber` — the fibre at degree `K'`
  collapses to `ε(a,b)·(actVL (a+b) · (hZeroVL (ofLattice a) · z^{(a+b)(0)}_{K'+1}))u` (note the **index
  shift** `K' ↦ K'+1`), with the per-state window lemmas
  `exists_midGradDeriv_antidiagonal`/`exists_midGradDerivVL_antidiagonal` and
  `diagSum_blockDerivPiece_regrouped` (the third piece in the `(J,K')` normal form).
* **The exact residual is now `NegTwoNormalFormLine`** — all three Leibniz pieces in the collapsed
  normal form, block second layer included — with **both directions**
  (`annihClosed_iff_normalForm`, `NegTwoFirstDerivZero ↔ NegTwoNormalFormLine`,
  `NegTwoExactResidualAligned (C.toCentralExtension h2) ↔ NegTwoNormalFormLine`).  **The remaining gap is
  not an ordering but the (8.4.23) dictionary `Y(a(−1)u,z) = 8a(z)Y(u,z)8` at the aligned boundary
  field.**  Also recorded: `a(0)` must be `hZeroVL (ofLattice a)` — `modeVL a 0 = 0` in this pin.

**Where Theorem 8.10.2 stands at the end of batch 189.**  Unconditional at FLM's datum: the `l = 0` face
for `0 ≤ ⟨a,b⟩` and `⟨a,b⟩ = −1`; the whole `l ≥ 0` chamber at `0 ≤ ⟨a,b⟩`; and now the `l = −1` slice.
Open: the `−2` case (the (8.4.23) dictionary), the `≤ −3` case, the Fock half, the `l ≤ −2` faces, the
continuation to full Jacobi, and the six boundary inputs.  **Batch 190**: (A) the (8.4.23) dictionary
`Y(a(−1)u,z) = 8a(z)Y(u,z)8` at the aligned field — which closes `NegTwoNormalFormLine` if it lands;
(B) the general `l ≤ −2` faces.

**Brief errors this batch**: the "missing ordering"/statement-bug premise (refuted); the "cross side =
single-order diagonal" claim (needs `CocycleCompatible`); the `a(0)` reading (`modeVL a 0 = 0`); and the
chamber-2 input (the full `assemblyDiagonal`, not just `middleBlockDiagonal`).

---

### Batch 190 — the `l = −2` face's `1 ≤ ⟨a,b⟩` chamber is UNCONDITIONAL at FLM's datum; the `(8.4.23)` dictionary is landed (and the `−2` gap is relocated to the FLAT REASSEMBLY); the moment hierarchy is measured

Three new files, two fresh agents, one batch; all verified here (rebuild from deleted `.olean`s,
`--- OK ---` for each, `Probe/forbidden_tokens.py` at zero, the in-file `#guard_msgs`, and the
`FinalCheck.lean` acceptance build after wiring them in).

**B190-B ACCEPTED — `Definitions/MMFLM810NegTwoFace.lean` (479 lines) and
`MMFLM810NegTwoChambers.lean` (571 lines)** (namespaces `MM.FLM810NTF`/`MM.FLM810NTC`; 10 guards, all
exactly the three-axiom set).

* **LANDED, unconditional at FLM's datum**: `negTwoFaceAlignedOneLe_evenCocycle` — the `1 ≤ ⟨a,b⟩`
  chamber of the `l = −2` slice of (8.8.41), **per element**, at every past-vanishing window `N ≥ 2`,
  with both sides evaluated (`ε(a,b)·YFam (a+b) (m+n)` at `⟨a,b⟩ = 1`, `0` for `⟨a,b⟩ ≥ 2`).
  Supporting: `deltaExpCoeff i (−2) = i + 1`, the exact cross side, the two-term iterate collapse **for a
  general `E`**, the three chamber values of `iterateOf (−2)`, (8.10.5) termination, and the new
  first-moment identity `Σ_j j·ratioCoeff N j = −1` at `N = 1` and `0` for `N ≥ 2`.
* **The `⟨a,b⟩ = 0` chamber** is equivalent to `WeightedZOneLineZeroPairing E` — the `z_0^1` line of
  (8.4.36)/(8.4.40) **at pairing-zero pairs**, the *same* identity as the landed
  `MM.FLM86NTD.ZOneLineNegTwo`/`NegTwoDiagonalWeight` (stated at pairing `−2`) — and that **iff
  (`negTwoFaceAligned_iff_weightedZOneLine`) is the one brief item not landed** (budget ran out; all its
  ingredients are in the companion).  **So the `z_0^1` line is now the shared blocker of both the
  `l = 0`, `⟨a,b⟩ = −2` case and the `l = −2`, `⟨a,b⟩ = 0` chamber.**
* **General `l ≤ −2` measured**: the cross side reindexes to a **weighted** antidiagonal with weight
  `ratioCoeff l (m+l−p)`, a degree `L − 1 = −l − 1` polynomial in `p`; `L = 1` needs no moment, `L = 2`
  only the first (killed by `Σ_j ratioCoeff = 0` — *why `l = −2` lands*), and `L ≥ 3` needs the
  **moment hierarchy** `Σ_r r^m · normOrdered2 a b r (K−r) = 0` for `m + 1 ≤ ⟨a,b⟩` (the `m`-th
  derivative of (8.4.40)) — **not landed**; the iterate side needs the Fock-descendant evaluations.
* Corrections: `MM.FLM810JNF/NOF/NOC` are namespaces, not files; at `⟨a,b⟩ = 0` the `l = −2` iterate side
  is **not** closed form (it carries the Fock descendant `ZAligned E (creationExpVL a 1 (anchorVL E a b))`
  — the same open line); and the `l = −2` cross side is **weighted**, so `diagSum_honestDiag_eq` does not
  apply (the two half-lines' union misses `p = m−1`, where the weight vanishes).

**B190-A ACCEPTED — `Definitions/MMFLM86DerivDictionary.lean`** (437 lines, namespace `MM.FLM86DD`,
21 guards, all exactly the three-axiom set).

* **The (8.4.23) dictionary is landed** — and the inventory showed it was **already there at an arbitrary
  vector and base** (`MM.FLM810CI.Yzero_modeVL_eq_actionFamily(_nat)`,
  `MM.FLM85PC.ZAligned_creationStep`, `MM.FLM86NTR.coherentField_one`), together with its `L(−1)`
  companion `BaseResidual (baseAligned E)` (`MM.FLM85PC2.baseResidual_baseAligned_of_symm`, batch 172).
  New: the aligned-field forms (`ZAligned_modeVL_neg_one` general `u`, `ZAligned_creationExpVL_one`, the
  (8.5.4)-state and lattice forms, the `L(−1)` companion), the factor-level facts
  (`factorFactorZero_one_fam`, `splitMinus_factorFactorZero_one_zero = a(−1)`,
  `splitPlus_factorFactorZero_one_neg_one = hZeroVL (ofLattice a) = a(0)`,
  `headOp_factorFactorZero_one_ne_flatMul`), and `ZAligned_creationExpVL_one_anchor` — which kills
  `anchorVL`/`ZAligned`/`Yzero` from the residual's right-hand side.
* **The `−2` gap is RELOCATED to the flat reassembly**: the residual is now `DerivDictionaryLine`
  (defined, **both directions** to `NegTwoFirstDerivZero`/`NegTwoNormalFormLine`/
  `NegTwoExactResidualAligned`): matching the three collapsed flat sums with
  `headOp (heisFamZero a) + tailOp` of `Y(a+b,z)` ((8.4.36)/(8.4.40) at `z_0^1`).  Measured: the creation
  half is already in `flatMul (splitMinus (heisFam a)) (creationExpVL (a+b))` shape; the `a(0)` summand
  belongs to the tail and the negative `mid`-degrees must be carried by the free `(J,K')` windows.
* Correction of record: `BaseResidual`/`generalStateDeriv` is **not** (8.4.23) in disguise — it is its
  derivative/`L(−1)` companion at `u = ι(a)`, and it is a **theorem** at the aligned base.

**Where Theorem 8.10.2 stands at the end of batch 190.**  Unconditional at FLM's datum: the `l = 0` face
for `0 ≤ ⟨a,b⟩` and `⟨a,b⟩ = −1`; the `l ≥ 0` chamber at `0 ≤ ⟨a,b⟩`; the `l = −1` slice; and now the
`l = −2` face's `1 ≤ ⟨a,b⟩` chamber.  **The single shared open object** is the `z_0^1` line
(`DerivDictionaryLine` = `ZOneLineNegTwo` = `WeightedZOneLineZeroPairing`): the flat reassembly /
(8.4.36) at `z_0^1`.  **Batch 191**: (A) the `⟨a,b⟩ = 0` equivalence plus the `z_0^1` line's flat
reassembly; (B) the moment hierarchy for `l ≤ −3`.

**Brief errors this batch**: namespaces vs files; "the iterate side is already computed" (false at
`⟨a,b⟩ = 0`); "two diagonal sums of the `(∗)` shape" for the weighted `l = −2` cross side; and the
(8.4.23) premise (it was already landed).

---

### Batch 191 — the `z_0^1` line's missing half is DIAGNOSED (the (8.4.19) tail contraction) and unified as `ZOneLineAllPairings`; the moment picture is CORRECTED (only the low moments survive; `N ≥ c` gives 0) and the `l = −3`, `2 ≤ ⟨a,b⟩` chamber is closed by the landed assembly

Two new files, two fresh agents, one batch; both verified here (rebuild from deleted `.olean`s,
`--- OK ---` for each, `Probe/forbidden_tokens.py` at zero, the in-file `#guard_msgs`, and the
`FinalCheck.lean` acceptance build after wiring them in).

**B191-A ACCEPTED — `Definitions/MMFLM86ZOneReassembly.lean`** (458 lines, namespace `MM.FLM86ZOR`,
10 guards, all exactly the three-axiom set).

* **The reassembly is NOT landed; its missing half is diagnosed**: the creation half is already in
  `flatMul (splitMinus (heisFam a)) (creationExpVL (a+b))` shape and the `a(0)` summand does belong in the
  tail, but placing it there requires **carrying `a(k)` (`k ≥ 0`) past `E^+_{a+b}` through the (8.4.19)
  contraction inside the free `(J,K')` windows** — no landed theorem performs that (only the per-half
  collapses exist).  Step-1 grep was measured **negative** (no landed reassembly).
* **The equivalence is landed**: `negTwoFaceAligned_iff_weightedZOneLine (C) (h2) (hs)`, both directions,
  via `zeroPairing_iff_weightedZOneLine` (under `CocycleCompatible`),
  `weightedHonestDiag_zero_eq_diagSumWeighted` (the `p ↦ −p−1` reindexing of the two half-lines at
  `⟨a,b⟩ = 0`, using `YFam_mul_eq_normOrdered2_of_form_eq_zero`) and `negTwoZeroWindow_weighted_eq_zero`
  (the weight `p+m` vanishes off the reindexed window; the gap `p = −m` is exactly the zero-weight point).
* **Correction of record**: `DerivDictionaryLine` (at `⟨a,b⟩ = −2`) does **not** imply
  `WeightedZOneLineZeroPairing` (at `⟨a,b⟩ = 0`) — the pairing is not a variable of either statement.
  The **unified** obligation is the new `MM.FLM86ZOR.ZOneLineAllPairings` (**the `z_0^1` line for ALL
  pairs**), and the file lands `zOneLineNegTwo_of_allPairings`, `weightedZOneLine_of_allPairings`,
  `derivDictionaryLine_of_allPairings`.  **So the single remaining obligation for both faces is the tail
  contraction making `ZOneLineAllPairings` a theorem.**
* Also recorded: `MM.FLM810NTF.negTwoFaceAligned_iff_weightedZOneLine` did **not** exist anywhere (only
  prose mentions it); it is now landed under `MM.FLM86ZOR`.

**B191-B ACCEPTED — `Definitions/MMFLM86MomentHierarchy.lean`** (491 lines, namespace `MM.FLM86MH`,
6 guards, all exactly the three-axiom set).

* **THE MOMENT PICTURE IS CORRECTED** (the batch-190-B measurement was wrong): the moments
  `Σ_r r^m · normOrdered2 a b r (K−r)` do **not** vanish for `m+1 ≤ ⟨a,b⟩`; `m = 0` is the assembly,
  `m = 1` the `z_0^1` line.  Correct: expanding by (8.4.25) and regrouping makes the `j`-sum produce the
  finite-difference sum `Σ_j ratioCoeff N j · (poly in j)`, and by
  `Σ_j ratioCoeff N j j^s = 0` for `s < N` **only the LOW moments `m ≤ c−1−N` survive**.
* Landed: `ratioCoeffSum_eq_binomCoeffFinDiff` (consuming the already-landed
  `MM.FLM86Y.binomCoeffFinDiff` core from `MMFLM86TaylorOperatorAux.lean` — the first draft re-proved an
  equivalent before finding it), the **new** vanishing case `binomCoeffFinDiff_eq_zero_of_lt` (`s < n`,
  not covered by the landed `n ≤ s` lemma), `binomCoeffFinDiff_eq_int`,
  `deltaExpCoeff_neg_eq_binomCoeff` (the `l = −c` weight is `C(m−p−1, c−1)`), the **master identity**
  `weightedHonestDiag_eq_sum_ratioCoeff`, and its specialisations: `weightedHonestDiag_eq_binomMoment`
  (`N ≤ c−1`), **`weightedHonestDiag_eq_zero_of_le` (`N ≥ c`: the weighted antidiagonal is 0 — no
  moment, no (8.4.40) line, no new residual)**, `weightedHonestDiag_eq_assembly` (`N = c−1`, the landed
  `assemblyDiagonal`), `weightedHonestDiag_eq_firstMoment` (`N = c−2`, the `z_0^1` line), and
  `negThree_weightedChambers` (**the `l = −3`, `2 ≤ ⟨a,b⟩` chamber: `ε(a,b)·YFam (a+b) (m+n)` at
  `⟨a,b⟩ = 2`, `0` for `⟨a,b⟩ ≥ 3` — closed by the landed assembly alone**).
* **Corrections**: the moment hierarchy as stated in the brief is false; `l = −3` with `2 ≤ ⟨a,b⟩` needs
  no hierarchy; and the `l = −3`, `⟨a,b⟩ = 1` chamber is **not** reachable today (it needs the `z_0^1`
  line at pairing 1, which is neither of the two named residuals).
* **The `l = −3` residual** (recommended next): (a) the general-`c` cross-side reindexing (mechanical,
  via `deltaExpCoeff_neg_eq_binomCoeff` + the reflection `C(−i−1, c−1) = (−1)^{c−1}C(i+c−1, c−1)`);
  (b) the `l = −c` analogue of `honestDiag_eq_zero_of_crossWindowOn`; (c) the iterate side at `l = −3`.

**Where Theorem 8.10.2 stands at the end of batch 191.**  Unconditional at FLM's datum: `l = 0`
(`0 ≤ ⟨a,b⟩`, `⟨a,b⟩ = −1`), `l ≥ 0` (`0 ≤ ⟨a,b⟩`), `l = −1`, `l = −2`'s `1 ≤ ⟨a,b⟩` chamber, and
`l = −3`'s `2 ≤ ⟨a,b⟩` chamber.  **The single shared obligation is `ZOneLineAllPairings`** (the `z_0^1`
line for all pairs), whose missing half is the (8.4.19) tail contraction.  **Batch 192**: (A) that tail
contraction; (B) the `l = −3` face's mechanical parts (a)--(c).

**Brief errors this batch**: "the ingredients, all landed" (the tail contraction was missing); "the
reassembly closes both faces at once" (needs the unified all-pairings statement); and the moment-hierarchy
claim (refuted; the corrected picture is the finite-difference sum).

---

### Batch 192 — the `l = −3` face at `2 ≤ ⟨a,b⟩` is UNCONDITIONAL at FLM's datum; the `(8.4.19)` premise is corrected (the contraction is against `E⁻`, not `E⁺`) and the annihilation/zero-mode laws are landed

Two new files, two fresh agents, one batch; both verified here (rebuild from deleted `.olean`s,
`--- OK ---` for each, `Probe/forbidden_tokens.py` at zero, the in-file `#guard_msgs`, and the
`FinalCheck.lean` acceptance build after wiring them in).

**B192-B ACCEPTED — `Definitions/MMFLM810NegThreeFace.lean`** (630 lines, namespace `MM.FLM810NTH`,
6 guards, all exactly the three-axiom set).

* **`negThreeFaceAlignedTwoLe_evenCocycle (C) (h2) (hs)`** — the `l = −3` face of (8.8.41) at
  `2 ≤ ⟨a,b⟩`, per element, per monomial, every window `N ≥ 3`, **unconditional at FLM's own datum**
  `C.toCentralExtension h2`.  Pairings `1` and `0` are not claimed.
* Supporting, all **general `c ≥ 1`** where possible:
  **(a)** `crossBracket_neg_nat_eq_sum` (under `CocycleCompatible`): the cross side is a weighted sum
  `Σ_p binomCoeff (m−p−1) (c−1) • honestDiag E a b p (m+n−c−p) w` over the two half-lines, with the
  reflection `binomCoeff_pos_eq_neg_pow` (from `FLM810SJ.binomCoeff_neg_succ_eq`) and
  `disjoint_negCWindows`;
  **(b)** `weightedHonestDiag_eq_zero_of_crossWindowOn` (general `c`): the face's own `CrossWindowOn` gives
  the **weighted** vanishing outside the half-lines (weight `0` in the gap `m−c < p < m`, while the honest
  diag is *not* zero there), plus the window-independence bridge `weightedHonestDiag_neg_three_eq`;
  **(c)** `crossIterate_ZAligned_lattice_neg_three` (general `E`): `i = 0,1,2` survive, with the chamber
  forms giving `ε(a,b)·YFam (a+b) (m+n)` at pairing `2` and `0` above.
* **Correction**: the brief's "(b) turns `CrossWindowOn` into the *honest* vanishing" is wrong — it yields
  the **weighted** vanishing, and the face additionally needs the window-independence bridge (landed).

**B192-A ACCEPTED — `Definitions/MMFLM86TailContraction.lean`** (641 lines, namespace `MM.FLM86TC2`,
17 guards, all exactly the three-axiom set).

* **`(8.4.19)` read verbatim off printed p. 192** (PDF index 246):
  `8α(z)Y(a,z)8 = 8Y(a,z)α(z)8 = α(z)⁻Y(a,z) + Y(a,z)α(z)⁺`, with (8.4.10)'s `α(z)^±` splitting.
* **THE PREMISE OF BATCH 191 IS REFUTED, and it is good news**: **there is no contraction past `E⁺`** for
  any `k ≥ 0` (`k ≥ 1` was already landed as `MM.FLM86TN.modeVL_comm_annihExp_pos`; `k = 0` is proved here:
  `hZeroVL_comm_annihExp`, `splitPlus_heisFamZero_comm_annihExp`).  The contraction (the `⟨a,b⟩` shift)
  lives against `E⁻` and was already landed (`MM.FLM84CT.modeVL_comm_creationExpVL_int`,
  `hZeroVL_comm_creationExpVL`); it is repackaged here as
  `splitPlus_heisFamZero_contraction_creation`.  What was genuinely missing at the **assembled** field was
  the **annihilation/zero-mode** law, now landed: `modeVL_YFam_add_pos`, `hZeroVL_YFam_add` (the zero mode
  differentiates the middle factor by the pairing), and the uniform
  **`splitPlus_heisFamZero_YFam_add : a(k)(Y(a,z)_j v) = Y(a,z)_j (a(k) v) + ⟨b,a⟩ Y(a,z)_{j+k} v`** for all
  `k ≥ 0`.
* Also landed: `ZAligned_anchor_head_tail` (the residual's right-hand side in the corrected
  `FLM85TA.headOp` + `FLM85Ac.tailOp` spelling) and **`zOneLineAllPairings_iff_headTail`**
  (`ZOneLineAllPairings` ⟺ that spelling, **both directions**, no weakening).
* **`ZOneLineAllPairings` is still the named residual**; what remains is the **flat reassembly** (the
  (8.10.8)-associativity step), named under the equivalent `DerivDictionaryLine`/`NegTwoNormalFormLine`/
  `NegTwoFirstDerivZero`/`ZOneLineNegTwo`.  Conditional on it, **both faces now close** by new
  compositions: `negTwoFirstDerivZero_of_allPairings`, `negTwoExactResidualAligned_of_allPairings`
  (`l = 0`, `⟨a,b⟩ = −2`) and `negTwoFaceAlignedZeroPairing_of_allPairings` (`l = −2`, `⟨a,b⟩ = 0`).
* Traps recorded: `headOp` must be `FLM85TA.headOp` (not `FLM85Ac.headOp`); `factorFactorZero` lives in
  `FLM810ZM`; `sub_eq_iff_eq_add'` (not `.mp`); on `Module.End` the `Sub`/`Add` diamond defeats
  `sub_eq_zero`/`sub_self`/`simp` (work at the vector level); repeated `rw` of one lemma errors on the
  second pass (`simp only` is the fix).

**Where Theorem 8.10.2 stands at the end of batch 192.**  Unconditional at FLM's datum: `l = 0`
(`0 ≤ ⟨a,b⟩`, `= −1`), `l ≥ 0` (`0 ≤ ⟨a,b⟩`), `l = −1`, `l = −2`'s `1 ≤ ⟨a,b⟩` chamber, and now
**`l = −3`'s `2 ≤ ⟨a,b⟩` chamber**.  **The single shared obligation is the flat reassembly**
(`ZOneLineAllPairings` ⟺ its head/tail spelling), which closes both the `l = 0`, `⟨a,b⟩ = −2` case and
the `l = −2`, `⟨a,b⟩ = 0` chamber.  **Batch 193**: (A) the flat reassembly, now with
`splitPlus_heisFamZero_YFam_add` in hand; (B) general `l ≤ −4` (the `l = −3` template + the moment
machinery).

**Brief errors this batch**: the `E⁺`-contraction premise (refuted; it is against `E⁻`); "honest
vanishing" for the window lemma (it is weighted); and the `headOp`/`factorFactorZero` locations.

---

### Batch 193 — general `l = −c`: the `ν ≥ c−1` chamber is UNCONDITIONAL for every `c`; and the `z_0^1` residual is landed ON THE LATTICE GENERATORS and proved F-LINEAR in the state (the Fock half is the only input left)

Two new files, two fresh agents, one batch; both verified here (rebuild from deleted `.olean`s,
`--- OK ---` for each, `Probe/forbidden_tokens.py` at zero, the in-file `#guard_msgs`, and the
`FinalCheck.lean` acceptance build after wiring them in).

**B193-B ACCEPTED — `Definitions/MMFLM810NegCFace.lean`** (583 lines, namespace `MM.FLM810NCF`,
12 guards, all exactly the three-axiom set).

* **`negCFaceAlignedGe_evenCocycle (C) (h2) (hs) {c} (hc : 1 ≤ c)`** — the general `l = −c` face of
  (8.8.41) at `ν := (form a b).toNat ≥ c − 1`, **unconditional at FLM's datum**, per element, for **every**
  `c ≥ 1`.  Supporting: `negC_weightedChambers` (assembly at `ν = c−1`, `0` for `ν ≥ c`),
  `crossBracket_ZAligned_lattice_neg_nat_eq_ite`, and the **general-`E`** iterate side
  `crossIterate_ZAligned_lattice_neg_nat` (+ `…_eq_sum_succ` for `ν ≤ c−2`: `k = c−1−ν` Fock descendants
  plus the boundary value), with `crossWindowOn_ZAligned_lattice_neg_nat`.
* **Exact residual `NegCBinomMomentAligned E c`** (the binomial-moment identity for `0 ≤ ν ≤ c−2`), with
  `negCFaceAligned_iff_binomMoment_evenCocycle` **both directions**.  Chambers: `ν ≥ c−1` unconditional;
  `ν = c−2` is the **first moment (`z_0^1` line)**, `1 ≤ ν ≤ c−3` the `z_0^{c−1−ν}` lines, `ν = 0` the
  `z_0^{c−1}` line.  So **`c = 1` is entirely unconditional** (the landed `−1` face), `c = 2` leaves only
  `ν = 0` (the landed `WeightedZOneLineZeroPairing`), `c = 3` leaves `ν ∈ {0,1}`.
* Corrections: `crossIterate_…_of_nonpos` was stated only at the even-cocycle datum (restated at general
  `E`; it also pins `N ≥ c`); the residual range is `0 ≤ ν ≤ c−2` (not `1 ≤ N ≤ c−3`); the iterate side
  needs pointwise state vanishing (not just `iterateOf_eq_zero`); and two cast/defeq traps.

**B193-A ACCEPTED — `Definitions/MMFLM86FlatReassembly.lean`** (579 lines, namespace `MM.FLM86FRA`,
8 guards, all exactly the three-axiom set).

* **LANDED: the `z_0^1` residual holds ON THE LATTICE GENERATORS, for ALL pairings**:
  `zOneLineAllPairingsOnGenerators (hs) (E)` — for all `a b m n c` there is a window with
  `Σ_{p∈s} p • normOrdered2 a b p (−m−n−p) (ι c) = ZAligned E (creationExpVL a 1 (anchorVL E a b)) (m+n) (ι c)`.
  Supporting (all new): `genFam_iotaVL`, `modeVL_iotaVL_eq_zero_of_pos`, and the index bridges
  `sum_Icc_eq_flatMul`, `sum_Icc_smul_eq_flatMul_deriv` (**the weight `p` IS the (8.4.12) derivative**),
  `sum_range_toNat_succ_eq_flatMul`.
* **LANDED: the residual is `F`-LINEAR in the state** — `ZOneLineAt` (at one pair/monomial/state),
  `zOneLineAllPairings_iff_forall_ZOneLineAt` (`Iff.rfl`), and `zOneLineAt_add`, `_smul`, `_zero`,
  `_of_mem_span`.  **So `ZOneLineAllPairings` is determined by its values on any spanning set of `V_L`;
  with the generator half, the only remaining input is the residual on the creation monomials
  `α₁(−n₁)…α_k(−n_k) ι(c)` — the Fock half.**
* **Corrections**: `splitPlus_heisFamZero_YFam_add` was **not** the missing tool (in the head/tail spelling
  the `a(k)` summands *already are* the tail; the generator case consumes only the landed fact that the
  tail's `k = 0` summand is the zero mode); the reassembly is **not** a reindexing inside the `(J,K')`
  windows (those belong to the `z_0^0`/firstDeriv line) — at an arbitrary state the flat pieces reach the
  action only through the **annihilation-pair contractions**, i.e. the Fock half, a (8.10.8)-associativity
  computation; and the `⟨a,b⟩ = −2` instance does not imply the general-state identity.

**Where Theorem 8.10.2 stands at the end of batch 193.**  Unconditional at FLM's datum: `l = 0`
(`0 ≤ ⟨a,b⟩`, `= −1`), `l ≥ 0` (`0 ≤ ⟨a,b⟩`), `l = −1`, and — for **every** `c ≥ 1` — the `ν ≥ c−1`
chamber of the `l = −c` face (which subsumes the earlier `l = −2`, `−3` chamber results).  **The single
shared obligation is the `z_0^1` line `ZOneLineAllPairings`, now reduced to its Fock half** (it holds on
the generators and is linear in the state).  **Batch 194**: (A) the residual at the first Fock level
`α(−n)ι(c)` (the `E⁺_aE⁺_b` contraction past one creation mode, with the landed
`MM.FLM810CFC.modeVL_mul_annihExp_sub`/`modeVL_annihFam_add` as the engine); (B) the higher `z_0^m` lines
(`m ≥ 2`, the `1 ≤ ν ≤ c−3` chambers) by the (8.4.40) chain.

**Brief errors this batch**: the `E⁺`-contraction framing; the residual range; the `crossIterate` generality;
and the claim that the flat reassembly is a window bookkeeping step.

---

### Batch 194 — the `z_0^1` line at the FIRST FOCK LEVEL is LANDED (with the contraction and the right-hand-side mode law); and the derivative ladder shows NO higher `z_0^k` line is easier than the `z_0^1` blocker

Two new files, two fresh agents, one batch; both verified here (rebuild from deleted `.olean`s,
`--- OK ---` for each, `Probe/forbidden_tokens.py` at zero, the in-file `#guard_msgs`, and the
`FinalCheck.lean` acceptance build after wiring them in).

**B194-A ACCEPTED — `Definitions/MMFLM86ZOneFockOne.lean`** (954/960 lines, namespace `MM.FLM86ZOF`,
14 guards, all exactly the three-axiom set).

* **LANDED, unconditional apart from the form symmetry `hs`**:
  **`zOneLineAt_modeVL_iotaVL`** — the `z_0^1` residual `MM.FLM86FRA.ZOneLineAt` at
  `v = α_{c'}(−R) ι(c)`, **every pairing, every `R ≥ 1`, every `(m,n)`**.  This is the first Fock level of
  the shared blocker.
* Engine landed alongside: **the contraction `normOrdered2_modeVL_sub`**
  (`normOrdered2 a b p q (α_c(−r)w) = α_c(−r)(normOrdered2 p q w) − ⟨c,a⟩·normOrdered2 (p+r) q w −
  ⟨c,b⟩·normOrdered2 p (q+r) w`, shift `+r` in **both** indices), whose direction was forced by
  consistency of the two laws (a priori `−r` was possible); its weighted-diagonal corollary
  `diagSumWeighted_modeVL`; **the right-hand-side mode law `ZAligned_modeVL`**
  (`Z_j(α_c(−r)w) = α_c(−r)(Z_j w) − ⟨c,a+b⟩·Z_{j−r}w + ε(a,b)·r·⟨a,c⟩·Y(a+b,z)_{r−j}w`); the window laws
  `headWindow_modeVL_shift`/`tailWindow_modeVL_shift`; and the generator collapse
  `diagSum_normOrdered2_iotaVL`.
* **The general creation monomial is NOT reached, and the reason is precise**: the single-mode proof closes
  because the contraction reduces both sides to the **two landed generator cases** (degrees `−m−n` and
  `−m−n+R`) plus the generator collapse; at the **next** mode the contraction produces the **unweighted
  diagonal sum at the level-one descendant `α_{c''}(−R₂)ι(c)`** — i.e. the **`z_0^0` line at a Fock
  descendant**, genuine new input, not a corollary of the generator case.  **The exact remaining
  sub-residual is that descended `z_0^0` line.**
* **Corrections**: the named engine `modeVL_mul_annihExp_sub`/`modeVL_annihFam_add` covers only the
  **left**-hand side (the right needs `ZAligned_modeVL`); the landed `headWindow_modeVL_sub`/
  `tailWindow_modeVL_sub` **cannot** be used as stated (they force the field's lattice label to equal the
  factor's, while the residual has field `Y(a+b,z)` and factor `a(z)`); and the shift direction (`+r` in
  both laws) was unstated.  Traps: `-r • x` parses as `(-r)•x` but prints like `-(r•x)`; `rw` instantiates
  a lemma **once per pass**, so repeating a name silently rewrites only the first instantiation (no error);
  the `-γ•H` vs `-(γ•H)` mismatch needs explicit `calc`/`show`.

**B194-B ACCEPTED — `Definitions/MMFLM86ZMoments.lean`** (650 lines, namespace `MM.FLM86ZM`, 12 guards,
all exactly the three-axiom set).

* **NO higher `z_0^k` line is easier than the `z_0^1` blocker**: the landed **derivative ladder**
  `momentSum_firstDeriv` / `zLineAt_firstDeriv` / **`zLineScheme_succ_firstDeriv`** is an
  **equivalence** — the `(j+1)`-st line of `X` at degree `K+1` (value `R`) ⟺ the `j`-th line of
  `d/dz₁X` at degree `K` (value `(j+1)·R`).  Hence order `k` **is** order 0 (assembly-shaped) of
  `firstDeriv^k (normOrdered2 a b)` — the derivative tower of the shared blocker, exactly batch 185's
  measurement at `j = 1`.
* Also landed: the Vandermonde expansion `binomCoeff_add_eq_sum` (and `binomCoeff_add_two` — "the
  quadratic weight expanded flat") and the flat reassembly `sum_binomCoeff_smul_eq_sum`.
* **The exact residual `ZLine E j`** (pairing-free, at the boundary field): `j = 0` is a **theorem**
  (`zLine_zero`, the landed (8.4.20) assembly), **`j = 1` is verbatim the shared blocker**
  (`zLine_one_iff_zOneLineAllPairings`, both directions), `j ≥ 2` is the new residual.
* **The chamber decomposition of `NegCBinomMomentAligned E c`**: `NegCMomentLine E c k := ZLine E k`
  restricted to pairings `N = c−1−k`; forward, each chamber consumes exactly its own line; backward,
  `negCBinomMomentAligned_of_zLine` (all lines `1 ≤ j ≤ c−1` suffice).  **Ledger**: `N ≥ c` zero; `N = c−1`
  the landed assembly; `N ≤ c−2` the `c−1` residual chambers, **all of the `z_0^k` form with
  `k = c−1−N`** — so every remaining chamber is decided by the same derivative tower, and none is closed
  by this batch.  Edge case `c = 1`: residual empty.
* Correction: the landed `z_0^2` statements are for the **one-variable** `openBoxField`/formal
  `shiftFamily`, not the honest three-factor `normOrdered2 a b` (and the `MMFLM86*` ones are conditional on
  the named gap `Shifted8436`/`SubstProductExpansion`); and the `ν = c−3` chamber needs Vandermonde's
  orders 0,1,2, not just `z_0^2`.

**Where Theorem 8.10.2 stands at the end of batch 194.**  The single shared obligation is still the
`z_0^1` line `ZOneLineAllPairings`, now known to be: **proved on the generators** (batch 193-A),
**`F`-linear in the state** (193-A), **proved at the first Fock level** (194-A), and **the easiest of all
the `z_0^k` lines** (194-B's derivative ladder).  The next input is the **descended `z_0^0` line**
(`α_{c''}(−R₂)ι(c)`).  **Batch 195**: (A) that descended assembly identity; (B) the general
creation-monomial induction (level `n` → `n+1`) for both lines together.

**Brief errors this batch**: the engine covers only the LHS; the `headWindow_modeVL_sub` reuse claim; the
shift direction; the "induction on the (8.5.4) list" understatement; and the `z_0^2` "already landed"
hint (wrong shape, one-variable).

---

### Batch 195 — 🎉 BREAKTHROUGH: the shared obligation `ZOneLineAllPairings` IS A THEOREM (the `z_0^1` line of (8.4.36)/(8.4.40) for all symmetric pairings); and the earlier "descended `z_0^0` line" premise is REFUTED

Two new files, two fresh agents, one batch; both verified here (rebuild from deleted `.olean`s,
`--- OK ---` for each, `Probe/forbidden_tokens.py` at zero, the in-file `#guard_msgs`, and the
`FinalCheck.lean` acceptance build after wiring them in).

**B195-B ACCEPTED — `Definitions/MMFLM86FockInduction.lean`** (348 lines, namespace `MM.FLM86FI`,
10 guards, all exactly the three-axiom set).  **This closes the campaign's single shared obligation.**

* **`zOneLineAllPairings_of_symmetric_form (hs) (E) : MM.FLM86ZOR.ZOneLineAllPairings bL form E` — a
  THEOREM**, from landed ingredients only.  The `z_0^1` line of (8.4.36)/(8.4.40) now holds at every
  **symmetric** pairing, with **no residual left**.  (Honest caveat, recorded in the file:
  `ZOneLineAllPairings` is *defined* without `hs`; only its symmetric-form instance is proved, which is
  the hypothesis of every landed ingredient and consumer — FLM (7.2.22)/(7.2.23).)
* **The induction chain**: `zOneLineAtLevel_zero` (the batch-193-A generator case) →
  **`zOneLineAt_modeVL`** (the step: weigh the level-`n` hypotheses at `(m,n)` and `(m−r,n)` against the
  **landed assembly** at `w` at degree `−m−n+r`, with the window
  `U = (s₁∪s₂∪s₃) ∪ (s₁∪s₂∪s₃)−r`, then `diagSumWeighted_modeVL` + `ZAligned_modeVL`) →
  `zOneLineAtLevel_succ` → `zOneLineAtLevel_all` (`Nat.rec`) → `zOneLineAllPairings_of_level`
  (`zOneLineAt_of_mem_span` + `spanningStatement`).
* **HEADLINE CORRECTION: the "descended `z_0^0` line" does NOT exist as new input.**  The landed
  `MM.FLM86AR.assemblyDiagonal` / `MM.FLM810NOC.assemblyDiagonal_antidiagonal` (batch 181) already proves
  the unweighted diagonal sum at **every state and every pairing** with the per-state (8.10.5) window;
  batch 194-A's `diagSum_normOrdered2_iotaVL` is that value law at `w = ι(c)`.  Hence
  `zZeroLineAtLevel_all : ∀ n, ZZeroLineAtLevel E n` needs no induction, and the level-`n` hypothesis
  carries the `z_0^1` half **only**.  The batch-195-A task (dispatched on the refuted premise) was
  **steered mid-flight and stopped**.
* Minor window correction: the step needs the **`−r`-shift of the assembly window** (`s₃ − r ∈ U`);
  batch 194-A's level-one proof could omit it only because at a generator every vanishing window gives the
  same sum.

**B195-A ACCEPTED (minor) — `Definitions/MMFLM86DescendedAssembly.lean`** (370 lines, 9 guards, all
exactly the three-axiom set).  With the headline refuted, the file's only non-redundant content is the
**window-level decomposition** `diagSum_normOrdered2_modeVL_of_lines` (exactly which windows the
contraction consumes: `(K,W)`, `(K+r,W)`, `(K+r,W+r)`), the **hypothesis-free explicit window**
`diagSum_normOrdered2_modeVL_iotaVL` (`Icc (⟨a,c⟩−r) (K+r−⟨b,c⟩)`) and the vanishing lemma
`normOrdered2_modeVL_iotaVL_eq_of_support`.  It adds **no new value law**; kept for the window bookkeeping
and the explicit witness.

**Consequences now available for free** (the landed one-directional reductions compose): the `l = 0`,
`⟨a,b⟩ = −2` case (`NegTwoFirstDerivZero`, `NegTwoExactResidualAligned` and every equivalent name); the
`l = −2` face's `⟨a,b⟩ = 0` chamber (`WeightedZOneLineZeroPairing`), hence the **whole `l = −2` face**;
and, for every `c`, the **`ν = c−2` chamber** of the `l = −c` face (`NegCMomentLine E c 1` from
`ZLine E 1`).  **Batch 196** harvests these (A) and attempts the general-`j` induction `ZLine E j`
(`j ≥ 2`), which would close `ResidueNegativeAligned` at the even cocycle entirely (B).

**Brief errors this batch**: the "descended `z_0^0` line" premise (refuted); the index normalization in
195-A's brief (`ε·YFam (a+b) (−K−1) = ε·genFam (a+b) K`, not `(K+1)`); and "the unweighted diagonal sum at
a descendant" is not a statement until the window is fixed (with the generator's own window it is false).
---

### Batch 196 — the general-`j` induction `ZLine E j` is LANDED conditional on the single `c`-independent base `ZLineGeneratorBase`; the `l = −2` face and every `ν = c−2` chamber become UNCONDITIONAL

**A — the harvest** (`Definitions/MMFLM86HarvestAllPairings.lean`, `Definitions/MMFLM86HarvestAllPairingsFaces.lean`).
With `ZOneLineAllPairings` a theorem (batch 195), the landed one-directional reductions compose with no
hypothesis left.  `negTwoFaceAligned_evenCocycle` closes the **whole `l = −2` face** at FLM's datum, and
`negCFaceAlignedFrom_subTwo_evenCocycle` closes the `ν ≥ c−2` chamber of **every** `l = −c` face.  The
sharpest new form is the equivalence
```
residueNegativeAligned_evenCocycle_iff_negThreeAndBelow :
  ResidueNegativeAligned (evenCocycle …) ↔ NegThreeAndBelowResidualAligned
```
i.e. the whole negative-pairing residue at the even cocycle is *exactly* the `⟨a,b⟩ ≤ −3` chamber: the
`l = 0` cases `⟨a,b⟩ ∈ {0,−1,−2}` are paid for, and no `l ≠ 0` face is left behind for those pairings.
This is a strict advance in accounting — the earlier "`⟨a,b⟩ ≤ −1`" residual is replaced by
"`⟨a,b⟩ ≤ −3`" plus a theorem.  Also harvested: every equivalent `z_0^1` residual name.

**B — the induction** (`Definitions/MMFLM86ZLineInduction.lean`, 873 lines).  The level induction on
`ZLine E j` is LANDED, conditional only on the named base:
```
theorem zLine_all_of_generatorBase (hs) (E) (h : ZLineGeneratorBase (F := F) bL form E) :
    ∀ i : ℕ, 1 ≤ i → ZLine (F := F) bL form E i
theorem zLineGeneratorBase_iff (hs) (E) :
    ZLineGeneratorBase (F := F) bL form E ↔ ∀ i : ℕ, 1 ≤ i → ZLine (F := F) bL form E i
theorem negCBinomMomentAligned_of_generatorBase (hs) (E) (h) {c} (hc : 1 ≤ c) :
    NegCBinomMomentAligned (F := F) bL form E c
```
Ingredients proved: the `j`-independent contraction `momentSum_modeVL_normOrdered2` at weight `C(r,j)`;
the Chu–Vandermonde identity `binomCoeff_sub_eq_sum_moment` (`C(p−r,j) = Σ_l C(−r,l)C(p,j−l)`);
F-linearity of the line values in the state `zLineAtVal_of_mem_span`; the step
`zLineAt_modeVL_of_modeLaw`; the level induction `ZLineAtAllLevel`/`_succ`/`_all`; and the spanning
assembly `zLineScheme_of_level`/`zLine_of_level`.  `zLine_one_unconditional : ZLine E 1` re-derives
batch 195-B without the base.  So **the whole `ZLine` tower for `j ≥ 2` is one base away**, and that base
is `c`-independent.

The base has exactly TWO halves:
```
def ZLineGeneratorBase (E) : Prop :=
  (∀ a b c₀ i K, ZLineAt (normOrdered2 a b) i K (iotaVL c₀) (zLineVal E a b i K (iotaVL c₀)))
  ∧ (∀ a b c i r, 1 ≤ r → ∀ K u, ZLineModeAt (F := F) bL form E a b c i r K u)
```
(i) the order-`i` line at the lattice generators, and (ii) the order-`i` **mode law** for the field of the
coherent state `E⁻_a(z)_i ι(ab)` through one creation mode `YFam`.  Both halves are landed for `i ≤ 1`
(`zLine_zero`, `zOneLineAllPairingsOnGenerators`; `modeVL_genFam_sub`, `ZAligned_modeVL`); `i ≥ 2` is new.
The converse direction of `zLineGeneratorBase_iff` is proved by linearising the contraction over the line
windows.

**`ResidueNegativeAligned` at the even cocycle is still NOT unconditional**: it needs `ZLine E j`,
`j ≥ 2`, hence `ZLineGeneratorBase`.

**Brief errors this batch** (three corrections, all consequential).
(a) `zLineAt_firstDeriv` is an **equivalence**, so it cannot manufacture the generator case from landed
material; order 0 of `firstDeriv^j(normOrdered2 a b)` is the same difficulty, because
`firstDeriv^j(normOrdered2)` is not a `normOrdered2` and `assemblyDiagonal_antidiagonal` is stated only for
`normOrdered2 a b`.
(b) The step needs the **lower-order** lines at degree `K+r`, not only order `j` at all monomials.
(c) The step also consumes the **value law at the descendant** (`ZLineModeAt`); `ZAligned_modeVL` covers
only `j = 1`.  Hence the base has two halves, not one generator case.
(d) Arithmetic trap (carried from 195): `zLineVal 0 K u = ε·genFam (a+b) K u`, since
`genFam u m = YFam u (−m−1)`; the first draft had the index off by one and the mode law would have been
false.

---

### Batch 197 — the order-2 line is reduced to exactly TWO statements (`zLine_two_iff`), and the Fock half is reduced to the `k = 2` display plus two named carrier inputs

**A — the truncated base and the exact order-2 residual** (`Definitions/MMFLM86ZLineBaseUpTo.lean`, 401
lines, `MM.FLM86ZLT`; `Definitions/MMFLM86ZLineTwo.lean`, 366 lines, `MM.FLM86ZLT2`).  Batch 196's
single `c`-independent base is now **truncated** without loss:

```
theorem zLineGeneratorBaseUpTo_iff (hs) (E) (J) :
    ZLineGeneratorBaseUpTo E J ↔ ∀ i : ℕ, i ≤ J → ZLine E i
theorem zLineGeneratorBase_iff_forall_upTo (hs) (E) :
    ZLineGeneratorBase E ↔ ∀ J, ZLineGeneratorBaseUpTo E J
```

both directions at the *same* `J` — neither uses an order above `J`, so the brief's feared failure of
one direction does not occur.  The chamber consumption is now per-chamber and sharp:

```
theorem negCBinomMomentAlignedAt_of_lines (hs) (E) {c nu J} (hc : 1 ≤ c)
    (hJ : ∀ j, 1 ≤ j → j ≤ J → ZLine E j) (hnu : nu ≤ c - 2) (hcJ : c - 1 - nu ≤ J) :
    NegCBinomMomentAlignedAt E c nu
theorem negCBinomMomentAlignedAt_of_zLine_two (hs) (E) ... :
    NegCBinomMomentAlignedAt E c nu      -- every chamber with c - 1 - nu ≤ 2, new chamber nu = c-3
```

and the order-two residual is **exactly two statements**:

```
theorem zLine_two_iff (hs) (E) : ZLine E 2 ↔ (order-2 line at the lattice generators) ∧ (order-2 mode law)
```

(the `i = 0, 1` halves being the landed `zLine_zero`, `zLine_one_unconditional`, `zLineModeAt_zero`,
`zLineModeAt_one`).  The file also lands the whole (8.4.16)/(8.4.12) **window-shift engine for the
derived factor** `a'(z) = (1/1!)(d/dz)a(z)`: the closed forms of its halves
(`splitMinus_factorFamilyZero_two`, `splitPlus_factorFamilyZero_two`), the head/tail discrepancies and
their two-component shifted forms (`headWindow_factorFamilyZero_two_modeVL_sub`/`_shift`,
`tailWindow_factorFamilyZero_two_modeVL_sub`/`_shift`), and the commutator
`[α_c(-r), splitPlus a'(z)]` giving the tail correction `-r(r+1)⟨c,a⟩ Y_{m+r+2}` at window `M ≥ r+1`.

**A did NOT land** `zLineModeAt_two`, the order-2 generator line, or `zLine_two`.  The order-2 mode
law's remaining step is pinned down in three parts in the file header: (1) the (8.5.5) recursion at
degree two, `ZAligned E (E⁻_a(z)₂ w) = (eps/2)·reflectFam(C' + D)` with
`C' = actionFamily (factorFactorZero 1 a) W`, `W = YzeroField baseAligned (E⁻_a(z)₁ iota(a+b))`,
`D = actionFamily (factorFactorZero 2 a) (baseAligned E (a+b))`; (2) the **two applied laws** — the
two-component `heisFamZero a` law for `W` (whose commutator is `ZAligned_modeVL` rearranged,
inhomogeneous, correction `r⟨a,c⟩` from the (8.4.19) `⁺`-tail) and the homogeneous law for `D`; (3) the
identifications `zLineVal 1 (K+r) u = eps B_{K+r-1} u`, `zLineVal 0 (K+r) u = eps genFam(a+b)_{K+r} u`,
`C(-r,2) = r(r+1)/2` and the coefficient identity `-⟨c,a⟩ r(r+1)/2 = -(the C(-r,2) term)`, whose
arithmetic closes exactly against the `ZLineModeAt` right-hand side.

**B — the Fock half at its shortest** (`Definitions/MMFLM86FockOneStep.lean` 333 lines,
`...Sharp.lean` 197, `...Two.lean` 281).  At a general `E` (no cocycle, no evenness), with two newly
named carrier inputs

```
def FockPairDatum E  := OneSidedCreatedPairs (ZAligned E) vacuumVL
def FockCreatedSpan E := Submodule.span F (Set.range (iteratedCreation (ZAligned E) vacuumVL)) = ⊤
```

both directions are landed:

```
theorem fockCreatedStep_iff_createdStep (E) (hpair) (hspan) :
    FockCreatedStep E ↔ IterateProductStepOnCreated (ZAligned E) convOneSided vacuumVL
theorem fockCreatedStep_iff_display8610List (E) (hpair) (hspan) :
    FockCreatedStep E ↔ Display8610List (ZAligned E) vacuumVL convOneSided
theorem fockCreatedStep_iff_display8610TwoOneSided (hnd) (hs) (E) (hpair) (hspan) :
    FockCreatedStep E ↔ Display8610TwoOneSided E
theorem coherentCreatedStep_of_twoOneSided (hnd) (hs) (E) (hpair) (hspan) :
    Display8610TwoOneSided E → CoherentCreatedStep E
```

so the `k = 2` display is **sufficient**, not merely necessary, and `CoherentCreatedStep` is now
consumable from a concrete display instead of the opaque one-step.  New mathematics: the conditional
additivity of the `dite`-carrier `convOneSided` (`convOneSided_add_right_of_pairFinite`,
`_smul_right_of_pairFinite`), `PairFinite` preserved along the span
(`pairFinite_shiftedField_of_mem_span`), and the **sharpness refutation**
`not_convOneSided_add_right_unconditional` (a compiled counterexample over `ℚ`: unconditional
additivity of `convOneSided` is FALSE, so the carrier input cannot be dropped).  Non-vacuity of the
datum is recorded (`fockPairDatum_created_nil`, `fockPairDatum_created_singleton_iotaVL`).

**Accounting corrections of record.**  (1) The Fock residual is **two-layered**: the created-state
evaluation (the `k = 2` display `Display8610TwoOneSided`, i.e. the (8.5.5) evaluation of (8.5.14))
**plus** the carrier datum `FockPairDatum` (which is strictly stronger than batch 182's
`ShiftedPairFiniteGenerators`: all created states and all factors, not the lattice generators only) and
the spanning `FockCreatedSpan`.  The ledger line "the Fock half is implied by the landed (8.6.10) gap
`IterateProductStep`" should be read as "equivalent to the `k = 2` display modulo those two named
carrier inputs".  (2) The brief's chamber lemma `negCBinomMomentAlignedAt_of_lines` is **false at
`c = 0`** (at `(c,ν) = (0,0)` the order is `k = 0` and the landed (8.4.20) assembly computes the left
side as `ε·YFam(a+b)(m+n+1)` against the stated `ε·YFam(a+b)(m+n)`); the landed form carries
`hc : 1 ≤ c`, which every face consumer has.  (3) The brief understated the order-2 mode law: its first
summand acts on the degree-one field, whose commutator with `α_c(-r)` is **not** homogeneous (the
(8.4.19) `⁺`-tail), so the landed homogeneous `actionFamily_modeVL` does not apply as it stands.

**Lead for the next batch (dispatcher).**  `MM.FLM85CA.createdStateSpan_eq_top`
(`MMFLM85ConvAdditive.lean:502`) **is** a landed proof that the created states span `V_L`, from
`hcreate : CreationProperty (toVertexOp Y) vac` and `hmode : ∀ p : ℕ+ × L, (Y (umode p)) (-p.1) = modeVL p.2 (-p.1)`
plus an explicit family `umode`.  So `FockCreatedSpan E` is very likely dischargeable at `ZAligned E`
(the landed (8.5.23) `creationPropertyAligned` supplies `hcreate`) rather than being a genuine new
residual; the (8.5.2) mode identification is the only hypothesis to locate.

---

### Batch 198 — 🎉 `ZLine E 2` IS A THEOREM: both halves of the order-two base landed (MODE2 + GEN2), and the `ν ≥ c−3` chamber of every `l = −c` face is now unconditional

**A — MODE2** (`Definitions/MMFLM86ModeLawTwo.lean`, 618 lines, `MM.FLM86MLT`): `zLineModeAt_two` at a
general `E` with only `hs`, i.e. the order-two mode law `ZLineModeAt E a b c 2 r K u` for all data and all
`1 ≤ r`.  The proof needed two things batch 197-A had not landed: the **two-component window laws for the
factor `a(z)` itself** (`headWindow_heisFamZero_two_modeVL_shift`, `tailWindow_heisFamZero_two_modeVL_shift`,
correction `r⟨a,c⟩Y_{m+r+1}`, bound `r ≤ M`) built from the landed `MM.FLM810CC.modeVL_headWindow_sub` /
`modeVL_tailWindow_sub`, and the generic applied law `actionFamily_two_modeVL_of` together with its two
instantiations: `actionFamily_factorFactorZero_one_two_modeVL` (the factor `a(z)` against the degree-one
coherent field, correction `+2r⟨a,c⟩B_{m+r+1}` — the (8.4.19) tail and the two-component `δ`-term ADD,
no cancellation) and `actionFamily_factorFactorZero_two_two_modeVL` (the derived factor, correction
`-r(r+1)⟨c,a⟩Y_{m+r+2}`, window `r+1 ≤ M`).  With `zAligned_creationExpVL_two_anchor` (the (8.5.5)
recursion at degree two), `zLineVal_two_eq` and `binomCoeff_neg_two : C(-r,2) = 2⁻¹(r+1)r`, the final
assembly closes exactly against the `ZLineModeAt` right-hand side.  Packaging: `ZLineTwoGenerator`,
`zLine_two_of_generator` (the order-two line now needs GEN2 ALONE) and
`negCBinomMomentAlignedAt_of_generatorTwo`.

**B — GEN2** (four files, 336 + 365 + 241 + 272 = 1214 lines, `MM.FLM86GLT`:
`MMFLM86GeneratorLineTwoAux/Eval/Comp/GeneratorLineTwo.lean`): `zLineGeneratorTwo` — the order-two line
`ZLineAt (normOrdered2 E a b) 2 K (iotaVL c) (zLineVal E a b 2 K (iotaVL c))` for all `a b c`, `K`, at a
general `E` with only `hs`.  Engine: the quadratic weight `2·C(T,2) = T(T-1)` and the second-order weight
bridge to `flatMul (deriv (deriv X))`; the window→`flatMul` bridges for the corrected (8.4.16) head against
a SHIFTED field (`headWindow_eq_flatMul_of_shift`/`_of_shift`, relating the source vector `w` and the
target vector `w'` separately); `splitMinus (factorFamilyZero 2) = deriv` of the creation half; the two
(8.5.5) evaluations at `iota(c)` (`momentSum_two_iotaVL`, `zLineVal_two_iotaVL`) and the composite
`:a(z)a(z)Y(ab,z):` action (the head needs the SQUARE of the creation half via `flatMul_assoc`).  The two
sides meet by `2·C(T,2) = T(T-1)`.  Also `zLine_two_iff_modeLaw : ZLine E 2 ↔ MODE2` (the GEN2 half is
discharged).

**Capstone (dispatcher), `Definitions/MMFLM86ZLineTwoClosed.lean`**: `zLine_two : ZLine E 2`
(unconditional), `zLineGeneratorBaseUpTo_two : ZLineGeneratorBaseUpTo E 2`,
`negCBinomMomentAlignedAt_of_two` — the chamber `c-1-ν ≤ 2` (the new chamber `ν = c-3` included) of every
`l = -c` face, at the single symmetric-pairing hypothesis.  No new mathematics: compositions of the two
halves.

**Campaign-level trap discovered by the capstone (record it).**  A declaration whose STATEMENT fails to
elaborate (here an unknown identifier: the dispatcher's file used `CentralExtension` without
`open FLMLattice`) is silently turned into a `sorryAx` term; `#print axioms` then reports
`[propext, sorryAx, Classical.choice, Quot.sound]` and the ONLY symptom is the `#guard_msgs` mismatch.
Two consequences: (a) always scan the whole build log for `error` lines, not just the final
`--- OK/FAILED`; (b) never trust `tools/build.sh … | tail` for the exit code (the pipeline reports
`tail`'s status — this masked a `DRIVER_EXIT=1` for one round).  A scratch probe that `#print axioms`
every public declaration without a guard docstring is the reliable check.

**Brief errors this batch.**  (a) "importing `MMFLM86ZLineTwo` gives `zLine_two_iff`" is false — that
lives in `MMFLM86ZLineBaseUpTo` (`MM.FLM86ZLT`); both imports are needed.  (b) The "landed engine" covered
only the DERIVED factor `(1/1!)(d/dz)a(z)`; the `a(z)` half (two window laws + generic applied law) was
the largest unrecorded piece of work.  (c) `M ≥ r+1` is needed for the derived factor only; the generic law
carries `r+1 ≤ M`.  (d) The final assembly needs three ring index identities applied in a specific order
(`(K+r)-2 = K-2+r`, `-(-(K+r))-1 = K-2+r+1`, `K+r = K-2+r+2`); a `simp` normalisation applies them in the
wrong order and leaves `K-2+r+2-2`.  (e) Two measured Lean traps: `rw`/`simp [neg_smul]` does not fire on
`(-A) • x` for `x : VL F L` (module instance diamond — use `module` or `(neg_smul A x).symm`), and
`rw [← smul_add]` can rewrite the OUTER `(1/2) • (…)` (use `(smul_add _ _ _).symm` in a `calc`).
(f) 198-B's derivative-ladder LEAD does not work: the order-one generator line is landed for
`normOrdered2` itself, not for its derivative; the direct route closed instead.

---

### Batch 199 — both halves of the general-`j` base are reduced to ONE named residual each; the left-hand side of the generator line is unconditional at EVERY `j`

**A — MODE-ALL** (`Definitions/MMFLM86ModeLawAllAux.lean`, 633 lines + `Definitions/MMFLM86ModeLawAll.lean`,
507 lines, namespace `MM.FLM86MLA`).  The order-`j` mode law for all `j` is now equivalent to one named
induction step `AnchorModeCommStep E`, with all the analytic machinery landed at a general `E` with only
`hs`:

* **the general-`n` (8.4.16)/(8.4.19) window law for the derived factor `(1/(n-1)!)(d/dz)^{n-1}a(z)`, for
  every `n >= 1`**: `splitMinus_factorFamilyZero_eq` (`splitMinus (factorFamilyZero n a) i =
  C(i+n-1,n-1) . alpha_a(-i-n)`), `splitPlus_factorFamilyZero_eq`, `splitPlus_factorFamilyZero_modeVL_sub`
  (the commutator fires exactly at `k = r+n-1` with coefficient `n.<c,a>C(-r,n)`), the head/tail window
  discrepancies and their shifted forms (tail repair `-n.<c,a>C(-r,n).Y_{m+r+n}`, window `r+n-1 <= M`),
  `actionFamily_modeVL_of_window` (the applied law with an EXPLICIT window bound) and
  `actionFamily_factorFactorZero_modeVL` (every `n >= 1`, instantiating the landed `n = 1, 2` cases);
* the (8.5.11) anchor field `A_m = YzeroField (baseAligned E) (E^-_a(z_0)_m iota(ab))` with
  `zLineVal_eq_anchorField : zLineVal j K u = A_j.fam (K-j) u`, the (8.5.5) recursion
  `anchorField_recursion`/`_apply` (`A_j.fam = (1/j) sum_{i<j} actionFamily (factorFactorZero (i+1) a) A_{j-1-i}`);
* the commutator form `AnchorModeComm` with the IFF `zLineModeAt_iff_anchorModeComm`, the base cases
  `anchorModeComm_zero`/`_one`, the triangle transpose `sum_range_triangle`, and the applied law with a
  finite family of correction fields `actionFamily_modeVL_of_sumWindow`;
* **`def AnchorModeCommStep E`** (the ONE named residual) with `anchorModeComm_all_of_step` and
  `zLineModeAt_all_of_step : AnchorModeCommStep E -> forall j a b c r, 1 <= r -> forall K u, ZLineModeAt ...`,
  and `anchorModeCommStep_of_le_two`: **the residual is a theorem for every `j <= 2`**, so every remaining
  failure sits at `j >= 3`.

Only the index bookkeeping of the step is unwritten, and the report verifies the coefficient count closes
exactly: `(1/j)(j-1-l)<c,a>C(-r,l+1) + (1/j)(l+1)<c,a>C(-r,l+1) = <c,a>C(-r,l+1)`.  The coefficient identity
used is `n.C(-r,n) = (-1)^n r.C(r+n-1,n-1)` (`nat_mul_binomCoeff_neg`), from
`MM.FLM86R.binomCoeff_neg_cast` + `MM.FLM84SM.binomCoeff_succ_mul`.

**B — GEN-ALL** (`Definitions/MMFLM86GeneratorLineAllAux.lean`, 281 lines + `...All.lean`, 306 lines,
namespace `MM.FLM86GLA`).  The order-`j` line at the lattice generators is NOT closed for `j >= 3`, but:

* the **left-hand side is evaluated unconditionally at every `j`**: `sum_Icc_binomCoeff_eq_flatMul_iterDeriv`
  (the order-`j` weight bridge `sum_{i in Icc 0 N} C(i,j) X_i (Y_{N-i} w) = (j!)^{-1} flatMul (iterDeriv j X) Y (N-j) w`),
  `normOrdered2_iotaVL_eq_zero_of_not_mem_window`, and `momentSum_iotaVL_flat` (the order-`j` moment at a
  generator over the flat window equals `flatGeneratorValue`);
* the target is **exactly equivalent** to one named residual: `zLineAt_iotaVL_iff_flat` (pointwise, no
  hypothesis) and `zLineGeneratorAll_iff_flat : ZLineGeneratorAll E <-> ZLineGeneratorFlat E`, plus
  `zLineGeneratorFlat_iff_zAlignedGeneratorFlat`;
* **`def ZAlignedGeneratorFlat E`** (the named residual) is exactly "the (8.5.5) recursion
  `coherentFieldNat_eq` evaluated at a lattice generator", open for `j >= 3`;
* `zLineAtGenerator_upTo_two` / `zLineAtGeneratorFlat_of_le_two`: the `j <= 2` slice, a machine-checked
  consistency test of the index conventions against batch 198's GEN2.

The report's hand computation says the missing induction closes and names its ingredients:
(i) `splitMinus (factorFamilyZero (r+1) a) = (r!)^{-1} iterDeriv r (splitMinus (heisFam a))`;
(ii) the `^+`-tail at `iota(c)` (`k != r` killed by `modeVL_iotaVL_eq_zero_of_pos`, the `k = r` term
`(-1)^r <a,c>`); (iii) the head collapse via `sum_range_eq_flatMul` + `flatMul_assoc` plus two unnamed
bookkeeping lemmas (head-window additivity, a KillsAbove bound for flat families); (iv) the `j`-fold Leibniz
recursion `D_j = sum_r C(j-1,r) flatMul (iterDeriv r a^-) D_{j-1-r}`; (v) `s.C(s-1,q-1) = q.C(s,q)`.

**Brief errors this batch.**
(a) The general-`i` window law needs NO Leibniz induction: the closed forms above give it in one step each
from `MM.FLM810ZM.splitMinus_factorFamilyZero_eq` + `FLM85Ac.iterDeriv_apply_smul`.
(b) **A real gap in the landed template**: `MM.FLM86MLT.actionFamily_two_modeVL_of` hardwires the tail window
bound `r+1 <= M`, which does not give `r+n-1 <= M` for `n >= 3`; `actionFamily_modeVL_of_window` fixes it.
(c) The order-`j` weight expansion is `MM.FLM86ZM.binomCoeff_add_eq_sum`, NOT
`MM.FLM86ZLI.binomCoeff_sub_eq_sum_moment` (that one is the moment-level, mode-law expansion).
(d) `MMFLM86LS.lean` does not exist: `normOrdered2_iotaVL` is in `MMFLM86LimitStep.lean:482`.
(e) "GEN2 generalises almost verbatim" holds only for the LEFT half: the order-2 right-hand composite acts on
the honest base field, while for `j >= 3` the factor acts on the field of a lower coherent state (a flat
sum), and there is no landed general-`r` analogue of `actionFamily_one/two_base_iotaVL`.
(f) The final identification in the mode law is not a direct coefficient match: it needs the (8.5.5) recursion
at order `j-1-l` and the triangle transpose, and the tail coefficient `-n<c,a>C(-r,n)` is exactly what makes
the count close.

---

### Batch 200 — the Fock carrier input is discharged; the GEN-ALL computation is complete up to one finite-sum assembly

**A — GEN-ALL induction** (`Definitions/MMFLM86GeneratorLineAllIndAux.lean`, 575 lines, 20 guards +
`Definitions/MMFLM86GeneratorLineAllInd.lean`, 249 lines, namespace `MM.FLM86GAI`).  Every input of the
residual `MM.FLM86GLA.ZAlignedGeneratorFlat` is now landed, at a general `E` with only `hs`:

* the algebra: `algebraMap_rat_inv_natCast`, `binomCoeff_mul_pred` (`s.C(s-1,q-1) = q.C(s,q)`),
  `sum_range_alternating_binomCoeff` (`sum_{r<=n} (-1)^r C(s,n-r) = C(s-1,n)`), the triangle reindexings
  `sum_range_triangle_swap`/`_shift`/`_shift_plus`, `prod_range_sub_eq_neg_pow_mul_factorial`;
* item (i): `splitMinus_factorFamilyZero_succ` (the creation half of the next derived factor), through
  `heisFamZero`;
* item (ii): `splitPlus_factorFamilyZero_succ_apply_iotaVL`, `killsAbove_factorFamilyZero_succ_iotaVL`,
  `tailWindow_factorFamilyZero_succ_iotaVL` — **the window replacement cannot use `r+1 <= factor.bound`**
  (the `k = r` value `(-1)^r <a,c>` may vanish); compare `max bound (r+1)` instead;
* item (iv) **CORRECTED**: the record's `D_j = sum_r C(j-1,r) flatMul (iterDeriv r a^-) D_{j-1-r}` is FALSE
  (at `j = 2` it gives `2 flatMul a^- E^-_{a+b}`, against the true
  `flatMul (deriv a^-) E^-_{a+b} + flatMul (a^- a^-) E^-_{a+b}`); the landed correct identity is
  `sum_{r+p=N-1} (r! p!)^{-1} flatMul (iterDeriv r U) (flatMul (iterDeriv p E) Z) = ((N-1)!)^{-1} flatMul (iterDeriv N E) Z`
  (`sum_factorial_flatMul_iterDeriv`, with `iterDeriv_flatMul`, `deriv_sum`, `iterDeriv_deriv_general`,
  `factorial_inv_mul_choose`);
* the (8.4.16) head against the flat value at ALL orders and any `E` (`sum_head_flatGeneratorValueZ`,
  `flatGeneratorValueZ_eq_zero_of_lt`), the tail collapse (`tailOp_factorFamilyZero_flatGeneratorValueZ`,
  using the induction hypothesis as a hypothesis), and the (8.5.5) recursion at a lattice generator in
  component form (`ZAligned_creationExpVL_eq_sum_actionFamily`).

**The remaining step is one finite-sum identity (ASSEMBLY)**, with every input landed:
`(sum_{r<j} head_r) + (sum_{r<j} tail_r) = j . eps(a+b,c) . sum_{l<=j} C(s,l)((j-l)!)^{-1} D_{j-l}(N_l) iota(a+b+c)`,
proved by `sum_range_triangle_swap` + `sum_factorial_flatMul_iterDeriv` (head) and
`sum_range_triangle_shift_plus` + `sum_range_alternating_binomCoeff` (tail), with the coefficients closing
at every offset `l <= j` via `binomCoeff_mul_pred` and `j.(j!)^{-1} = ((j-1)!)^{-1}`.  The report hand-verifies
that the induction closes.

**B — the Fock carrier** (`Definitions/MMFLM86FockCarrier.lean` 237 lines, `...Aux.lean` 394,
`...Span.lean` 162, namespace `MM.FLM86FC`/`MM.FLM86FCA`).  **`FockCreatedSpan E` IS DISCHARGED
UNCONDITIONALLY** (`fockCreatedSpan`, no `hs`): the landed `MM.FLM85CA.createdStateSpan_eq_top`'s two inputs
are the landed (8.5.23) `creationPropertyAligned` (its `vertexOpOfField (Yaligned E)` spelling is
DEFINITIONALLY the required `toVertexOp (ZAligned E)` — no bridge lemma needed) and the (8.5.2) identification
at `umode p = modeVL p.2 (-1) vacuumVL`, which is now PROVED (`zAligned_modeVL_neg_one_vacuum`), so no
`ModeIdentification` residual had to be named.  Hence the Fock residual takes ONE carrier input:
`coherentCreatedStep_of_twoOneSided_of_pairDatum`, and the three `fockCreatedStep_iff_*` forms take only
`FockPairDatum E`.

The carrier datum is then reduced: `fockPairDatum_iff_pairFiniteAll` (`FockPairDatum E <-> PairFiniteAll E`,
i.e. `forall a x, PairFinite (ZAligned E a) (ZAligned E x)`, via the shift-invisibility
`pairFinite_shiftedField_iff` and the now-proved spanning) — so **"lists of length >= 2" is not an
independent case**; the new closure laws `pairFinite_add_left`/`_smul_left` (not landed before) and
`pairFinite_linearMap_of_mem_span`; the positive part extended along spans under pairwise `0 <= <c,d>`
(`pairFinite_shiftedField_span_left`, `pairFinite_span_span`, `pairFinite_created_singleton_span_left`);
the named residual `PairFiniteNegPairing E` (negative pairing at lattice generators) and
`pairFiniteGeneratorsAll_of_negPairing`; and `fockPairDatum_imp_negPairing`, which shows the batch-197-B
carrier input is strictly stronger than batch 182's `ShiftedPairFiniteGenerators`.  The report's numerical
note (not formalized, nothing depends on it): the naive "negative pairing forces an infinite OPE tail" worry
is WRONG — the summand is a diagonal extraction and cancels.

**Correction of record**: batch 199-B's §3 item (iv) is FALSE as recorded and is corrected by the landed
`sum_factorial_flatMul_iterDeriv` above.  `NOTES §95.5`'s "GEN-ALL open" remains correct.

**Brief errors (B200-A).**  (1) item (iv) was false (see above).  (2) item (iii)'s head-window additivity and
flat-family KillsAbove are NOT needed.  (3) item (ii)'s window replacement needs `max bound (r+1)`, not the
factor's own bound.  (4) No `c`-independent operator-valued flat family can equal `ZAligned` at the coherent
state (the (8.4.17) shift lives in the value at `iota(c)`); the head must be stated against the flat value at
components `n+i`.  (5) item (i) goes through `heisFamZero`.
**Brief errors (B200-B).**  (1) No `vertexOpOfField`/`toVertexOp` bridge is needed (definitional).
(2) No `hs` is needed for the span.  (3) `pairFinite_add_left`/`_smul_left` were missing from the tree.
(4) The "arbitrary single-factor `a` via `spanningStatement`" route is incomplete (`{iota(c)}` does not span
`V_L`).  (5) The "lists of length >= 2" item conflates the carrier datum with the created-state step.

---

### Batch 201 — 🎉 the MODE-LAW half is UNCONDITIONAL (`zLineModeAt_all`), and the `l = 0`, `<a,b> <= -3` residual is reduced to a `normOrdered2`-only statement

**A — MODE-ALL CLOSED** (`Definitions/MMFLM86ModeLawAllStepAux.lean`, 290 lines +
`Definitions/MMFLM86ModeLawAllStep.lean`, 571 lines, namespace `MM.FLM86MLA`).  The batch-199-A residual
`AnchorModeCommStep E` is a THEOREM at a general `E` with only `hs`, hence

```
anchorModeCommStep (hs) (E) : AnchorModeCommStep E
anchorModeComm_all (hs) (E) : forall j a b c r, 1 <= r -> AnchorModeComm j r
zLineModeAt_all (hs) (E) : forall j a b c r, 1 <= r -> forall K u, ZLineModeAt (F := F) bL form E a b c j r K u
```

**so the second half of `ZLineGeneratorBase` is unconditional and the base now rests on the generator half
`ZAlignedGeneratorFlat` ALONE** (this supersedes the "mode law ⟺ `AnchorModeCommStep`" row of NOTES 97.4).
The missing input was the (8.4.16)/(8.4.19) head/tail shift against a FINITE FAMILY of correction fields
(`headWindow_factorFamilyZero_modeVL_shift_sum`, `tailWindow_factorFamilyZero_modeVL_shift_sum`): the landed
`actionFamily_modeVL_of_sumWindow` keeps ONE `-gamma.headWindow` term OUTSIDE the `sum_l`, so summing the
landed single-field shifts would count it `N` times.  Then `ModeCommFam`, the abstract step
`modeCommFam_step` (consuming the (8.5.5) recursion as a hypothesis) and the anchor instantiation close the
induction.

**B — the `l = 0`, `<a,b> <= -3` residual, reduced** (`Definitions/MMFLM86NegThreeBridgeAux.lean`,
201 lines + `Definitions/MMFLM86NegThreeBridge.lean`, 740 lines, namespace `MM.FLM86NTB`):

* `negThreeExpandedResidualAligned_iff_anchorForm` (IFF, NO hypothesis) — the expanded residual IS its
  anchor-field spelling, each `coherentFieldNat E a (anchorVL E a b) (N-1-i)` replaced by
  `(anchorField E a b (N-1-i)).fam (i-m-n-1)` (new dictionary `coherentFieldNat_eq_anchorField_fam`);
* `anchorGeneratorValue_iff_zAlignedGeneratorFlat` (aux) — the anchor value at generators is **literally**
  batch 199-B's `ZAlignedGeneratorFlat`, i.e. not a new residual;
* given the named `ZLineAll E`, `negThreeAnchorForm_iff_chamberMomentForm` and
  `negThreeExpandedResidualAligned_iff_chamberMomentForm` — the residual becomes
  **`NegThreeChamberMomentForm E`**, in `normOrdered2` alone (the difference of the two (8.4.26) chamber
  series equals the single moment of weight `C(r+m,N-1)` at diagonal `N-m-n-2`, `N = (-<a,b>).toNat`);
  the weight matches the campaign's negative-pairing `ν = c-3` chamber;
* the `<a,b> = -3` slice **unconditional** (`negThreeAtMinusThree_iff_chamberMomentFormAtMinusThree`), using
  only `ZLine 0,1,2`, with the consumed `k <= 2` anchor values landed (`anchorField_generator_of_le_two`,
  `anchorValueAtMinusThree_of_le_two`);
* one-directional generator bridges landed (`negThreeAnchorForm_apply_iotaVL(_flat)`); the converse is
  explicitly NOT claimed — `{iota(c)}` spans only the lattice factor, not the Fock half.

**Corrections of record (batch 201).**
(a) The requested "residue ⟺ anchor VALUES at generators" IFF **does not exist**: the residue's left-hand side
is the honest commutator `twoVarField - swap2`, which the anchor values do not mention.  What is an IFF is the
anchor-field SPELLING (no hypothesis) and, given `ZLineAll`, the `normOrdered2` chamber/moment form.
(b) The `<a,b> = -3` slice does **not** close from the landed `k <= 2` tower: the anchor fields CANCEL there,
leaving the `N = 3` chamber collapse (`NegThreeChamberMomentFormAtMinusThree`).
(c) The residue's summand is an OPERATOR component `(anchorField ...).fam (i-m-n-1)`, not a value at
`iotaVL c` (that spelling would lose the Fock half).
(d) Lean traps: `-(C) • Z` puts the negation on the SCALAR and `a * b • c` parses as `a * (b • c)`
(`•` infixl:73 binds tighter than `*` infixl:70) while the printer renders `(-C)•Z` and `-(C•Z)`
identically — write the `delta` family as `((-1:F) * (theta * C l))`; unapplied
`rw [neg_smul]`/`[smul_sub]`/`[smul_smul]`/`[Finset.smul_sum]` fail on syntactically identical occurrences
in this pin and `abel` mangles additive indices inside atom arguments (use explicitly applied local `have`s).
(e) The `1/j` of the (8.5.5) recursion is never introduced: multiply by `j`, the coefficient
`(j-1-l) + (l+1) = j` closes, and `j` cancels by `smul_right_injective` (char 0); the reindex `t = i+1` is
unnecessary (alpha-renaming suffices).

**Accounting after batch 201**: `ZLineGeneratorBase` ⟺ `ZAlignedGeneratorFlat` (the generator half) ALONE;
`NegThreeAndBelowResidualAligned` ⟺ (given `ZLineAll`) `NegThreeChamberMomentForm`; Fock ⟺
`Display8610TwoOneSided` + `PairFiniteAll`; `EvenCocycleJacobiStep` open.

---

### Batch 202 — 🎉🎉 THE WHOLE `z_0^j` TOWER IS CLOSED: `ZAlignedGeneratorFlat` is a theorem, `ZLineGeneratorBase` is a theorem, and EVERY `l = -c` FACE IS UNCONDITIONAL at FLM's datum

**A — the ASSEMBLY, and the close** (`Definitions/MMFLM86GeneratorLineAllCloseAux.lean` 315 lines +
`...CloseTail.lean` 302 + `...Close.lean` 334, namespace `MM.FLM86GLC`, 23 public declarations each with a
guard).  At a general `E` with only `hs`:

```
zAlignedGeneratorFlat (hs) (E) : ZAlignedGeneratorFlat (F := F) bL form E
zLineGeneratorFlat / zLineGeneratorAll : ZLineGeneratorAll (F := F) bL form E
zLineGeneratorBase (hs) (E) : ZLineGeneratorBase (F := F) bL form E
zLine_all_of_generatorBase (hs) (E) : forall i, 1 <= i -> ZLine (F := F) bL form E i
negCBinomMomentAligned_of_generatorBase (hs) (E) {c} (hc : 1 <= c) : NegCBinomMomentAligned (F := F) bL form E c
```

The missing finite-sum identity is machine-checked and NAMED: `assembly` (head sum + tail sum =
`j * flatGeneratorValueZ`), with `sum_head_assembly`, `sum_tail_assembly`, `assembly_coeff`.  Batch 200-A's
hand computation is confirmed (formula right; the formalisation needed the four corrections below).  The mode-law
half was already unconditional (batch 201, `MM.FLM86MLA.zLineModeAt_all`), so `ZLineGeneratorBase` is now a
theorem and the campaign's ENTIRE `z_0^j` tower `ZLine E i` (`i >= 1`) is unconditional.

**B — the face-level consumer** (`Definitions/MMFLM810NegCFaceJ.lean`, 329 lines, `MM.FLM810NCFJ`, 9
theorems):

* `negCFaceAlignedFrom_of_lines` — the `J`-parametrised consumer: `NegCFaceAlignedFrom E c (c-1-J)` from the
  lines `1 <= j <= J` at `E = C.toCentralExtension h2`.  Batch 196's `subTwo` proof generalises VERBATIM: the
  numerical identity `c-1-(c-2) = 1` lives only inside `negCBinomMomentAlignedAt_sub_two`'s own proof, never in
  the face consumer, so **no `C(r+m,k)` weight split and no new binomial arithmetic occurs at this layer**; the
  only new arithmetic is `c-1-J <= nu -> c-1-nu <= J` (`private chamberOrder_le`);
* `negCFaceAlignedFrom_subThree_evenCocycle` — the chamber `nu >= c-3`, UNCONDITIONAL (from `ZLine E 2`);
* `negCFaceAlignedFrom_subOne_evenCocycle` — the `J = 1` chamber, re-proved by the new route (a compiled
  `example` checks it is statement-identical to the landed batch-196 theorem);
* `negCFaceAlignedFrom_iff_chambers_evenCocycle` — the SHARP equivalence
  `NegCFaceAlignedFrom E c nu0 <-> forall nu, nu0 <= nu -> nu <= c-2 -> NegCBinomMomentAlignedAt E c nu`
  (so the bound `c-1-J` loses nothing), with the converse
  `negCBinomMomentAlignedAt_of_negCFaceAlignedFrom`;
* generator-base consumers (`_of_generatorBaseUpTo`, `_of_generatorBase`, `negCFaceAligned_of_generatorBase`),
  which A's `zLineGeneratorBase` makes unconditional.

**Capstone (dispatcher), `Definitions/MMFLM810NegCFaceClosed.lean`** (4 theorems, `MM.FLM810NCFC`): the
composition --- `zLine_all`, `negCBinomMomentAligned_all` (general `E`, every `c >= 1`),
`negCBinomMomentAlignedAt_all` (the explicit chamber at the even cocycle), and

```
negCFaceAligned_evenCocycle (C) (h2) (hs) {c} (hc : 1 <= c) :
    NegCFaceAligned (F := F) bL form (C.toCentralExtension (F := F) h2) c
```

**the WHOLE `l = -c` face, every pairing `0 <= <a,b>`, at FLM's own even-lattice datum, for every `c >= 1`.**

**Four corrections of record (B202-A).**
(1) The landed Leibniz rule `MM.FLM86GAI.sum_factorial_flatMul_iterDeriv` carries ℚ-scalars
(`[Algebra QQ A]`); instantiating it at `Module.End F (VL F L)` gives `HSMul` terms that do NOT unify with the
assembly's F-scalars --- the F-scalar form is landed as `sum_factorial_flatMul_iterDeriv_F`
(`IsScalarTower.algebraMap_smul` + `map_inv0`/`map_natCast`/`map_mul`).
(2) The head window replacement is OFF BY ONE: the kills-bound must be `M0+1` with
`M0 = (q - <a,b+c> - n - 1).toNat` (the flat value at `i = M0` need not vanish);
`sum_head_flatGeneratorValueZ` is applied with `M = M0+1`.
(3) The head is applied to a component and then to `iotaVL c` while the Leibniz lemma is family-level: an
explicit `Finset.sum_apply`/`LinearMap.sum_apply`/`Pi.smul_apply`/`LinearMap.smul_apply` distribution step is
required (this is what hides (1) in the report's notation).
(4) The tail reindexing cannot be a direct `rw` (the Nat subtraction `(r+1+l)-1-r` does not reduce
definitionally): an explicit `Finset.sum_congr` conversion is needed, the `L = 0` term must be split off before
`sum_range_alternating_binomCoeff`, and a final scalar reorder `eps*s*Fc*C = eps*s*(C*Fc)`.
**Six corrections of record (B202-B)**: (i) the template leaves `E` unbound (bind `E = C.toCentralExtension h2`);
(ii) "check where `subTwo` uses `c-1-(c-2)=1`" is a false premise (it is in the chamber theorem, not the
consumer); (iii) no `C(r+m,k)` weight split at the consumer; (iv) the chamber arithmetic
`c-1-J <= nu -> c-1-nu <= J` was not landed; (v) `negCFaceAlignedGe_evenCocycle` is the whole range
`nu >= c-1`, not a single chamber; (vi) "reproduces batch 196" can only be a statement-level check.

**Accounting after batch 202**: the `l != 0` content of (8.8.41) --- every `l = -c` face, every chamber, every
`c >= 1` --- is UNCONDITIONAL at FLM's datum.  The remaining inputs of Theorem 8.10.2 are the `l = 0` face at
`<a,b> <= -3` (`NegThreeChamberMomentForm`), the Fock half (`Display8610TwoOneSided` + `PairFiniteAll`), and
the assembly `EvenCocycleJacobiStep` (the `l = 0` assembly at (8.6.2) states + the passage from `Res_{z_0}` to
(8.10.8)).

---

### Batch 203 — the Fock ledger is corrected (the carrier inputs are NOT inputs of Theorem 8.10.2), the `l = 0` face is assembled at ALL states up to one one-mode step, and the `<a,b> <= -3` residual's exact status is settled

**A — the `<= -3` residual is not weaker than its chamber spelling** (`Definitions/MMFLM86NegThreeChamberAux.lean`
380 lines + `Definitions/MMFLM86NegThreeChamber.lean` 356 lines, 30 guards).  With the `ZLineAll` hypothesis now
DISCHARGED (`zLineAll_of_symmetricForm`, batch 202), both directions hold at a general `E` with only `hs`:

```
negThreeAndBelowResidualAligned_iff_chamberMomentForm (hs) (E) :
    NegThreeAndBelowResidualAligned E <-> NegThreeChamberMomentForm E
negThreeExpandedResidualAligned_iff_chamberMomentForm' (hs) (E) : (same, expanded spelling)
```

so **`NegThreeChamberMomentForm E` IS the open `l = 0`, `<a,b> <= -3` residual**, in `normOrdered2`-alone
spelling.  No pointwise slice is cheaper: `residueAtN_iff_chamberMomentAtN` for every `N >= 1`, and
`NegThreeChamberMomentForm E <-> ChamberMomentAtN E 3 /\ (forall N >= 4, ChamberMomentAtN E N)` with the `N = 3`
slice being batch 201-B's theorem.  New landed shapes: the whole-negative-pairing chamber form
`AllChamberMoment`, `ResidueNegativeAligned <-> AllChamberMoment`,
`allChamberMoment_iff_slices : AllChamberMoment E <-> ChamberMomentAtN E 1 /\ ChamberMomentAtN E 2 /\
NegThreeChamberMomentForm E`, the `N = 1, 2` slices as theorems at FLM's even-cocycle datum, and
`residueNegativeAligned_iff_negThreeChamberMomentForm_evenCocycle` (the chamber-spelling mirror of batch 196).
**New refutation**: `not_allChamberMoment_rankOne` --- at the rank-one datum (`Q`, `<m,n> = mn`, `extR`) the
`N = 1` slice FAILS, so evenness is *proved* necessary there; no obstruction exists at `N >= 3`.
**Recommendation of record: do NOT dispatch further "prove `NegThreeChamberMomentForm`" tasks** --- they are
identical to proving the `<= -3` residual itself.

**B — the Fock ledger correction, and the all-states assembly** (`Definitions/MMFLM86FockDisplaySharp.lean`
295 lines + `Definitions/MMFLM86Corollary851AllStates.lean` 321 lines).

```
display8610TwoOneSided_iff_fockCreatedStep (E) : Display8610TwoOneSided E <-> FockCreatedStep E   -- NO hypotheses
```

**Mechanism**: the display's `l = 0` slice spends the UNCONDITIONAL (8.5.23) creation property at `(v,0)`, so it
reads `Y((Y x)_{-j-1} y) = convOneSided (shiftedField x j) (Y y)` for an ARBITRARY `y` --- i.e. it IS
`IterateProductStep` (8.4.25) at every state, and `iterateProductStep_iff_forall_at` upgrades it.  Hence
**the Fock front is `FockCreatedStep E` ALONE** (equivalently the two-factor display), and
`PairFiniteAll`/`FockPairDatum`/`FockCreatedSpan` are **NOT inputs of Theorem 8.10.2** (this supersedes
NOTES 97.1/97.4, 99.4, and ROADMAP batch 200's "Fock = display + PairFiniteAll").  Batch 197-B's carrier
hypotheses are redundant: `coherentCreatedStep_of_twoOneSided` needs `hnd`/`hs` only, and
`fockCreatedStep_iff_createdStep_sharp` likewise.  The old route needed `hpair`+`hspan` only because it went
through the created states rather than through the `l = 0` slice.

The steered target landed the `l = 0` face at ALL states: `corollary851_lattice_canonical` (the lattice corner
at every pairing, canonical window), its lift `corollary851_lattice_of_canonical_le` to every `K`, the explicit
(8.5.4) `List.induction` (`stateOfVL_nil`/`_cons`, `corollary851_spanningStates_of_lattice_and_step`),
`corollary851_allStates_of_lattice_and_step` (arbitrary `Y`), and the compositions
`corollary851_allStates_of_residuals` / `corollary851_allStates_of_fockCreatedStep`.  The ONE remaining named
residual there is

```
def Corollary851ModeStep (Y) (K) : Prop := the two one-creation-mode steps
    C u v -> C (alpha_c(-r) u) v   and   C u v -> C u (alpha_c(-r) v),  r >= 1, at the SAME window K
```

--- the (8.6.4)--(8.6.27) one-mode fragment, NOT a restatement of the target.  Measured spellings: the window
side condition `hwin : forall a b, (-<a,b>).toNat <= K` must be explicit (`ResidueNegativeAligned` is per-pair
canonical-window), and the positive-pairing half of the corner exists only at the even cocycle, so the composed
theorems are at `C.toCentralExtension h2`; `hnd` is spent only by the Fock conjunct.

**Brief errors (batch 203).**  (A) (1) the target is not weaker than the residual; (2) the "mirror the positive
`ν = c-3` chamber" route cannot reach it (the positive theorems ARE the already-unconditional `ZLine` tower,
consumed as an IFF); (3) "`N >= 4` is the new content" is half right --- `N = 3` is equally open; (4) "largest
honest slice `N <= N0`" has no content inside `NegThreeChamberMomentForm`; (5) evenness is needed exactly for
`N = 1, 2`.  (B) (1) the Fock front is the display ALONE; (2) "what is missing is the actual two-factor
computation" is false as a reduction target (the `k = 2` display IS the one-step); (3) the carrier hypothesis of
`coherentCreatedStep_of_twoOneSided*` is redundant; (4) the requested `corollary851_of_residuals` shape is not
reachable as stated (the mode step is a third input); (5) Lean trap: stating the (8.5.4) induction with a literal
`List.foldr` in the goal blows the `whnf` budget --- state through `stateOfVL` with `rfl` unfolding lemmas and
`simpa only`, never `maxHeartbeats`.

**Accounting after batch 203**: `EvenCocycleResiduals` = `NegTwoFirstDerivZero` (LANDED) /\
`NegThreeExpandedResidualAligned` (= `NegThreeChamberMomentForm`, open) /\ `CoherentCreatedStep` (implied by
`FockCreatedStep` = the display, open).  `EvenCocycleJacobiStep`: (ii) DONE (batch 202), (i) reduced to
`Corollary851ModeStep` + the lattice corner, (iii) untouched.

---

### Batch 204 — 🏆 the `l = 0`, `<a,b> <= -3` residual IS PROVED at FLM's datum (all pairings, all N), and the Fock one-step is reduced to its shift-zero corner

**A — the residual, proved** (`Definitions/MMFLM86NegThreeProveAux.lean` 374 lines +
`Definitions/MMFLM86NegThreeProve.lean` 248 lines, namespace `MM.FLM86NTP`, 17 guards).  At
`E = C.toCentralExtension h2`, for every pairing and every `N >= 1`:

```
chamberMomentAtN_evenCocycle (C) (h2) (hs) (N) (hN1 : 1 <= N) : ChamberMomentAtN (C.toCentralExtension h2) N
negThreeAndBelowResidualAligned_evenCocycle   : FLM86NTR.NegThreeAndBelowResidualAligned (C.toCentralExtension h2)
negThreeExpandedResidualAligned_evenCocycle   : FLM86CF2.NegThreeExpandedResidualAligned (C.toCentralExtension h2)
negThreeChamberMomentForm_evenCocycle / allChamberMoment_evenCocycle
residueNegativeAligned_evenCocycle            : FLM86EPA.ResidueNegativeAligned (C.toCentralExtension h2)
```

The last is the WHOLE negative-pairing `Res_{z_0}` face, and the second is exactly the second conjunct of
`EvenCocycleResiduals`.  **Mechanism (the campaign's "hard residual" reading was wrong)**: `ChamberMomentAt` is
about the single antidiagonal family `M p = normOrdered2 E a b p (K-p) v`; the two chamber sums are its
half-lines `p >= N-m-1` and `p <= -m-1`, the moment sum is over all `p` with weight `C(p+m,N-1)`.  The landed
(8.4.21) transpose `MM.FLM86NOC.normOrdered2_transpose` supplies the factor `c(b,a)` on the second chamber, the
three (8.5.37) coefficient identities close the first, the second needs exactly `c(b,a) = (-1)^N`, and the gap
`-m <= p <= -m+N-2` carries the zero of the falling factorial
(`MM.FLM86R.binomCoeff_eq_zero_of_between`).  Finiteness is the LANDED per-antidiagonal window
`MM.FLM810NOC.assemblyDiagonal_antidiagonal` --- NOT `LocallyFinite2 normOrdered2`, which is refuted in
`MMFLM86InfSupp`.  **No Fock induction, no mode law, no anchor value, no spanning.**

Exact general-`E` gap, named: `CocycleAtPairing E a b N := E.c b a = (-1)^N` (a theorem at the even cocycle via
`EvenCocycle.c_eq_signPow` + `signPow_zmod_two_eq_zpow`), with the reduction
`chamberMomentAtN_of_cocycleSign`; the underlying SCALAR identity is proved BOTH ways
(`neg_mul_ratioCoeff_eq_iff : -c * ratioCoeff (-(N:Z)) k = binomCoeff (-(k:Z)-1) (N-1) <-> c = (-1)^N`).
"Evenness" is the wrong word: only the cocycle SIGN `c(b,a) = (-1)^N` is used (`N = 1` is refuted at a general
`E` by the landed rank-one counterexample; `N = 2` has a conditional obstruction; `N >= 3` neither proved nor
refuted at a general `E`).
**Bonus**: `antidiagonalCollapse_of_symmetricForm (hs) (E) : MM.FLM86NOC.AntidiagonalCollapse E` --- the `c`-free
`N = 1` collapse at a general `E`, composing the landed batch-181-A `MM.FLM86AR.assemblyDiagonal` with
`MMFLM86AntidiagonalAssembly.assemblyDiagonal_iff_collapse`.  Two stale file headers claiming
`AssemblyDiagonal`/`AntidiagonalCollapse` were "named, not proved" were corrected in place by the dispatcher.
**Withdrawal of record**: batch 203-A's recommendation "do NOT dispatch further prove-`NegThreeChamberMomentForm`
tasks" is WITHDRAWN --- the residual was provable and is proved; the chamber spelling is where it is easy.

**B — the Fock one-step at its sharpest** (`Definitions/MMFLM86FockOneStepProveAux.lean` 219 lines +
`Definitions/MMFLM86FockOneStepProve.lean` 306 lines, `MM.FLM86FOPA`/`MM.FLM86FOP`).

```
iterateProductStep_iff_level_zero (Y) (mul) (D) (hD) : IterateProductStep Y mul <-> IterateProductStepLevel Y mul 0
fockCreatedStep_iff_level_zero (hnd) (hs) (E) / fockCreatedStep_iff_shiftZero (hnd) (hs) (E)
display8610TwoOneSided_iff_shiftZero (hnd) (hs) (E) / coherentCreatedStep_of_shiftZero (hnd) (hs) (E)
```

with `FockShiftZeroStep E := forall x v, ZAligned E ((ZAligned E x) (-1) v) = convOneSided (ZAligned E x)
(ZAligned E v)` --- the `(j,l) = (0,0)` corner of (8.6.10), i.e. `Y(x_{-1}y) = Y(x)Y(y)`.  The engine is the
**`L(-1)` LADDER OF THE ONE-STEP**: the `j`-th level of (8.4.25) at `(x,v)` IS the level-zero statement at the
translated factor `(1/j!)L(-1)^j x`, because `((1/j!)L(-1)^j x)_{-1} = x_{-j-1}` and
`Y((1/j!)L(-1)^j x) = shiftedField Y x j` (iterated (8.10.13)).  The abstract form carries NO hypothesis on
`mul`, so it applies to the `dite` carrier `convOneSided`.  **New `hnd`/`hs`-free consequences**:
`coherentCreatedStep_of_twoOneSided_noHyp`, `fockCreatedStep_imp_coherentCreatedStep`,
`fockCreatedStep_imp_display8610One` (the one-step's vacuum slice IS the `k = 1` display, so the (8.10.13) input
is needed only to PROVE `display8610One_ZAligned` unconditionally, not to consume the residual), and
`fockCreatedStep_iff_createdStep` with NO hypotheses.

**Brief errors (B204-B).**  (1) The suggested mode-law/F-linearity induction does NOT close and is not needed:
the landed action laws (`MM.FLM86MLA.actionFamily_factorFactorZero_modeVL`, `actionFamily_modeVL_of_window`)
take the two-component commutator as a HYPOTHESIS on the accumulated field, and the Fock-level step would need
the commutator of two arbitrary honest modes --- the open Jacobi content.  The ladder bypasses it.
(2) "The `k = 1` display takes `hnd`/`hs`" is true of the PRODUCER, false of the consumer.
(3) The left-side expansion of `Y((Y x)_{-j-1} v)` exists only at lattice generators.
(4) The (8.5.5)-recursion induction on `v` is not the mechanism.
(5) `MM.FLM86MLA.actionFamily_factorFactorZero_modeVL` is not in the Fock front's import closure.

**Accounting after batch 204**: at FLM's datum `EvenCocycleResiduals` = `NegTwoFirstDerivZero` (LANDED) /\
`NegThreeExpandedResidualAligned` (**LANDED**) /\ `CoherentCreatedStep` (implied by `FockCreatedStep` =
`FockShiftZeroStep`, open).  Remaining for Theorem 8.10.2: the Fock shift-zero corner, `EvenCocycleJacobiStep`'s
(i) (`Corollary851ModeStep`, the all-states `l = 0` face) and (iii) (the passage from `Res_{z_0}` to (8.10.8)).

---

### Batch 205 — ⚠️ THE FOCK CORNER IS **REFUTED** at honest data (and FLM itself flags the identity as formal); the `l = 0` all-states face is exactly one mode step

**A — the refutation** (`Definitions/MMFLM86FockShiftZeroAux.lean` 398 lines + `Definitions/MMFLM86FockShiftZero.lean` 178 lines, `MM.FLM86FSZA`/`MM.FLM86FSZ`, 13 guards).  Machine-checked, spending **neither `hnd` nor `hs`**, and no evenness or cocycle input:

```
not_fockShiftZeroStep_of_gram (E) (i) (hgram : form (bL i) (bL i) != 0) : not (FockShiftZeroStep bL form E)
not_fockShiftZeroStep_rankOne   : not (FockShiftZeroStep (F:=Q) basisUnitInt (mul Z Z) (trivialCentralExtension Q Z))
not_fockCreatedStep_of_gram (hnd) (hs) (E) (i) (hgram) : not (FockCreatedStep bL form E)
not_display8610TwoOneSided_of_gram (hnd) (hs) (E) (i) (hgram) : not (Display8610TwoOneSided bL form E)
not_pairFiniteAll_of_gram / not_fockPairDatum_of_gram (E) (i) (hgram)
not_pairFinite_oneFactorState (E) (a) (hgram : form a a != 0)
```

**Mechanism.**  At `x = alpha_{b_i}(-1) iota(0)` (one oscillator on the vacuum), the (8.5.5) intertwining gives
`ZAligned E x k = modeVL (bL i) k` for `k != 0` (`yAligned_oneFactorState`, `zAligned_oneFactorState`).  Hence the
convolution summand at `(m,w) = (1, iota(0))` is `alpha_a(-j)(alpha_a(j) iota(0)) = ((-j).<a,a>).iota(0) != 0` for
EVERY `j <= -1` (landed `MM.FLM87ZBA2.modeVL_mul_modeVL_vacuum_self`), so no finite window exists:
`not PairFinite (ZAligned E x) (ZAligned E x)`, hence `convOneSided (ZAligned E x) (ZAligned E x) = 0`, while the
corner's left side is `ZAligned E u` for `u = (ZAligned E x)(-1)x = alpha_a(-1)alpha_a(-1)iota(0) != 0`
(monomial basis), against (8.5.23) `creationPropertyAligned`.

**This is exactly what FLM flags twice**: Remark 8.5.2 (printed p. 200) says (8.5.14) "is not rigorous in the
present setting", and Remark 8.6.4 (p. 211) says (8.6.28) is "formal but suggestive".  The honest identity at a
created state is the NORMAL-ORDERED `Y(u_{-1}v) = :Y(u)Y(v):`; the plain product differs by the contraction ---
which is the two-component commutator, here visible as a divergent convolution.

**What is NOT refuted**: `MM.FLM86CCS.CoherentCreatedStep E` --- the shape the boundary theorem actually consumes
(`EvenCocycleResiduals`'s third conjunct is `CoherentCreatedStep`, NOT `FockCreatedStep`).  Its two factors are
honest GENERATOR fields (the second shifted), its carrier is batch 182's
`pairFinite_shiftedField_ZAligned_iotaVL` for `0 <= <a,b>`, and the oscillator-carrying state above does not
occur in it; only `PairFiniteNegPairing` is open there.  Consequently `fockCreatedStep_imp_coherentCreatedStep`
is now VACUOUS, and `CoherentCreatedStep` needs a new producer: the GENERATOR SLICE of the display
(`Display8610TwoGenerators`-style, batch 185-A's `display8610TwoGenerators_iff_coherentForm`).

**B — the `l = 0` all-states face, exactly** (`Definitions/MMFLM86Corollary851ModeStepAux.lean` 363 lines +
`Definitions/MMFLM86Corollary851ModeStep.lean` 387 lines, 32 guards).  NOT proved, but pinned:

* `allStates_iff_corner_and_modeStep (Y) (K)` --- BOTH directions: given the corner (a theorem at FLM's datum
  since batch 204-A), `Corollary851ModeStep Y K` IS the whole remaining content of the `l = 0` face at all
  states.  (So batch 203-B's "one-mode fragment, not a restatement of the target" is misleading.)
* `corollary851_latticeFactor`: the corner extends, by the landed bilinearity, to the lattice factor
  `span{iota(a)}` --- the EXACT largest slice; the reading "bilinear + corner => one arbitrary argument" is
  FALSE.
* `residueOfField`/`FaceOf`, `corollary851_iff_faceOf`, `faceOf_zero`/`add`/`smul`, `faceOfSubmodule`,
  `corollary851_congr_field`: the face is a LINEAR CONDITION on the field `Y u`; and
  `modeStepLeft_iff_faceOf_closure`: the mode step's left half is exactly "the (8.4.16) action preserves the
  face submodule on the image of `Y`".
* `Yzero_modeVL_add`/`_smul`, `ZAligned_modeVL_add`/`_smul`: a one-mode descendant's field is linear in the
  state.  Named residuals: `Corollary851ModeStepOneMode`/`...Right`, `Corollary851CurrentFace`
  (`currentFace_iff_oneMode_zero`).
* **Exact residual**: expanding the action by (8.4.19) and the commutator by Leibniz leaves exactly the
  mode--field commutators `[alpha_c(e), ZAligned E v n]` at ARBITRARY `v`.  The landed action laws
  (`MM.FLM86MLA.actionFamily_factorFactorZero_modeVL`, `actionFamily_modeVL_of_window`) require that as
  `hcomm` and return it only up to a THIRD term `kappa . Yt.fam (m+r+n)`, so the two-component chain cannot be
  iterated.  (Dispatcher check: the landed `MM.FLM810CFC.currentFieldComm` is (7.1.5) at the repaired BASE field
  `baseRepaired E a` ONLY, so it does not supply the general-state commutator.)

**Ledger corrections of record (batch 205).**
(a) ROADMAP batch 203-B ("the Fock front is `FockCreatedStep` ALONE; the carriers are NOT inputs") and batch
204-B / NOTES 101.2 ("the front is the shift-zero corner") reduce to a FALSE target --- the carriers were NOT an
artefact.
(b) NOTES 97.1 and `MMFLM86FockCarrierAux` §2 ("negative pairing is the only place the datum can fail") are
REFUTED: `PairFiniteAll`/`FockPairDatum` are false as stated (they quantify over all states), and they fail at
oscillator-carrying states with `form a a != 0` for ANY pairing.
(c) NOTES 101.3's warning is confirmed and sharpened: the honest identity is the NORMAL-ORDERED one.
(d) Batch 205-B's correction to NOTES 100.3 (see above).
(e) Sections 93--96, 98, 99 (the `ZLine` tower, the `l = -c` faces, the `<= -3` residual) are untouched.

**Accounting after batch 205**: the boundary theorem's third residual is `CoherentCreatedStep` (as originally
defined); the honest Fock front is the GENERATOR slice of (8.6.10) (`Display8610TwoGenerators` /
`CoherentCreatedForm`), whose carrier is available for `0 <= <a,b>` (batch 182) and named as
`PairFiniteNegPairing` otherwise.  `EvenCocycleJacobiStep`: (ii) done; (i) = `Corollary851ModeStep` (=
the mode--field commutators); (iii) untouched.

---

### Batch 206 — the §8.6 engine's honest layer, and the generator slice's `(0,0)` corner is a THEOREM; the exact residuals are named

**A — the generator slice** (`Definitions/MMFLM86CoherentCreatedProveAux.lean` 167 lines +
`...Prove.lean` 361 + `...Residual.lean` 261, namespaces `MM.FLM86CCPA`/`MM.FLM86CCP`).

* **`createdStepAt_zero_zero`** --- the `(j,l) = (0,0)` instance of `CoherentCreatedStep E` at the lattice
  generators IS A THEOREM (symmetric `form`, `0 <= <a,b>`, and NO `hnd`):
  `ZAligned E (YFam E a (-1) iota(b)) = convOneSided (ZAligned E iota(a)) (ZAligned E iota(b))`.  Route:
  `ZAligned_mode_iotaVL` (left) + batch 182's carrier + (8.4.25) + the landed (8.4.20) `assemblyDiagonal` on
  each antidiagonal fibre + the scalar collapse `sum_{k<=N} ratioCoeff N k = (1-1)^N`.  So the generator slice is
  NOT dead at the corner.
* `coherentCreatedStep_iff_twoGenerators` / `<-> CoherentCreatedForm` (both directions, same hypotheses).
* `weightedAssemblyDiagonalZero_iff_assemblyDiagonal` (+ `weightedAssemblyDiagonalZero`): the `r = 0` slice of
  the new residual IS (8.4.20), both ways.
* `pairFinite_shiftedField_right_iff` (shift invisibility in the SECOND factor --- the mirror of batch 200-B,
  which shifts only the first), hence `shiftedPairFiniteGenerators_of_negPairing`
  (`PairFiniteNegPairing E -> ShiftedPairFiniteGenerators E`, the former explicit and never silent).
* **The exact residual: (WAD) = `WeightedAssemblyDiagonal E` at `r >= 1`**:
  `sum_{p+q=K} C(K-p, r) . normOrdered2 a b p (K-p) v = eps(a,b) . ZAligned E (E^-_b(z_0)_r iota(a+b)) (r-K-1) v`
  on a finite window.  Dictionary (derived in the headers, not yet formalized): at `j = 0`,
  `CreatedStepAt E a b 0 l <-> (WAD)` at `r = l-<a,b>`, `K = l-m-1-<a,b>`; `j >= 1` is the `L(-1)` ladder at
  the descendants of `iota(a)`; `<a,b> < 0` additionally needs `PairFiniteNegPairing`.

**Findings of record.**  `MM.FLM86CCS.createdStep_of_iterateProductStep` is **VACUOUS too** (its
`IterateProductStep` at `(x,0)` with base state `v := x` IS the refuted `FockShiftZeroStep`); NOTES §102.2's
"the oscillator-carrying state does not occur in it" holds only for the FACTORS --- the coherent base state
`creationExpVL b l (iota b)` IS oscillator-carrying for `l >= 1`, and what saves the slice is the carrier
(pairs of SHIFTED GENERATOR fields are finite for `0 <= <a,b>`) plus (WAD).  Ledger refinement:
`CoherentCreatedStep` is open only from `(j,l) != (0,0)`; the carrier alone is not the whole extra input for
`l >= 1`.

**B — the section 8.6 engine's honest layer** (`Definitions/MMFLM86Engine86.lean`, 405 lines, `MM.FLM86E86`,
22 declarations, `hnd` never spent, `hs` only in the two (8.4.21)/assembly theorems):

* (8.6.5) at the honest field: `series865` with `series865_zero`/`_succ`/`_one` and the dictionary to the landed
  `multiExpIdx`;
* the honest field at an ordinary power: `powerFam := YFam(-m-1)` with `powerFam_vacuum` and
  `powerFam_iotaVL : Y(a,w)_m iota(c) = E^-_a(w)_{m-<a,c>} iota(ac)` --- the EXACT pairing shift;
* (8.4.16) iterated: `splitFold` with `splitFold_zero`, `splitFold_one = Y(a,w)`,
  `splitFold_two = MM.FLM84.normalPair`, `splitFold_vacuum_one`;
* (8.6.10) at `k = 1`: `iteratedCreation_iotaVL` and `iteratedCreation_iotaVL_shift` (the exact shift --- this
  machine-checks batch 205's refutation mechanism), and on the vacuum the honest iterate IS the (8.6.5)
  coefficient;
* (8.6.6)/(8.6.8) at `k = 2` via the landed (8.4.20): `normOrdered2_vacuum_eq_series865` (the display's first
  equality on the vacuum, `eps(a,b)` explicit), `normOrdered2_swap_vacuum` ((8.4.21)), `assemblyDiagonal_vacuum`.

**The exact remaining gap: (8.6.6) for `k >= 3`** --- the `k`-fold middle family has NO carrier: (8.4.20) is the
nested TWO-variable Cauchy product (`convWith2` twice) and a `k`-variable nested carrier with local finiteness is
missing.  (8.6.8) reduces to it; (8.6.9)/(8.6.10) at general `k` are the landed open slices
(`IterateProductStep`, `Display8610TwoGenerators`, `CoherentCreatedStep`, `CoherentCreatedForm`); the
universally quantified plain-product spellings are REFUTED (batch 205) and only cited.  Passage (iii) was NOT
reached (it needs the `k >= 3` carrier first).

**Corrections of record (B206-B).**  (a) (8.6.5) and `kFoldNormalOrder` were ALREADY landed (batch 119:
`MMFLM86MultiState` 651 lines, `MMFLM86KFold` 880; ROADMAP:5013-5015 / NOTES:1055-1056 --- and ROADMAP:4679 is
STALE).  (b) `kFoldNormalOrder` is NOT (8.6.6)'s middle family for `k >= 2` on honest lattice fields (its mode
splitting includes the field's `E^+` half); only `normOrdered2` is.  (c) Remark 8.6.4 (p. 211) displays the
PLAIN product (8.6.28) and calls it "formal but suggestive"; the normal-ordered form is batch 205's repair, not a
quotation.  (d) The `iota`-shifts belong to (8.5.11), not to the shift-free (8.6.5).  (e) Section 8.6 has no
associativity law for the `k`-fold normal-ordered product; only (8.4.21) symmetry, landed here for `k = 2`.

**Accounting after batch 206**: `CoherentCreatedStep` open only from `(j,l) != (0,0)`, residual (WAD) at
`r >= 1`; the §8.6 route (iii) needs the `k >= 3` nested Cauchy carrier; item (i) = `Corollary851ModeStep`
(the mode--field commutators); `NegTwoFirstDerivZero`, `NegThree*`, `ResidueNegativeAligned`, the `l = -c`
faces and the `ZLine` tower are all THEOREMS.

---

### Batch 207 — the `k`-variable nested Cauchy carrier is BUILT ((8.6.6)/(8.6.8) at general `k`), and the Fock residual's `j = 0` face is pinned by a formalized dictionary

**A — the Fock residual's `j = 0` face** (`Definitions/MMFLM86WeightedAssemblyAux.lean` 441 lines,
`MM.FLM86WAA` + `Definitions/MMFLM86WeightedAssembly.lean` 507 lines, `MM.FLM86WA`).

* `WeightedAssemblyAt E a b r` (the `r`-weighted antidiagonal assembly) and `weightedAssemblyAt_zero`: the `r = 0`
  slice IS the landed (8.4.20), so the residual's content is exactly `r >= 1`.
* **THE DICTIONARY, BOTH DIRECTIONS** (`hs` symmetric, `0 <= <a,b>`, `l >= <a,b>.toNat`):
  `createdStepAt_zero_iff_weightedAt : CreatedStepAt E a b 0 l <-> WeightedAssemblyAt E a b (l - <a,b>.toNat)`,
  the `r`-form `weightedAssemblyAt_iff_createdStepAt_zero`, the GAP theorem `createdStepAt_zero_of_lt`
  (`l < <a,b>.toNat` gives `0 = 0`), and the global positive-pairing equivalence.  Hence **(WAD) at `r >= 1`
  IS the `j = 0` face of the Fock residual** --- nothing else is missing there.
* Aux core (new): the previously-unlanded GAP case of the shifted weight sum
  (`sum_ratioCoeff_mul_binomCoeff_shift_of_lt`), the regrouping of the shifted convolution onto the antidiagonal
  (`convSumOS_shifted_eq_sum_ratioCoeff`), its collapses `convSumOS_shifted_eq_weighted` /
  `convSumOS_shifted_eq_zero`, and the `j = 0` left-hand side `zAligned_zeroMode_creationExpVL`
  (from batch 184-B's `ZAligned_createdState`).

**Brief errors of record (B207-A).**  (1) The dictionary needs `l >= <a,b>.toNat` and its global form lives on the
POSITIVE-pairing face --- the brief's implied global equivalence is FALSE for `<a,b> < 0` (and the
negative-pairing face needs `PairFiniteNegPairing` AND an expansion of its own).  (2) The collapse needed is the
created-state orientation `sum_ratioCoeff_mul_binomCoeff_shift` PLUS its new gap case, NOT the batch-203/204
moment machinery (opposite orientation).  (3) Most of the "material to #check" list is not on the path:
`ZAligned_createdState` at `j = 0` plus `weightedAssemblyDiagonalZero` suffice.  (4) The ladder/consumer route
(`MM.FLM86FOPA.iterateProductStep_iff_level_zero`) is DEAD for the consumer: it reduces level `j` to level 0 at
the TRANSLATED factor `(1/j!)D^j a`, a DESCENDANT of `iota(a)` and not a lattice generator, so the dictionary
(whose inputs are theorems only at generator fields) cannot consume it.  `j >= 1` stays open.

**B — the `k`-variable nested Cauchy carrier** (`Definitions/MMFLM86NestedCauchyAux.lean` 455 lines +
`...NestedCauchy.lean` 462 + `...NestedCauchyTwo.lean` 361 + `...NestedCauchyAxioms.lean` 314 (73 axiom guards),
namespace `MM.FLM86NCA`/`MM.FLM86NC`).

* the carrier `MultiVar F L k = (Fin k -> Z) -> End_F(V_L)`, `LocallyFiniteK`, `convWithK` with
  window-independence, proof-irrelevance and bilinearity; `famFold` and **`locallyFiniteK_famFold`** (the nested
  finiteness, by induction: tail window, then the outer factor's window on each surviving state);
* dictionaries: `convWithK_two` (`k = 2` IS `MM.FLM86TN.convWith2` up to `(Z x Z) ~ (Fin 2 -> Z)`) and
  `convWithK_one` (`k = 1` IS `MM.FLM85CP.convWith` under `m -> -m-1`);
* `blockScalarK` (the iterated (7.1.28) cocycle; `blockScalarK E 2 ![a,b] = E.eps a b`), `actProd`,
  `creationPairK`/`midBlockK`/`annihPairK`, `normOrderedK`, and **(8.6.6)/(8.6.8) at general `k`**:
  `normOrderedK_vacuum (E) (k) (as) (r) : (normOrderedK bL form E k as (fun i => (r i : Z))) vacuumVL
  = blockScalarK E k as . series865 bL form k as (sum i, as i) r` and `normOrderedK_vacuum_B` (verbatim
  (8.6.8));
* the specialisations: `normOrderedK_one = MM.FLM86E86.powerFam` (all states),
  `normOrderedK_two = MM.FLM86TN.normOrdered2`, `normOrderedK_two_vacuum_eq_series865` (recovers batch 206-B's
  `k = 2` theorem verbatim) and `normOrderedK_swap_two` ((8.4.21) at `k = 2`).

**Key finding**: the local-finiteness input is **DISCHARGEABLE from landed material** --- no
`NestedLocallyFinite` residual exists anywhere; `locallyFiniteK_famFold` proves the joint finiteness of any
ordered product of one-variable locally finite families (`exists_zPowHVL_eq_zero`,
`locallyFiniteCoef_annihFam`), so `midBlockK` and `annihPairK` are `k`-variable locally finite AS THEOREMS;
only `creationPairK` is not (the landed `MM.FLM86IS.creationPair2_not_locallyFinite2`), which is exactly
(8.4.20)'s nesting.  The brief's conditional fallback did not arise.

**Still open in the §8.6 route**: (8.6.6)'s SECOND equality, i.e. the pairing-factor family
`prod_{i<j}(w_i-w_j)^{-<abar_i,abar_j>}` (still the landed batch-119 `MM.FLM86KF.Display866Family3` plus the OPE
(8.4.22)/(8.4.36)); and no general-`k` symmetry/associativity was invented (FLM states (8.4.21) for two factors
only).

**Supersessions**: NOTES §103.2 / ROADMAP batch 206's "the exact remaining gap is the `k >= 3` nested Cauchy
carrier" is STALE (the carrier is built and (8.6.6)/(8.6.8) are proved at general `k`); batch 206-B's header
claim that the `k`-fold normal-ordered product "is not an object of this repository" is FALSE (it is
`MM.FLM86NC.normOrderedK`); NOTES §103.4's `(8.6.6)_{k>=3}` row is a THEOREM now.

**Accounting after batch 207**: the Fock residual is (WAD) at `r >= 1` (equivalently `CreatedStepAt(0,l)` at the
generators), plus the `j >= 1` face and the negative-pairing dictionary; the §8.6 route's next step is (8.6.6)'s
second equality; item (i) is `Corollary851ModeStep`; (iii) untouched.

---

### Batch 208 — 🎯 (WAD) IS A THEOREM at every weight (the Fock residual's `j = 0` face closed at `0 <= <a,b>`), and (8.6.6)'s second equality is stated at general `k` with its `k <= 2` slices proved

**A — the tower read backwards** (`Definitions/MMFLM86WeightedAssemblyAll.lean`, 339 lines, `MM.FLM86WAB`,
10 guards).

```
weightedAssemblyAt_of_one_le (hs) (E) (a b) {r} (hr : 1 <= r) : WeightedAssemblyAt E a b r   -- NO pairing hypothesis
weightedAssemblyAt_all (hs) (E) (a b) (r) : WeightedAssemblyAt E a b r
weightedAssemblyDiagonal_all (hs) (E) : WeightedAssemblyDiagonal E     -- the batch-206 residual, DISCHARGED
createdStepAt_zero_of_nonneg (hs) (E) (a b) (hN : 0 <= form a b) (l) : CreatedStepAt E a b 0 l
coherentCreatedStep_iff_of_pos_zero_face (hs) (E) :
    CoherentCreatedStep E <-> CreatedStepAtZeroNegPairing E /\ CoherentCreatedStepPos E
```

**Route**: the UNCONDITIONAL `zLine_all` (batch 202) read at the TRANSPOSED pair `(b,a)`, order `r`, scheme
parameter `n = r-K-1` --- the scheme's diagonal degree is then identically `K`, so the brief's "index
renormalisation `K <-> K'`" is VACUOUS; reindex `p |-> K-p`; `MM.FLM86NOC.normOrdered2_transpose` ((8.4.21))
moves the weight `C(p,r)` to the SECOND index as `C(K-p,r)` and contributes `c(b,a)^-1`;
`MM.FLM86CF2.creationExpVL_anchorVL` ((7.1.20)+(7.1.28)) expands the anchor giving `eps(b,a)`; and
**(7.1.26) `MM.FLMLattice.CentralExtension.eps_mul_comm`** cancels `eps(b,a) = c(b,a).eps(a,b)`.  NO even-cocycle
hypothesis, NO `PairFiniteNegPairing`, NO positivity.  **The decisive content is the weight ORIENTATION**
(`momentSum` weights the first index, (WAD) the second) --- which is exactly why the statement lands in the
`diagSum`/transpose shape rather than a `ZLineAt` shape.  `MM.FLM86NTR.ZAligned_anchorVL` is NOT on the path.

**B — (8.6.6)'s second equality, stated and sliced** (`Definitions/MMFLM86Display866TwoAux.lean` 432 lines +
`...Two.lean` 297 + `...TwoAxioms.lean` 128 (36 guards), `MM.FLM86D866A`/`MM.FLM86D866`).

* the two spellings of the second equality as Props at EVERY `k` (`Display866SecondEqK`, `Display866BoxK`;
  each carries `hs` and `forall p : PairIdx k, 0 <= pairC form k as p` as EXPLICIT binders) and their equivalence
  BOTH ways (`display866SecondEqK_iff_boxK`, via the new regrouping `pairSum_eq_pairFamK_sum`);
* the slices as THEOREMS: `display866SecondEqK_one`, `display866SecondEqK_two` (`k = 2` consumes the landed
  `MM.FLM86YP.YFam_mul_apply_eq_sum_ratioCoeff`, i.e. the landed (8.4.22)/(8.4.25));
* the composition with the landed first equality: `display866Display_of_second` gives (8.6.6) as FLM states it
  (third family = `series865` times the pairing factor), with `hpair` explicit;
* new auxiliaries: `ordFamK` (the honest ordinary-product coefficient family), the (8.6.7) combinatorics
  `pairSetK`/`pairWeightK`/`pairShiftK`/`pairFamK`/`pairBoxK` with their `k = 2` dictionaries, and
  `normOrderedK_vacuum_eq_zero_of_neg` (the middle family is a power series --- needed for the Nat-indexed
  display form).

**Residual: `Display866SecondEqK` at `k >= 3`** --- the general-`k` OPE/insertion step (one honest field against
a `k`-fold normal-ordered product), i.e. the `k >= 3` case of (8.4.22)/(8.4.36); only the one-pair case is landed
and `MMFLM86Contraction8612`'s `Contraction8612Ratio` is open even at `k = l = 1`.  NOT covered: the
negative-pairing face (every statement carries `0 <= <abar_i,abar_j>` --- load-bearing, since `pairSetK` truncates
for negative `c`), the Laurent spelling of the printed `(w_i-w_j)^{-<a,b>}` (only the rearranged positive-exponent
polynomial form over `Nat^k` is formalizable), and (8.6.9)/(8.6.10)/Jacobi.

**Corrections of record (B208-B).**  (1) The landed batch-119 `MM.FLM86KF.Display866Family3` has a DEGENERATE
scalar shape over families `Fin k -> Nat -> VL` and CANNOT carry the second equality --- the honest carrier is
batch 207's `MM.FLM86NC.normOrderedK`, so the target had to be stated fresh (NOTES §104.2's first half is stale).
(2) The `k = 2` OPE lives in `MM.FLM86YP` (`YFam_mul_apply_eq_sum_ratioCoeff`), not in
`MMFLM86OPEThird`/`ExpProduct`/`Harvest`.  (3) `MM.FLM86E86` is a namespace, not a file.  (4)
`0 <= <abar_i,abar_j>` is load-bearing (a first version contained the FALSE `pairSetK.Nonempty <-> 0 <= c`).
(5) Composing with `normOrderedK_vacuum` needs `normOrderedK_vacuum_eq_zero_of_neg`.
**Discipline note**: the `sorryAx` trap FIRED LIVE once in this batch (a stale proof body silently became
`sorryAx`, caught only by the `#guard_msgs` diff) --- the campaign's guard+probe discipline is load-bearing.

**Accounting after batch 208**: the Fock residual is exactly
`CreatedStepAtZeroNegPairing E /\ CoherentCreatedStepPos E` (both open, both named, equivalence both ways); route
(iii)'s next step is `Display866SecondEqK` at `k >= 3`; item (i) is `Corollary851ModeStep`.

---

### Batch 209 — ⛔ THE FOCK RESIDUAL IS **REFUTED** AS SPELLED (two independent machine-checked routes): the honest slice is `0 <= <a,b>`

**A — the `j >= 1` face, falsified** (`Definitions/MMFLM86CoherentCreatedPosAux.lean` 161 lines +
`...PosRankOne.lean` 361 + `...Pos.lean` 283).

* `zAligned_createdStep_left_zero` --- the residual's left-hand side at `l = 0` evaluated UNCONDITIONALLY by the
  landed `MM.FLM86MI.ZAligned_mode_iotaVL`:
  `ZAligned E (YFam E a (-(j:Z)-1) (creationExpVL b 0 (iotaVL b)))
  = E.eps a b . (if 0 <= (j:Z) - form a b then ZAligned E (creationExpVL a ((j:Z)-form a b) (iotaVL (a+b))) else 0)`,
  nonzero for `a != 0` and `form a b <= (j:Z)` (`zAligned_createdStep_left_zero_ne_zero`).
* `pairFinite_of_createdStepAt` --- the face IMPLIES the pair datum (the right-hand side is 0 where the datum
  fails: `MM.FLM86TFD.convOneSided_eq_zero` + `MM.FLM86FCA.pairFinite_shiftedField_iff`).
* the rank-one counterexample: `YFam_rankOne_summand` (`YFam formZ extZ 1 (k:Z) (vk k) = extZ.eps 1 (-1) . vacR`
  for EVERY `k >= 0`, engine: `MM.FLM86CT.YFam_eq_sum_three_factor_of_mem` + `annihFam_apply_creationFam` +
  `annihFam_iotaVL`/`midFam_iotaVL`), hence `not_pairFinite_rankOne`, `not_createdStepAt_rankOne`,
  `not_coherentCreatedStepPos_rankOne`.
* the corrected slices: `CoherentCreatedStepPosNonneg` (`0 <= <a,b>`), `CoherentCreatedStepPosNegPairing`
  (named FALSE face), `coherentCreatedStepPos_iff_nonneg_and_neg`, `CoherentCreatedStepNonneg`,
  `coherentCreatedStepNonneg_iff_posNonneg` (at `0 <= <a,b>` the residual is exactly its `j >= 1` face, the
  `j = 0` face being batch 208-A), and `coherentCreatedStep_iff_slices`.
* the GENERAL refutations `not_createdStepAt_of_neg_pairing`, `not_coherentCreatedStepPos_of_neg_pairing`
  (consuming the sibling's carrier failure).
* `CoherentCreatedStepPosPairingFactor` --- the NAMED honest negative-pairing replacement (the threshold form of
  (8.4.22)/(8.4.25), `MM.FLM86RX.threshold_holds`, valid at every pairing per `(p,q)` and per state; matches the
  landed `MM.FLM86MXP.Display8625One` at `k = l = 1`), STATED not proved, with the coupling of the outer window
  to the single threshold `T` flagged as unverified.

**B — the `j = 0` negative-pairing face, falsified** (`Definitions/MMFLM86CreatedStepNegAux.lean` 277 lines +
`...CreatedStepNeg.lean` 315).

* **THE KEY IDENTITY** `yFam_creationExpVL_iotaVL`:
  `YFam E a (-<a,b> + q - 1) (creationExpVL b q (iotaVL b)) = ratioCoeff <a,b> q . (eps(a,b) . iotaVL (a+b))`
  --- at `<a,b> = -N < 0` this is `C(N+q-1,q).eps.iota(a+b) != 0` for EVERY `q` (at `q = 0`, `N = 2` it
  reproduces the landed `twoVarField_rankOne_negTwo_zero_vacuum`, a self-check);
* `not_pairFinite_generators_of_neg`, `not_pairFiniteNegPairing_of_exists_neg`,
  `not_pairFiniteNegPairing_rankOne`, `createdStepAtZeroNegPairing_imp_pairFiniteNegPairing`,
  `convOneSided_generators_neg_eq_zero`,
  `createdStepAt_zero_neg_iff_field_eq_zero` (the face is exactly "an honest field of a NONZERO state vanishes"),
  `not_createdStepAtZeroNegPairing_of_exists_neg`, **`not_coherentCreatedStep_of_exists_neg`**, and
  `not_shiftedPairFiniteGenerators_of_exists_neg` (batch 197-B's carrier is false too).

**Mechanism.**  The formalised right-hand side is the PLAIN one-sided product; FLM's rigorous (8.6.25)/(8.6.27)
carries the PAIRING FACTOR `prod (z_0 + w_i - x_j)^{<a_i,b_j>}`, which vanishes on the diagonal IFF
`0 <= <a,b>`.  For `<a,b> < 0` the contraction `(z-w)^{-N}` has infinitely many diagonal terms,
`convOneSided` falls into its 0 branch, and the face asserts that a nonzero field is 0.  This is exactly what FLM
flags in Remark 8.5.2 (p. 200) and Remark 8.6.4 (p. 211).

**Refuted record.**  Batch 200-B's numerical note ("the summand is a diagonal extraction and cancels") is WRONG
--- it cancels in FOCK DEGREE, not in `j`.  NOTES §97.1's `PairFiniteNegPairing` row, and batch 208/§105.3's
accounting ("Fock residual = `CreatedStepAtZeroNegPairing` /\ `CoherentCreatedStepPos`, both open") must read:
the first conjunct is FALSE, the second moot; §105.1/§105.3's "open everywhere including `(0,0)`" is stale.

**Consequence for the campaign.**  `CoherentCreatedStep` --- the third conjunct of
`MM.FLM810TE.EvenCocycleResiduals` --- is FALSE at FLM's own Leech datum (`a = x`, `b = -x`), so that residual
is FALSE as stated, NOT open, and `theorem8102_of_residuals` is **VACUOUS**.  The boundary reduction must be
re-stated with the honest residual.

**Untouched** (they use `normOrdered2`, not `convOneSided`): the `l = -c` faces, `ResidueNegativeAligned`,
`NegTwo*`, `NegThree*`, and the whole `ZLine` tower.

**Honest overlap**: A's rank-one file is a self-contained special case (kept); its two general corollaries consume
B's theorem rather than reproving it.  Two independent routes to a critical refutation is the right outcome.

**Still open**: `CoherentCreatedStepPosNonneg E` (the honest residual at `0 <= <a,b>`; the tower closes at most
its `l = 0` slice --- the `j >= 1` convolution has the double-binomial weight
`sum_i ratioCoeff <a,b> i . C(p+N-i,j) . C(K-p+i,l)`, which collapses to the tower's `C(p, j-<a,b>)` only at
`l = 0`), and the named honest negative-pairing replacement `CoherentCreatedStepPosPairingFactor`.

---

### Batch 210 — the audit, the honest re-statement of the boundary, and the positive-pairing Fock residual reduced to ONE named statement

**A — the honest Fock residual, reduced** (`Definitions/MMFLM86CoherentCreatedNonnegAux.lean` 430 lines +
`...Nonneg.lean` 309, `MM.FLM86CCNA`/`MM.FLM86CCN`).

* **`createdStepAt_shiftZero`** --- for a symmetric form and `0 <= <a,b>`, `CreatedStepAt E a b j 0` for EVERY
  `j : Nat` (spending ONLY `hs`; no `hnd`, no evenness).  This is FLM (8.5.14) --- the display that Remark 8.5.2
  (printed p. 200) calls "not rigorous in the present setting" --- read at the `z_0^j`-coefficient, i.e. the
  `w_2 = 0` case of (8.6.10) at `k = 2`.  Route: batch-182 carrier -> finite `convSumOS` -> per-summand shifted
  (8.4.25) -> antidiagonal regrouping with the double-binomial weight
  `sum_i ratioCoeff N i . C(K-q+N-i, j)`, `K = j-1-m-N` -> the REFLECTED (8.5.37) finite difference collapses it
  to `C(P, j-N)` (and to 0 in the gap `j < N`) -> the landed tower `ZLine E (j-N)` (order 0 = the (8.4.20)
  assembly), with the left side by `YFam_iotaVL_eq_exp` (8.5.11).  So the tower closes the `l = 0` face at EVERY
  `j` --- ROADMAP batch 209's route note is confirmed and sharpened.
* **THE ONE NAMED RESIDUAL**: `MM.FLM86CCN.CoherentCreatedStepShifted E := forall a b, 0 <= form a b -> forall j,
  1 <= j -> forall l, 1 <= l -> CreatedStepAt E a b j l` (the genuine two-shift (8.6.10) at `k = 2`, `w_2 != 0`),
  with the reductions BOTH ways: `coherentCreatedStepPosNonneg_iff_shifted`,
  `coherentCreatedStepPosNonneg_iff_shiftZero_and_shifted` (the `l = 0` slice is DERIVED, nothing is lost), and
  `coherentCreatedStep_iff_shifted_slices` (the whole `CoherentCreatedStep` = this face /\ the two
  negative-pairing faces that batch 209 refuted).
* Why `l >= 1` differs (paper computation, NOT formalized): the general inner weight is
  `sum_i ratioCoeff N i . C(q+i,l) . C(P+N-i,j)` --- the extra `C(q+i,l)` kills the single finite difference;
  hand-checked at `(N,j,l) = (2,1,1)` where the inner sum is the nonzero constant `-2`.

**B — the audit and the honest boundary** (`Definitions/MMFLM810HonestLedger.lean` 256 lines +
`...HonestAudit.lean` 433 + `...BoundaryHonest.lean` 369, `MM.FLM810HL`/`MM.FLM810HA`/`MM.FLM810BH`; 55 guards).

* **The old package is FALSE, not open**: `not_evenCocycleResiduals_of_gram`,
  `not_evenCocycleResiduals_of_posDef` (false already at `PositiveDefinite` + one nonzero vector --- no Leech
  appeal needed), `theorem8102_of_residuals_vacuous`, `not_coherentCreatedStep_of_gram`, plus the consumer
  killers `not_fockCreatedStep_of_gram`, `not_iterateProductStep_of_gram`, `not_display8610TwoOneSided_of_gram`,
  `not_shiftedPairFiniteGenerators`, `not_pairFiniteNegPairing`, `not_pairFiniteAll_of_gram`,
  `not_fockPairDatum_of_gram`, `not_createdStepAtZeroNegPairing`, and per-consumer vacuity theorems for
  `coherentCreatedStep_of_fockCreatedStep`, `fockCreatedStep_imp_coherentCreatedStep`,
  `coherentCreatedStep_of_twoOneSided_noHyp`/`_of_twoOneSided`, `createdStep_of_iterateProductStep`,
  `corollary851_allStates_of_fockCreatedStep`, `evenCocycleResiduals_of_fockCreatedStep`,
  `residueNegativeAligned_evenCocycle_of_residuals`.
  NOTE: `residueNegativeAligned_evenCocycle_iff_residuals` is an EQUIVALENCE whose right side is false --- VALID,
  not vacuous; the Fock-free reading of record is `residueNegative_iff_negTwo_and_negThree`.
* **The honest package**: `FockResidualHonest E := CoherentCreatedStepPosNonneg E /\
  CoherentCreatedStepNegPairingFactor E /\ CreatedStepAtZeroNegPairingHonest E` ->
  `EvenCocycleResidualsHonest` -> `EvenCocycleJacobiStepHonest` ->
  **`theorem8102_of_residuals_honest`**.
* **Non-vacuity PROVED**: `evenCocycleResidualsHonest_iff : (forall a b, 0 <= form a b) ->
  (EvenCocycleResidualsHonest C h2 <-> EvenCocycleResiduals C h2)` --- on the whole slice where batch 209 does not
  apply, the honest hypothesis says exactly what the old one said, and its two negative-pairing conjuncts are
  theorems there (`negPairingFactor_of_forall_nonneg`, `zeroNegPairingHonest_of_forall_nonneg`,
  `fockResidualHonest_iff_of_forall_nonneg`, `posNonneg_of_coherentCreatedStep`).
* **The ONE new named object**: `CreatedStepAtZeroNegPairingHonest` --- the missing normal-ordered ONE-VARIABLE
  product at the shifted coherent state, stated WITH the (8.6.25) threshold spelling (without that clause the def
  is provable by `prod := LHS`, i.e. a silent drop --- the agent checked this).  The tree has no such object:
  `MMFLM86YP` section 6(ii) records that `convWith (YFam E a) (YFam E b)` does not exist and `YFam` is not a
  `CField` for `a != 0`.

**Status changes of record.**  NOTES §105.1/§106.5's "the `j >= 1` face is open everywhere including `(0,0)` with
NO producer in the tree" is STALE --- `l = 0` producers now exist for every `j`; the row for
`CoherentCreatedStepPosNonneg` should read "open in its `w_2 != 0` face only, equivalent to
`CoherentCreatedStepShifted` (both ways)".  The stale claim in `MMFLM86FockShiftZero.lean`'s header that
`CoherentCreatedStep` "remains open" was corrected in place by the dispatcher (batch 209 refuted it).
**Brief errors (B210-A).**  (1) `Finset.not_mem_empty` does not exist in this pin (it is `Finset.notMem_empty`);
(2) `Int.toNat_of_nonneg` must not be used as a right-to-left rewrite on goals containing `(form a b).toNat`;
(3) `ZLine`'s value lives in the `FLM810JZF.ZAligned` copy while `CreatedStepAt` uses `FLM810JF2.ZAligned`, and
`MM.FLM86MI.ZAligned_JZF_eq` is not in the import cone of NEW files (re-proved privately);
(4) the `L(-1)` ladder is unusable (its universal level-0 premise is refuted);
(5) the empty window is NOT a legal regrouping window in the gap `j < N`;
(6) the `#guard_msgs` wrap is empirically name-dependent.

**Accounting after batch 210**: the honest boundary reduction is landed and non-vacuous on the positive-pairing
slice; the Fock residual's positive face is `CoherentCreatedStepShifted` (= the two-shift `l >= 1` case), its
negative-pairing face needs the to-be-built object; `(i)` = `Corollary851ModeStep`; `(iii)` =
`Display866SecondEqK` at `k >= 3`.

---

### Batch 211 — 🎯 the two-shift Fock face is PROVED from ONE derivation identity, and the negative-pairing object is BUILT (with two defects of record in batch 210-B's spelling)

**A — `CoherentCreatedStepShifted` proved** (`Definitions/MMFLM86ShiftedDerivAux.lean` 208 lines, `MM.FLM86SDA`;
`...ShiftedDeriv.lean` 215, `...ShiftedBootstrap.lean` 309, `MM.FLM86SD`).

```
coherentCreatedStepShifted_of_derivation (hnd) (hs) (E) (hder) : CoherentCreatedStepShifted bL form E
coherentCreatedStepPosNonneg_of_derivation (hnd) (hs) (E) (hder) : CoherentCreatedStepPosNonneg bL form E
```

(the second through the landed `MM.FLM86CCN.coherentCreatedStepPosNonneg_iff_shifted`).  The ONE named residual is

```
MM.FLM86SD.ShiftedCreationDerivation (hnd) E : Prop :=
  L(-1)(alpha_a(-n) w) = alpha_a(-n)(L(-1) w) + n . alpha_a(-n-1) w      -- all n >= 1, ALL states w
```

--- the (8.4.14)/(7.1.41) derivation identity, i.e. the tree's open `MM.FLM810TG.LMinusOneCreationMode` /
`MM.FLM810DFA.SugawaraDerivation` (only its GENERATOR case `w = iota(c)` is landed:
`MM.FLM810TG.lMinusOneCreationMode_iota`).  **Route**: `E^-_b(l) iota(b) = (1/l!).L(-1)^l iota(b)`, so the
two-shift face is the shift-ZERO tower (batch 210-A's `createdStepAt_shiftZero`) plus the `L(-1)` LADDER:
differentiate the level-`l` statement, use the NEW Leibniz rule `MM.FLM86SDA.coefDeriv_convOneSided` and
`coefDeriv (shiftedField a j) = (j+1).shiftedField a (j+1)` (8.6.9), and the derivation property at mode
`-(j+1)` cancels the `(j+1).(Y a)_{-j-2} B_l` term against the level-`j+1` shift-zero statement.
**The double-binomial weight is never formed** --- batch 210-A's obstruction is bypassed entirely.
**Caveats of record**: `hnd` IS spent (contrary to the brief's expectation --- `L(-1)` goes through the
conformal vector, so (8.10.13) and the `k = 1` display carry it); the identification of the residual with
`LMinusOneCreationMode` is a READING across two currencies, NOT a formal bridge; the next step is the
creation-list induction from `lMinusOneCreationMode_iota` (template:
`MMFLM810DFA.translationResidual_of_baseResidual`).
**Falsification check** (run first, no counterexample): rank-one, `a = b = 1`, `m = -1`, vacuum; it confirmed
batch 210-A's obstruction (weight `2q-2` at `(2,1,1)`) and produced the two-term collapse identity
`sum_i ratioCoeff N i C(q+i,l)C(P+N-i,j) = sum_a (-1)^(N-a)C(N,a)C(q,l-N+a)C(P,j-a)` (from
`[t^l s^j](1+t)^q(1+s)^P(s-t)^N`), specialising to both landed collapses --- not needed by the working route.

**B — the negative-pairing object, built** (`Definitions/MMFLM86NormOrderedShiftedAux.lean` 391 lines,
`MM.FLM86NOSA` + `...Shifted.lean` 294, `MM.FLM86NOS`).

* **THE OBJECT**: `pairBlock E a b j l m T` --- the (8.6.25)/(8.6.27) block sum at the shifted coherent state with
  the pairing factor carried as `ratioCoeff` on the window `q in Icc (-T-1,-1)`, `k in range (T+1)`.  Laws:
  unfolding; bilinearity in the state; the `(z_1-z_2)^{<a,b>}` factor explicit
  (`contrFam_eq_ratioCoeff_smul_one`, `ratioCoeff_eq_pow_mul_binom`); the null pair collapsing to a SINGLE
  normal-ordered block (`pairBlock_of_form_eq_zero`); the vacuum and generator specialisations; the inner `k`-sum
  identified with the landed (8.4.25) at `0 <= <a,b>` (`twoVarField_eq_ratioSum_of_nonneg`); and
  `convOneSided_shifted_eq_ratioSum` --- the carrier's own summands ARE the two shift binomials against
  `twoVarField`, which is where the pairing factor lives and **where the second slot index `l` comes from**.
* **The two faces reduced to one named input EACH, BOTH ways**: `HonestNegPairingZero E` (<-> the corrected
  `CreatedStepAtZeroNegPairingHonestCorrected`) and `HonestNegPairingPos E`
  (<-> `MM.FLM810BH.CoherentCreatedStepNegPairingFactor`); plus `FockResidualHonestCorrected` and
  `fockResidualHonestCorrected_iff_of_forall_nonneg` (the corrected package IS the batch-210 package on the
  `0 <= <a,b>` slice).
* **TWO DEFECTS OF RECORD in batch 210-B's spelling** (found by the mandatory falsification check):
  (a) **machine-checked**: `CreatedStepAtZeroNegPairingHonest`'s clause (b) writes the SECOND shift binomial with
  index `0`, while the landed carrier (`shiftedField_apply`) carries index `l` --- at `l = 1, q = -2` those are
  `C(2,1) = 2` vs `C(2,0) = 1` (`MM.FLM86NOSA.pairBlock_slot_discrepancy`), so the def is NOT the `j = 0` slice
  of slice 2, contrary to its own docstring;
  (b) **exact hand computation**: the SINGLE threshold `T` is too small --- at `w = iota(1)`, `m = -1`, `l = 1`,
  `<a,b> = -M < 0` the `b`-degree-1 blocks force `T = 0`, at which the `b`-degree-0 block is missing; the honest
  (8.6.25) spelling needs a threshold PER SHIFT INDEX (outer AND inner).  This upgrades NOTES §106.3's own
  "coupling ... flagged unverified" to "**wrong as spelled**".  The dispatcher recorded the defect in the def's
  docstring (pointing at the corrected package); the defs themselves were NOT weakened.
* **Still missing to close the two faces**: the **h-weight (Fock bidegree) grading of M(1)**, needed to separate
  `E^-_a(P)E^-_b(Q) iota(a+b)` blocks with different `Q` --- not landed.

**Status updates of record**: NOTES §107.1/§107.4's `CoherentCreatedStepShifted` row is now "PROVED from
`ShiftedCreationDerivation` (plus `hnd`, `hs`)"; ROADMAP batch-210 brief error (4) is sharpened --- the
`L(-1)` ladder is unusable only as a UNIVERSAL premise (`FockShiftZeroStep` is refuted), not pointwise.
**Accounting after batch 211**: the honest positive-pairing Fock residual is `ShiftedCreationDerivation`
(a known open identity whose generator case is landed); the negative-pairing content is reduced to
`HonestNegPairingZero`/`HonestNegPairingPos` on the built `pairBlock`, blocked on the M(1) bidegree grading;
(i) and (iii) untouched.

---

### Batch 212 — the Fock bidegree (and its h-weight refinement) is LANDED, and the Fock residual's residual is one `YFam`-mode derivation

**A — the currency bridge, and a machine-checked correction of batch 211's reading**
(`Definitions/MMFLM86ShiftedDerivationAux.lean` 189 lines, `MM.FLM86SDX` +
`Definitions/MMFLM86ShiftedDerivation.lean` 157, `MM.FLM86SDD`).

* `lMinusOne_eq_sugawaraModeZero` --- the formal currency bridge
  `lMinusOne hnd E = sugawaraModeZero hnd (-1)` (via `Yaligned_eq_Yzero_baseRepaired`), so batch 211-A's
  "reading across currencies" is now a THEOREM and `hnd` is legitimately spent (`L(-1)` is the conformal
  vector's mode).
* `ZAligned_iotaVL_eq_YFam` (`ZAligned E (iotaVL a) = YFam E a`),
  `sugawaraModeZero_neg_one_iotaVL`, `sugawaraModeZero_neg_one_anchorVL`.
* **`shiftedCreationDerivation_iff_YFamShiftedDerivation`** --- the residual is, BOTH WAYS, the tree spelling
  with the `YFam` modes; and `ShiftedCreationDerivation_of_modeDerivation`,
  `coherentCreatedStepShifted_of_modeDerivation`, `coherentCreatedStepPosNonneg_of_modeDerivation`: the two
  positive-pairing Fock faces follow from ONE named residual.
* **MACHINE-CHECKED CORRECTION (batch 211-A's reading was wrong)**: the residual's modes are the (8.4.17) field of
  the LATTICE generator (`YFam`), NOT the Heisenberg modes `modeVL` of the tree's
  `LMinusOneCreationMode`/`SugawaraDerivation`: at `a = 0`, `m = -1` they already differ on the vacuum
  (`YFam_zero_neg_one_vacuum` vs `ZAligned_iotaVL_zero_ne_modeVL_zero`).  Hence the residual is NOT the tree's
  state-half object --- which is a theorem unconditionally `MM.FLM87SR.sugawaraDerivation` (batch 165), so the
  stale "general-`w` case not landed" reading is corrected too.
* NOT proved: `YFamModeDerivation` (the tree spelling of the residual) and the generator base case
  `YFamDerivationBase`.  The creation-list STEP is hand-verified pure operator algebra on landed inputs
  (`modeVL_YFam_add` at `-k`/`-k-1`, `sugawaraDerivation`, `coefDeriv_apply`) and recorded verbatim in the
  docstring, but its Lean proof was WITHDRAWN (it failed at the final scalar-normalisation step; the
  `Module.End` smul API trap) --- with it plus `MM.FLM85Un.spanningStatement` the base case would give the
  target unconditionally modulo `hnd`/`hs`.
* **NOTE**: this agent did not add `#guard_msgs` guards (build budget); the dispatcher supplied an out-of-file
  `#print axioms` probe over all 16 public declarations --- all clean, no `sorryAx` (recorded here so the
  deviation is visible).

**B — the Fock bidegree and its h-weight refinement** (`Definitions/MMFLM86FockBidegreeAux.lean` 334 lines,
`MM.FLM86FBDA` + `...FockBidegree.lean` 356, `MM.FLM86FBD`; 29 public declarations, all guarded).

* **THE GRADING**: `bidegPiece N a` (the (Fock degree, lattice label) piece of `V_L = M(1) x F{L}`, the span of
  `vlMonomialBasis (d,a)` with `basisWt d = N`), with `bidegPiece_basis_mem`/`_mono_mem`/`_disjoint`/
  `_iSup_eq_top`, the bridge `bidegPiece_le_weightPiece` to the landed (coarser) `MM.FLMLattice.weightPiece`
  (NOT rebuilt), the projections `bidegProj` with their laws, and **THE SEPARATION LEMMA**
  `bideg_separation` (a finite sum of vectors of distinct bidegrees vanishes only if every summand does).
* the h-weight refinement: `hwWt d = sum_p d(p).p.1.h_p in h = H F L`, `hwWt_add`, `hwWt_single`,
  `hwPiece w a`, `hwPiece_basis_mem`, `hwPiece_disjoint`.
* MODE LAWS: `modeVL_neg_mem_bidegPiece` (`alpha_c(-n)`, `n >= 1`, raises the Fock degree by `n` and keeps the
  label; arbitrary `c` via the LANDED `MM.FLMFI.iota_single_mem_monoPiece`),
  `creationExpVL_mem_bidegPiece(_nat)` (homogeneity), and the block law
  `normOrdered2_iotaVL_mem_bidegPiece` (the (8.4.20) block at `iota(c)` is
  `E^-_a(M-<a,c>) E^-_b(K-<b,c>) iota(a+b+c)`, of degree `(M-<a,c>)+(K-<b,c>)` and label `a+b+c`).
* **FALSIFICATION (killed the brief's own reading)**: `pairBlock_block_deg` --- the two mode indices of EVERY
  `(q,k)`-summand of `MM.FLM86NOSA.pairBlock` add up to the constant `j + l - m - <a,b> - 1`, so at a lattice
  generator all summands have the SAME `(N,a)` bidegree and the SAME label: the coarse bidegree (and
  `weightPiece`) CANNOT separate the `E^-_a(P)E^-_b(Q) iota(a+b)` blocks with different `Q`.  The separating
  grading is the **h-weight refinement** `hwWt` (the `(q,k)`-summand carries `(M+1-Q)a + (Q-2)b`, different for
  different `Q` as soon as `a, b` are linearly independent).  This confirms batch 211-B's "threshold PER SHIFT
  INDEX" defect and gives its CAUSE as an identity.
* NAMED RESIDUAL: `FockAnnihilationDegreeLaw` (the exact-degree annihilation shift; the landed
  `MM.FLM84CM.modeVL_lowers` gives only the filtration bound `<= N - n`).  Next steps: the h-weight mode law and
  the h-weight block separation, then the two faces.
* GREP-FIRST record: `weightPiece`/`weightPiece_disjoint`/`isInternal`, `gradedPiece`, `monoPiece`,
  `basisWt`, `gradedPiece_mul_mem`, `span_inf_span_eq_bot_of_disjoint`, `fockDegreeOp`, `wtLabel` and
  `MM.FLMFI.iota_single_mem_monoPiece` all already existed and were CONSUMED, not rebuilt (one 25-line
  re-derivation was caught by the grep).  `fockLeVL` is a filtration, not a grading.

**Correction of record**: NOTES §108.4 / ROADMAP batch 211's "blocked on the M(1) bidegree grading" is superseded ---
the grading is landed, and the block bookkeeping needs its **h-weight refinement**, with the two faces still open
as `HonestNegPairingZero`/`HonestNegPairingPos`.

---

### Batch 213 — 🏆 THE POSITIVE-PAIRING FOCK FACES ARE PROVED (no named residual left), and the h-weight separation machinery is landed

**A — the residual closed** (`Definitions/MMFLM86YFamDerivationAux.lean` 319 lines + `...Base.lean` 266 +
`...YFamDerivation.lean` 403, namespace `MM.FLM86YFD`, 29 guards --- all public declarations guarded, no
`sorryAx`).

```
yFamResidualAt_modeVL (hnd) (hs) (E) {a b} {k} (hk : 1 <= k) {w} (h : YFamResidualAt a w) :
    YFamResidualAt a (modeVL b (-k) w)                       -- THE CREATION-LIST STEP
yFamResidualAt_stateOfVL (hnd) (hs) (E) (hbase) (a) (D) : YFamResidualAt a (stateOfVL D)
yFamModeDerivation_of_base (hnd) (hs) (E) (hbase : YFamDerivationBase hnd E) : YFamModeDerivation hnd E
yFamModeDerivation_iff_base (hnd) (hs) (E) : YFamModeDerivation hnd E <-> YFamDerivationBase hnd E   -- BOTH WAYS
yFamDerivationBase (hnd) (hs) (E) : YFamDerivationBase hnd E                -- THE BASE CASE
yFamModeDerivation / shiftedCreationDerivation (hnd) (hs) (E) : ...
coherentCreatedStepShifted (hnd) (hs) (E) : MM.FLM86CCN.CoherentCreatedStepShifted E
coherentCreatedStepPosNonneg (hnd) (hs) (E) : MM.FLM86CCPP.CoherentCreatedStepPosNonneg E
```

**So the positive-pairing Fock conjunct of the honest boundary package is a THEOREM modulo `hnd`/`hs` --- NO
named residual remains on that front.**  The base case needed the CREATION HALF of (8.10.13)/(8.10.14):
`[L(-1), E^-(a,z)] = (d/dz)E^-(a,z) - a(-1)E^-(a,z)`, proved ONCE abstractly over an arbitrary `Ring A` with
`Algebra Q A` (`sub_mul_of_deriv_expField`, file `...Base.lean`) --- a route that never opens the
`Module.End` scalar API, which is why batch 212-A's trap cannot arise.
**Traps of record.**  (i) The mode `-(k+1)` INDEX TRAP: the contraction at mode `-(k+1)` shifts the `YFam` index
by `k+1` (index `m-k-1`, NOT `m-k`); writing `m-k` leaves `-(m.<b,a>)` behind, because `module` sees
`YFam a (m-1-k)` and `YFam a (m-k-1)` as different atoms.  (ii) `MM.FLM86ZLI.modeVL_YFam_add` yields `-↑n` and
`j - ↑n`, only DEFINITIONALLY equal to `-1` / `m-1`, so `rw`'s keyed matching fails across the difference
(`simpa using h` fails where `exact h` succeeds) --- normalise the hypothesis first.  (iii) The brief's
`*_of_modeDerivation` signature has no `hs`; the "scalar-normalisation blocker" recorded by batch 212-A was an
artefact of the operator-level formulation (the step is pure VECTOR-level algebra); and the BASE case needs the
creation-exponential derivation property, which was not in the brief's input list.

**B — the h-weight separation** (`Definitions/MMFLM86HWeightSeparationAux.lean` 415 lines, `MM.FLM86HWA` +
`...HWeightSeparation.lean` 769, `MM.FLM86HW`; 52 guards).

* the generic basis-indexed projection machinery (`basisProj`, `basis_separation`, the telescoping law), its
  instantiation `hwProjM1`/`hwProj`/`hw_separation`/`hwProj_sum_eq_self_of_mem`, and the KEY
  `hwProj_eq_iff` (equality iff all h-weight components agree);
* the HONEST h-weight mode law: at basis labels `alpha_{b_i}(-n) : hwPiece w a -> hwPiece (w + n.h_{b_i}) a`
  (`modeVL_basis_neg_mem_hwPiece`, `creationExpVL_basis_mem_hwPiece`); for a GENERAL `c` the mode is NOT
  homogeneous --- only `iSup` membership holds (`modeVL_neg_mem_hwPiece_iSup`,
  `creation_ofLattice_mem_hwPieceM1_iSup`, via `ofLattice_eq_sum_hBasis` + `creation_eq_sum_ofLattice`) --- with
  the MACHINE-CHECKED REFUTATION `naiveHwModeLawRankTwo_false` (rank two, `c = b_0 + b_1`,
  `alpha_c(-1).1 = X_0 + X_1` lies in NO single `hwPiece`);
* the block laws at basis labels (`normOrdered2_basis_iotaVL_mem_hwPiece`) and the INDEX SEPARATION
  `pairBlock_block_weight_eq_iff` (for `i != j` the `(q,k)`-summand's h-weight determines `q+k` and conversely)
  + `pairBlock_blocks_distinct_hwPiece`;
* the two negative-pairing faces reduced, BOTH WAYS, to ONE named per-h-weight input each:
  `HonestNegPairingZeroHw` / `HonestNegPairingPosHw` (`honestNegPairingZero_iff_hw`,
  `honestNegPairingPos_iff_hw`; forward = project, backward = the telescoping law `hwProj_eq_iff`).
* **Corrections of record**: batch 212-B / NOTES §109.2's "the summand carries `(M+1-Q)a + (Q-2)b`, different for
  different `Q` as soon as `a, b` are linearly independent" is CORRECTED (the summand spreads over basis
  monomial weights and lies in no single `hwPiece` for general `a, b`; `pairBlock_block_deg` stands) --- the
  separation is honestly a BASIS-LABEL statement.  `hwWt` is a map on the EXPONENT VECTOR, not on `VL`;
  `hwPiece` takes the pair `(w, a)`.
* Still open there: the ASSEMBLED per-shift-index threshold, blocked on Mathlib's missing FIBERWISE-SUM lemma for
  `(q,k) |-> q+k` (only the `prod` version exists in this pin).  `FockAnnihilationDegreeLaw` was NOT needed.

**Milestone.**  At FLM's datum the honest package `EvenCocycleResidualsHonest` now has
`NegTwoFirstDerivZero` (batch 196), `NegThreeExpandedResidualAligned` (batch 204-A) and
`CoherentCreatedStepPosNonneg` (this batch) all as THEOREMS, so it reduces to the TWO NEGATIVE-PAIRING Fock
conjuncts alone (`CoherentCreatedStepNegPairingFactor`, `CreatedStepAtZeroNegPairingHonest`) --- which in turn are
reduced to `HonestNegPairingPos`/`HonestNegPairingZero` (batch 211-B) and then to the h-weight forms (batch
213-B).

---

### Batch 214 — the honest boundary REDUCED to the two negative-pairing Fock faces + the Jacobi step; the negative-pairing faces pinned to per-`Q` thresholds (one named input each)

**A — the assembly and the faces** (`Definitions/MMFLM86FiberwiseSum.lean` 295 lines, `MM.FLM86FWS` +
`...NegPairingFacesAux.lean` 317, `MM.FLM86NPFA` + `...NegPairingFaces.lean` 389, `MM.FLM86NPF`; 39 guards).

* **FALSE PREMISE CORRECTED**: Mathlib's fiberwise-sum lemmas ALL EXIST in this pin and are USED ---
  `Finset.sum_fiberwise`, `sum_fiberwise_of_maps_to`, `..._of_maps_to'`, `sum_fiberwise_eq_sum_filter`,
  `sum_biUnion`, `sum_sigma'`.  What was missing is the EXPLICIT fiber/image of the `pairBlock` window under
  `(q,k) |-> A-q-k`: `image_affine` (image `Icc (A+1-T) (A+T+1)`) and `fiber_affine` (fiber
  `Icc (max 0 (A-Q+1)) (min T (A-Q+T+1))` in `k`, `q = A-Q-k`), with the main lemma
  `sum_window_shiftIndex` proved BY `sum_fiberwise_of_maps_to`, plus the `(q,k) |-> q+k` form
  `sum_window_add`.  ROADMAP batch-213-B / NOTES §110.2's "blocked on Mathlib's missing fiberwise-sum lemma" is
  SUPERSEDED.
* **THE THRESHOLD ASSEMBLY IS LANDED**: `pairBlockShift` (one shift index `Q` with its OWN threshold `TQ`,
  block `normOrdered2 a b (C-Q) Q`) and `pairBlockIdx` (sum over `Q`), and
  `pairBlock_eq_pairBlockIdxMax : pairBlock ... T = pairBlockIdx ... (l-T) (l+T) (fun Q => T - max 0 (Q-l))`
  --- the landed single-`T` object IS the per-`Q` object with DIAGONAL thresholds
  (`diagThreshold_of_le`, `diagThreshold_coupled`).
* **THE FACES at the corrected object**: `HonestNegPairingZeroIdx`/`PosIdx` (arbitrary windows/thresholds),
  with `honestNegPairing*_iff_diag` (their diagonal slices are EXACTLY the batch-213-B faces, both ways) and
  `honestNegPairing*Idx_iff_hw` (both ways to the per-h-weight inputs).  `...Diag_imp_idx` shows the correction
  is a genuine WEAKENING (the landed single-`T` faces are stronger and false).
* **FALSIFICATION**: the landed single-`T` forms (`HonestNegPairingZero`/`ZeroHw`/`Pos`...) are FALSE on batch
  211-B's exact instance (`l = 1`, `m = -1`, `j = 0`, `<a,b> = -M`: the honest coefficients are `1` at `Q = 1`
  and `M` at `Q = 0`, but the diagonal family gives the SAME threshold `T` to both).  The coefficient core is
  machine-checked (`diagCoeff_pair_ne_one_one`: the diagonal coefficients `(T+1)(T+2)/2`, `T(T+1)/2` are never
  both `1`; `perQ_thresholds_attain_pair` attains `(1,1)`).  The OPERATOR-level refutation is NOT landed --- it
  needs LINEAR INDEPENDENCE of the distinct blocks `normOrdered2 a b (C-Q) Q`, i.e. the converse of
  `pairBlock_blocks_distinct_hwPiece` --- and is reported as the exact residual.
* So one named input per face remains (`HonestNegPairingZeroIdxHw`/`HonestNegPairingPosIdxHw`), plus that
  linear-independence lemma for the refutation.

**B — the REDUCED honest boundary** (`Definitions/MMFLM810BoundaryReduced.lean`, 521 lines, `MM.FLM810BR`,
5 defs + 12 theorems, 17 guards).

* **NO spelling mismatch**: `NegTwoFirstDerivZero` (hs only), `NegThreeExpandedResidualAligned` (hs, C, h2) and
  `CoherentCreatedStepPosNonneg` (hnd, hs) all apply at exactly `E = C.toCentralExtension h2`.
* `EvenCocycleResidualsHonestReduced C h2 := CoherentCreatedStepNegPairingFactor E /\
  CreatedStepAtZeroNegPairingHonest E`, with `evenCocycleResidualsHonest_iff_reduced` (BOTH ways) and
  **`theorem8102_of_negPairingFock (hnd) (hs) (C) (h2) (S) (hpd) (hW) (hfin) (hneg) (hzero) (hstep) : Theorem8102`**
  (+ the bundled `theorem8102_of_residualsHonestReduced`).
* Non-vacuity: `evenCocycleResidualsHonestReduced_of_forall_nonneg` and
  `evenCocycleResidualsHonestReduced_iff_of_forall_nonneg : Reduced <-> EvenCocycleResiduals`.
* Ledger of the step: `EvenCocycleJacobiStepReduced` + its iff, content named as
  **(i) `MM.FLM86C851.Corollary851ModeStep`, (iii) `MM.FLM86D866.Display866SecondEqK` at `k >= 3`**.
* Corrected-currency variants (batch-211-B's defect of record): `EvenCocycleResidualsHonestCorrected`,
  `EvenCocycleResidualsHonestReducedCorrected`, `EvenCocycleJacobiStepCorrected`,
  `theorem8102_of_honestNegPairingCorrected`.
* **HONEST LIMIT**: absolute satisfiability of the two negative-pairing conjuncts is NOT known; at `hpd` (positive
  definite) the `0 <= <a,b>` non-vacuity slice is UNAVAILABLE (`(x,-x)` is a negative pairing); and the two
  reduced packages (the 210-B spelling and the corrected one) are NOT shown equivalent.

**Note**: `AGENTS.md`'s "tools/build.sh is not usable on this machine" is STALE --- it has run every build in this
campaign fine.

---

### Batch 215 — the general OPE/insertion step is LANDED (all lengths), and the negative-pairing faces are pinned to their off-vacuum form (with the operator-level refutation of the old spelling)

**A — the negative-pairing faces** (Definitions/MMFLM86NegPairingFinalAux.lean 309 lines, MM.FLM86NPFX +
...Final.lean 274 + ...FinalRefute.lean 369, MM.FLM86NPF2; 31 guards).

* **The vacuum slice is a THEOREM**: honestNegPairingZeroIdx_vacuum / PosIdx_vacuum (+ their Hw forms) --- at
  w = vacuumVL and m >= 0 the honest left-hand side is 0 (FLM (8.10.7) first half, MM.FLM810TA2.creationRegularAligned)
  and the right-hand side is 0 by the EMPTY window.
* Each face is reduced, BOTH WAYS, to one strictly smaller named input: HonestNegPairingZeroIdxHwNonvac /
  HonestNegPairingPosIdxHwNonvac (the vacuum case excluded), via honestNegPairing*IdxHw_iff_nonvac.
* **The block independence** (part 2): hw_separation_of_injOn, blockShift/blockShiftWeight, blockShift_mem_hwPiece,
  blockShiftWeight_injOn, and the CONVERSE of pairBlock_blocks_distinct_hwPiece: **normOrdered2_blocks_no_cancel**
  (distinct-Q blocks cannot cancel), normOrdered2_blocks_coeff_unique, normOrdered2_blocks_combination_ne.
  Limits: needs linearly independent labels (i != j); single-block nonvanishing is an explicit hypothesis.
* **OPERATOR-LEVEL REFUTATION of the single-T ancestors** (batch 214-A could only refute their coefficients):
  not_honestNegPairingZero_rankOne (+ Diag/Hw and the three Pos twins) at the rank-one datum (L = ZZ,
  <m,n> = -mn, a = b = 1, l = 0, m = j, w = vacuumVL, so C = 0): rankOne_pairBlock_vacuum_eq gives
  pairBlock = (sum_{k<=T} binomCoeff (-k-1) j) . anchorVL with scalar T+1 > 0 (j = 0) resp. -(T+1)(T+2)/2 < 0
  (j = 1) --- nonzero for EVERY T, while the honest left side is 0.
* **Corrections**: (i) the operator-level refutation did NOT need the block independence (at C = 0 only the Q = 0
  block survives) --- ROADMAP batch 214 / NOTES 111.2's account is superseded; the independence lemma is landed
  anyway as the tool for the off-vacuum residual.  (ii) The corrected per-Q faces are NOT refuted and the
  correction is not vacuous.  (iii) Two attempted further reductions are recorded as NON-reductions (basis-state
  restriction; single-shift collapse).

**B — the general insertion step** (Definitions/MMFLM86Display866ThreeAux.lean 506 lines, MM.FLM86D866T +
...Insert.lean 219 + ...Three.lean 248; 32 guards).

* **powerFam_nestedExp --- THE GENERAL OPE/INSERTION STEP, a THEOREM for every length**:
  powerFam a p (E^-_{b_1}(x_1)...E^-_{b_l}(x_l) iota(c)) = sum_{k in box} (prod_j ratioCoeff <a,b_j> k_j) .
  E^-_a(p - <a,c> + sum_j k_j)(... iota(ac)), with only hs and 0 <= <a,b_j>.  At l = 1 this is the landed k = 2
  slice (MM.FLM86YP.YFam_mul_apply_eq_sum_ratioCoeff), at l = 0 the landed (8.5.11) shift.  This is exactly what
  the batch-208-B ledger called the general OPE/insertion step (the k >= 3 case of (8.4.22)/(8.4.36)); residual
  (i) of MMFLM86YFamProduct's header is closed and Contraction8612Ratio is NOT on the path.
* Supporting: the l-fold (8.4.19) contraction (annihExp_nestedExp), the general-label commutations, the MidLabel
  sector and its preservation, the nestedExp carrier of (8.6.5) and its laws (nestedExp_eq_or,
  series865_eq_nestedExp), piFinset_succ/sum_piFinset_succ.
* The series spelling Display866SeriesK with the BOTH-WAY equivalence display866SecondEqK_iff_seriesK, the
  composition display866Display_of_series (giving (8.6.6) for the new range) and the k <= 2 slices.
* **ONE NAMED RESIDUAL: Display866SeriesStep k** --- the k -> k+1 ASSEMBLY, i.e. the finite regrouping
  PairIdx (k+1) = PairIdx k + Fin k with the induced splittings of pairWeightK/pairShiftK/blockScalarK.  Its OPE
  half is PROVED; only the finite-index bookkeeping remains.  Reductions: display866SeriesStep_iff (step <-> the
  (k+1)-st case given the k-th) and display866SecondEqK_three_of_step : Display866SeriesStep 2 ->
  Display866SecondEqK 3.
* FALSIFICATION CHECK: a Python rank-one lattice model, positive pairings at k = 3 ((1,1,1),(1,2,3),(1,1,2),
  (2,3,1)) and k = 4 ((1,1,1,1)): 189/189 index tuples agree, 0 mismatches; a negative pairing (2,-1,1)
  mismatches, confirming 0 <= <abar_i,abar_j> is load-bearing.

**Ledger deltas**: NOTES 105.2/111.4 and ROADMAP batch 214 item (iii) are STALE in their diagnosis --- the
general insertion step is landed; (iii) now reduces to Display866SeriesStep (with k = 3 from the landed k = 2
slice + the k = 2 step).  NOTES 111.4's block-independence row is LANDED, and the per-face input is refined to
the OFF-VACUUM residuals.

---

### Batch 216 — 🎉 item (iii) IS CLOSED (Display866SeriesStep is a theorem for every k), and item (i) is REFUTED as spelled (the window is a TRUNCATION, not a parameter) and replaced by the window-free form

**A — (iii) closed** (Definitions/MMFLM86Display866SeriesStepAux.lean 656 lines + ...SeriesStep.lean 671, 70 guards).

* **display866SeriesStep : Display866SeriesStep bL form E k --- a THEOREM for every k**, hence
  **display866SecondEqK_all : Display866SecondEqK at every k >= 1** (with the unconditional
  display866SecondEqK_three, display866SeriesStep_two, display866SeriesK_three).  The previously conditional
  display866SecondEqK_three_of_step is superseded.
* Machinery: the finite-index decomposition PairIdx (k+1) = PairIdx k + Fin k (newPairIdx/tailPairIdx with an
  exhaustive case analysis), the assembly map pairFunJoin (eta/injective/mem), the reindexing of pairShiftK's four
  filtered sums, pairWeightK_join, pairShiftK_join_zero/_succ, pairSetK_succ/sum_pairSetK_succ, nestedExp_smul;
  step_left (induction hypothesis + the landed powerFam_nestedExp), step_left_filter (a negative creation index
  kills the terms, creationExpVL_eq_zero_of_neg), step_right (the filter decomposition of the target), and the term
  identification (the (7.1.28) cocycle landing on anchorVL).  Two kernel small-instance cross-checks against the
  landed k = 2 slices.
* **SO JACOBI LEDGER ITEM (iii) IS CLOSED**: Display866SecondEqK holds at every k >= 1.  NOTES 112.1/112.4 and
  ROADMAP batch 215-B's 'ONE NAMED RESIDUAL' line are STALE.
* Brief errors: piFinset_succ/sum_piFinset_succ are NOT the assembly's regrouping tools (they regroup over a
  VARIABLE index Fin (l+1); the assembly needs the PAIR index PairIdx (k+1), which did not exist and is landed
  here); and def-defined filter predicates (or a local Classical.propDecidable) break statement-level
  DecidablePred/defeq matching against the imported Display866SeriesK --- abbrev predicates are the fix.

**B — item (i) refuted as spelled; corrected ledger landed** (Definitions/MMFLM86Corollary851ModeStepProveAux.lean
227 lines + ...Prove.lean 463, 21 guards).

* **THE REFUTATION** (only hs, arbitrary E): not_corollary851ModeStep_zero and not_corollary851ModeStep_one ---
  the same-window Corollary851ModeStep is FALSE at K = 0 and K = 1.  Mechanism: corollary851_vacuum_left shows the
  hypothesis C vacuumVL w K is FREE at every K ((8.5.3) Y(1,z)=1 makes the iterate supported at i = -1, outside the
  window), while the conclusion C v v K fails for v = oneFactorState a (c = a, r = 1): the (1,-1) commutator is the
  (7.1.5) central term <a,a>.1 != 0, the K = 0 residue is the EMPTY sum and the K = 1 residue is (Y v)_0 v = 0 (the
  k = 0 component is the (7.1.33) zero mode, which kills the Fock-descended vacuum: zAligned_oneFactor_zero,
  oneFactor_zero_apply, commutator_oneFactor).
* **KEY STRUCTURAL FACT**: the window of Corollary851 is a TRUNCATION, not a parameter ---
  iterateOf_eq_zero_of_corollary851_succ shows any C u v K -> C w v K forces the K-th iterate of (w,v) to vanish
  (the coefficient is deltaExpCoeff K K = (-1)^K, a unit in char 0).  Hence the same-window step is STRICTLY
  STRONGER than FLM (8.5.13), and a consumer whose hwin forces K = 1 (some pairing -1, none <= -2) uses an
  UNSATISFIABLE hypothesis.
* **CORRECTED LEDGER LANDED**: Corollary851ModeStepExists (WINDOW-FREE: C u v K -> exists K', C (mode u) v K'),
  Corollary851ModeStepExistsAt (hypothesis window pinned), corollary851ModeStep_imp_existsAt,
  corollary851ModeStepExists_iff_forall_existsAt (BOTH ways), the window-threaded (8.5.4) induction
  corollary851_spanningStates_exists_of_lattice_and_step, and corollary851_spanningStates_exists_of_residual_and_step
  (the corner supplied by the landed batch-204-A theorem).  **So (i) has exactly ONE named input:
  Corollary851ModeStepExists (ZAligned E).**
* Content of that residual: (1) the mode--field commutators [alpha_c(e), ZAligned E v n] at ARBITRARY v --- in
  particular the ANNIHILATION modes alpha_b(n), n >= 0, are evaluated NOWHERE in the tree (the landed (7.1.5)
  lemmas are generator-field and creation-mode only); (2) the (8.6.25)/(8.6.27) residue matching, where batch 215's
  powerFam_nestedExp is already a piece.  Batch 213-A's YFamModeDerivation/zLineModeAt_all do NOT close this, and
  the batch-204-B dead end is confirmed.
* Ledger corrections: NOTES 100.3/111.1/111.4/112.1 and ROADMAP batches 203/205 name the SAME-WINDOW spelling as
  residual (i) --- that must become the window-free (or explicitly windowed) form; and ROADMAP 205-B/NOTES 102.3's
  identification of the missing input with the mode--field commutators ALONE misses the independent window defect.

**Accounting after batch 216**: the campaign needs ONLY (i) Corollary851ModeStepExists (the two pieces above),
the two off-vacuum negative-pairing Fock faces (HonestNegPairingZeroIdxHwNonvac / HonestNegPairingPosIdxHwNonvac),
and the six boundary inputs hnd/hs/S/hpd/hW/hfin.

---

### Batch 217 — item (i) reduced to THREE named residuals; the two negative-pairing faces reduced (both ways) to their BLOCK form

**A — item (i)** (Definitions/MMFLM86Corollary851ModeStepExistsAux.lean 375 lines + ...Exists.lean 321,
MM.FLM86C851P).

* BOTH-WAY reduction to one named input per half: Corollary851ModeStepExists Y <-> ModeStepLeftExists Y /\
  ModeStepRightExists Y, and each half <-> the package of its fixed-window slices
  (corollary851ModeStepLeft/RightExists_iff_forall_existsAt).
* the all-states reading named (Corollary851AllStatesExists := forall u v, exists K, C u v K),
  corollary851ModeStepExists_of_allStates, and corollary851ModeStepExists_iff_allStates (BOTH ways, modulo the
  named lift Corollary851SpanningToAllStates and the landed batch-204-A corner).
* NEW mode--field theorems: the uniform (7.1.5) base-field law for every nonzero mode index (modeVL_YFam_add_ne,
  merging the landed creation and annihilation halves), modeVL_iotaVL_eq_zero_of_pos, and **ModeFieldCommInput**
  (the (8.6.10) commutator [alpha_c(e), Y(v)_n] = sum_j binom(e,j) Y(alpha_c(j)v)_{n+e-j} at ARBITRARY v) PROVED
  at the lattice generators (FLM (8.6.38)) AND on their span (modeFieldCommAt_of_mem_span) --- so its open content
  is exactly the OFF-GENERATOR (8.5.4) states.
* THREE NAMED RESIDUALS: (1) ModeFieldCommInput off the generator span ((8.6.9)/(8.6.10) for oscillator states);
  (2) ResidueInsertion8625 (the (8.6.25)/(8.6.27) field insertion/residue matching); (3)
  Corollary851SpanningToAllStates --- NOT bookkeeping: of_cutoff can raise a window only when the iterate vanishes
  from it on, and the landed (8.10.5) truncation is VECTORWISE (ComponentTruncation per pair), the state-level
  vanishing being landed only at lattice pairs (stateVanishesBelow_ZAligned).
* CORRECTIONS (measured): the claim "the ANNIHILATION modes alpha_b(n), n >= 0, are evaluated NOWHERE in the tree"
  (ROADMAP 216-B / NOTES 113.3) is STALE --- batch 192-A landed them at the honest assembled field
  (MM.FLM86TC2.modeVL_YFam_add_pos, MM.FLM86TC2.hZeroVL_YFam_add, MM.FLM86TC2.convWith_modeVL_add_left, plus
  MM.FLM84CT.modeVL_comm_creationExpVL, MM.FLM86TN.modeVL_comm_annihExp_pos, MM.FLM86TC2.hZeroVL_comm_annihExp);
  what is open is only the arbitrary-STATE commutator.  Name corrections: Corollary851ModeStepExists is
  MM.FLM86C851P (not MM.FLM86C851); all-states is MM.FLM86C851.*; the residue is MM.FLM85Co.corollary851Residue and
  the delta coefficients MM.FLM85Br.deltaExpCoeff.

**B — the two negative-pairing faces in block form** (Definitions/MMFLM86NegPairingNonvacAux.lean 350 lines,
MM.FLM86NPNA + ...NonvacAnchor.lean 193 + ...Nonvac.lean 378, MM.FLM86NPN).

* pairBlockIdx_eq_blockCombination: pairBlockIdx over [lo,hi] IS a finite block combination over Icc lo hi;
  dropping the window gives the named inputs HonestNegPairing*IdxHwNonvacBlock --- off the proved vacuum slice,
  a FINITE set s of shift indices with ONE truncation integer per index, the coefficient of Q being a partial sum
  over the FORCED window [max 0 (l-Q), Tfun Q] --- with BOTH-WAY reductions
  honestNegPairing*IdxHwNonvac_iff_block (backward by [min' s, max' s] + blockCoeff_eq_zero_of_lt inside
  Finset.sum_subset).
* lhsIdx_anchor_apply_of_neg: at l = 0 the faces' LHS IS eps(a,b).coherentFieldNat at the anchor iota(a+b) of
  degree j - <a,b>, at EVERY pairing (no positivity!); the two anchor inputs are equivalent to the l = 0 slices
  (...Block_anchor_iff).  Plus pairBlockIdx_iotaVL + blockOfShift_iotaVL_eq_zero_of_lt_left/_right (the RHS at a
  lattice generator with its support window Icc (<b,c>) (C - <a,c>)) and creationExpVL_eq_YFam_vacuum
  (E^-_b(z_0)_l iota(b) IS the honest mode YFam E b (-(l)-1) on the vacuum).
* FALSIFICATION RECORD (Probe/b217b_falsification.py, a NECESSARY condition only): at the rank-one negative datum
  the honest vector lies in the block span with reachable nonneg-integer partial-sum coefficients (pattern: c = C
  on each pair {Q, C-Q}, C/2 at the middle) for C = 1..12; the single-T diagonal family fails already at C = 2
  (batch 214-A/215-A's diagCoeff_pair_ne_one_one in this currency).
* TRAPS OF RECORD: (1) a first-draft "refutation" was an ARITHMETIC ERROR (E^-_2(r) is NOT 2^r h_r --- the
  doubling gives (E^-_1)^2); (2) hwProj/hwProj_eq_iff live in MM.FLM86HWA, normOrdered2_vacuum in MM.FLM86MI;
  (3) MM.FLM86D866T.powerFam_nestedExp needs 'import Definitions.MMFLM86Display866Insert'; (4) rw with an instance
  at j := 0 produces the key -(up-arrow 0)-1 which does NOT match -1 (state a have first --- batch 213's
  zero-coercion trap); (5) by_cases on a goal not P -> Q leaves not P in both branches; (6) a '/-! ##' header
  missing its '-/' reports 'unterminated comment' at EOF.
* DISPATCHER NOTE: the fresh-build verification initially FAILED on B's three files (the guards referenced names
  from the agent's own late write); after the final write all three build clean (3x OK, 0 error, 0 sorryAx).
  Re-verify an agent's files AFTER its last write, always by deleting the oleans.

**Accounting after batch 217**: the campaign needs (i)'s three residuals (ModeFieldCommInput off-generator,
ResidueInsertion8625, Corollary851SpanningToAllStates), the Fock four named block/anchor inputs (equivalent to the
two faces), and the six boundary inputs.

---

### Batch 218 — (i) reduced to ONE named annihilation input (its zero half closed); and the Fock negative-pairing j=0 face is REFUTED in every spelling (block / per-Q / Idx), with the FREE-coefficient form as the corrected residual

**A — (i) to one input** (Definitions/MMFLM86ModeFieldCommAux.lean 202 + ...Zero.lean 403 + ...ModeFieldComm.lean 140 +
...Ann.lean 477).

* modeFieldCommInput_iff_step (BOTH WAYS); modeFieldCommStep_zero_clause + **modeFieldCommZeroHalf: the ZERO-MODE
  half of (8.6.9)/(8.6.10) at EVERY state is a THEOREM**;
* modeFieldCommInput_iff_annHalf / _iff_annStep (BOTH WAYS): the off-generator content of the input is EXACTLY ONE
  named statement --- **MM.FLM86MFC.ModeFieldCommAnnStep E**, the one-oscillator ANNIHILATION clause;
* machinery landed and reusable: modeVL_mul_splitMinus_factorFamilyZero (the (7.1.5) scalar defect e.<c,b> firing
  only at i = e-k with coefficient C(i+k-1,k-1)), the ^+ half contributing NOTHING (modeVL_mul_splitPlus_*),
  modeVL_headWindow_ann / modeVL_tailWindow_ann through both (8.4.16) halves, and sum_scalar_term (the summed defect
  = C(e,K).K.<c,b> at the shifted index --- exactly the extra RHS term from [alpha_c(K), alpha_b(-K)] = K.<c,b>).
* Brief errors: the MM.FLM86YFD derivation identity is the Sugawara L(-1) one (NOT the (7.1.5) mode-field
  commutator); powerFam_nestedExp routes to ResidueInsertion8625, not to this target; the circularity warning about
  actionFamily_*_modeVL was CORRECT and honoured; MM.FLM86TC2's annihilation laws are at GENERATORS only; the
  correct name is MM.FLM86C851E.ModeFieldCommInput.  Traps: rw does NOT rewrite under a Finset.sum binder with a
  forall-lemma (use Finset.sum_congr + show/ring); section variables bL/form become the FIRST explicit args;
  LinearMap.one_apply does not exist (Module.End.one_apply); smul_mul does not exist (mul_smul_comm/smul_mul_assoc);
  Int.cast vs Nat.cast need Int.cast_natCast.

**B — the Fock negative-pairing j = 0 face REFUTED** (Definitions/MMFLM86NegPairingBlockProveAux.lean 295 +
...Aux2.lean 182 + ...Prove.lean 312, MM.FLM86NPBP).

* THE COUNTEREXAMPLE (rank-one, L = ZZ, <m,n> = -mn, a = b = 1, trivial cocycle, (l,j,m,w) = (0,0,1,iota(1)) ---
  OFF the vacuum slice, rankOne_iotaVL_one_ne_vacuum by bidegree separation):
    rankOne_lhs : ZAligned E (YFam E 1 (-1) (creationExpVL 1 0 (iotaVL 1))) 1 (iotaVL 1)
                    = -(eps(1,1)*eps(2,1)) . rankOneVec
    rankOne_rhs (s) (Tfun) : exists K, 0 <= K and (blockCombination ...) (iotaVL 1)
                    = (K*(eps(1,1)*eps(2,1))) . rankOneVec
  so the identity forces -1 = K >= 0.  The evaluation uses (8.10.13) L(-1)iota(a) = alpha_a(-1)iota(a) at a = 2,
  the translation law Y(L(-1)v,z) = (d/dz)Y(v,z), and (8.5.11); at iota(1) the blocks vanish except Q = 0 and
  Q = -1 (which COINCIDE), and blockCoeff at j = l = 0 is a window CARDINALITY.
* REFUTED (machine-checked): not_honestNegPairingZeroIdxHwNonvacBlock_rankOne (217-B's block form),
  not_honestNegPairingZeroIdxHwNonvac_rankOne (215-A), not_FLM86NPF_HonestNegPairingZeroIdxHw_rankOne (214-A).  The
  j >= 1 (Pos) face is NOT refuted by this method.
* WHY 217-B's CHECK MISSED IT: its script tested w = vacuumVL with C = -m = 1..12 (m <= -1), where the honest
  coefficient -m/2 is POSITIVE and every block at m > 0 vanishes; at w = iota(c) with m > 0 the honest ordered-pair
  coefficient is -m/2 < 0 while blockCoeff is a cardinality.  Re-run: Probe/b218b_falsification.py.
* CORRECTED FORM LANDED: MM.FLM86NPBP.HonestNegPairingZeroIdxHwNonvacBlockFree --- the same statement with an
  ARBITRARY finite coefficient family coef : ZZ -> F in place of blockCoeff ... (Tfun Q); implied by both the
  (refuted) block form and the 215-A residual, and NOT refuted (rankOne_free_attained with s = {0},
  coef = fun _ => -1).  It is the new named residual.  The defect is exactly the FORCED partial-sum coefficient.
* **Consequence**: MMFLM810BH's negative-pairing j = 0 conjunct (CreatedStepAtZeroNegPairingHonest and relatives)
  may be UNSATISFIABLE, so that boundary hypothesis is likely vacuous too; the honest statement must be re-derived
  from FLM (8.6.25)/(8.6.27) directly.  FLM's own displays are NOT in question --- our reading of their right-hand
  side is.
* Ledger corrections: ROADMAP batch 217 / NOTES 114.2/114.4's "four named block/anchor inputs" are FALSE (replace
  by the free form); 215-A's "the per-Q faces are NOT refuted" and 217-B's "the check does NOT refute" are
  superseded (both tested only w = vacuumVL, m <= -1); NOTES 114.1's "off the generator span" is now exactly
  ModeFieldCommAnnStep (both ways), with the zero half a theorem.

**Accounting after batch 218**: the campaign needs (i) = ModeFieldCommAnnStep + ResidueInsertion8625 +
Corollary851SpanningToAllStates; the Fock side = the free form (or a re-derived honest statement); and the six
boundary inputs.

---

### Batch 219 — 🎯 ModeFieldCommInput IS PROVED (all states); the honest (8.6.25)/(8.6.27) coefficient is a BINOMIAL; and the honest Fock boundary package is REFUTED (the boundary hypothesis is VACUOUS)

**A — the honest coefficient** (Definitions/MMFLM86NegPairingHonestAux.lean 321 + ...Aux2.lean 250 +
...Honest.lean 500).

* **THE ANSWER**: reading FLM (8.6.25)/(8.6.27) (printed p. 210; Remarks 8.5.2 p. 200, 8.6.4 p. 211) at
  A = iota(a), B = iota(b) by extracting [w^0 x^0] (which returns the landed (8.5.11) value) and then [z_0^N] of
  the normal-ordered product gives
    ZAligned E (YFam E a (-1) iota(b)) m w = sum_Q binomCoeff (C-Q) N . :Y(a,b):(C-Q, Q) w,
    C = blockFirstIdx 0 0 m <a,b> = -m - <a,b> - 1,   N = -<a,b> > 0,
  the coefficient attached to the FIRST (a-)index P = C-Q.  This is NOT the (8.5.37) fiber sum / blockCoeff (a
  cardinality) and NOT the pairBlock partial ratioCoeff sum.  At N = 1 the coefficient is the integer C-Q;
  collecting the equal ordered pairs (creationExpVL_comm) gives the symmetric average C/2 = -m/2 --- exactly
  batch 218-B's measurement and 217-B's vacuum pattern.  Machine-checked for EVERY mode m and w = iota(c)
  (rankOne_lhs_all_modes, rankOne_coeff_determined_range, rankOne_coeff_determined).
* **NEW REFUTATIONS** (machine-checked at (a,b,l,m,w) = (1,1,0,1,iota(1))): rankOne_ne_pairBlock;
  not_honestNegPairingZero_rankOne (batch 211-B's "corrected slot"); not_honestNegPairingZeroAtZero_rankOne;
  **not_createdStepAtZeroNegPairingHonest_rankOne** (batch 210-B's def PROVED FALSE, not merely "possibly
  unsatisfiable"); not_createdStepAtZeroNegPairingHonestCorrected_rankOne; not_fockResidualHonest_rankOne;
  not_fockResidualHonestCorrected_rankOne; and **not_evenCocycleResidualsHonest_of_toCentralExtension** --- so
  theorem8102_of_residuals_honest (batch 210-B) and theorem8102_of_negPairingFock (batch 214-B) are VACUOUS, and
  the Fock ledger must be re-stated on the corrected coefficient.
* **CORRECTED RESIDUAL**: HonestNegPairingZeroCoeff E --- free support s, DETERMINED coefficient
  honestBlockCoeff = binomCoeff (C-Q) N; attained at the falsifying datum (rankOne_coeffDet_attained), implying the
  batch-218 free form on the l = 0 slice.  Corrected package
  FockResidualHonestCoeff = CoherentCreatedStepPosNonneg /\ HonestNegPairingPos /\ HonestNegPairingZeroCoeff.
  Slices 1 and 2 SURVIVE (slice 2 is under suspicion: its spelling uses the same pairBlock/cardinality object).
* Not proved: HonestNegPairingZeroCoeff at general (a,b,w); the general-N coefficient (derived in the Aux header,
  only N = 1 machine-checked --- general N needs the open (8.6.27) dictionary); the l >= 1 slices; the j >= 1 face.
* Brief errors: MM.FLM86NTP.residueNegativeAligned_evenCocycle does NOT resolve (the landed proved residue is
  MM.FLM86NTP.negThreeExpandedResidualAligned_evenCocycle); batch 210-B's object pins the second (8.5.37) slot at 0
  while pairBlock pins it at l (so the implication holds only at l = 0); API traps: LinearMap.sum_apply before
  Module.End.mul_apply, sub_smul needs explicit args, Int.toNat(-1) = 0.

**B — ModeFieldCommInput PROVED** (Definitions/MMFLM86ModeFieldCommAnnProveAux.lean 187 + ...AnnProve.lean 323).

* **modeFieldCommAnnStep is a THEOREM**, hence modeFieldCommAnnInput and **modeFieldCommInput** --- the full
  (8.6.9)/(8.6.10) mode--field commutator at ARBITRARY states, spending only hs.  Item (i)'s first residual is
  CLOSED.  Landed bridges: YtjFam, Yzero_modeVL_fam, Yzero_modeVL_window, hZeroVL_modeVL_comm_apply,
  sum_state_term.
* BRIEF ERROR (mine): the criterion "thence to Corollary851ModeStepExists" is FALSE --- no landed theorem reduces
  ModeFieldCommInput to it and nothing consumes ModeFieldCommInput as a hypothesis; the three (i) residuals are
  PARALLEL, and the bridge is the (8.6.4)--(8.6.27) residue matching (= ResidueInsertion8625).
* Technical: the tail window must be max (X.bound x) (X.bound (modeVL c e x)); the head window a FINITE SUM of
  descendant bounds, not a single X.bound x; orientation headOp_eq_headWindow / tailOp_eq_tailWindow.

**Accounting after batch 219**: Fock = HonestNegPairingZeroCoeff (j = 0) + HonestNegPairingPos (j >= 1, under
suspicion) + the general-N coefficient; (i) = ResidueInsertion8625 + Corollary851SpanningToAllStates (+ the
mode-field-to-mode-step bridge); the six boundary inputs.  EvenCocycleResidualsHonest / FockResidualHonest /
FockResidualHonestCorrected are REFUTED and must not be consumed; the honest boundary has to be re-stated on
FockResidualHonestCoeff.

---

### Batch 220 — slice 2 (j >= 1) is REFUTED too; the honest residual is HonestNegPairingCoeff, whose proof reduces to ONE vertex-algebra-free reindexing; ResidueInsertion8625 is only the finiteness half

**A — slice 2 refuted** (Definitions/MMFLM86NegPairingCoeffGeneralAux.lean 414 lines + ...CoeffGeneral.lean 225).

* **not_honestNegPairingPos_rankOne**: HonestNegPairingPos is FALSE at the rank-one datum
  (a,b,j,l,m,w) = (1,1,1,0,3,iota(1)) --- NON-VACUUM, NON-GENERATOR.  The honest LHS is iotaVL 3 (via the landed
  zAlignedGeneratorFlat + the ZAligned_JZF_eq bridge; no new input), while pairBlock ... 1 0 3 T (iotaVL 1)
  = K_T . iotaVL 3 with K_T = -(sum_{k in Icc 1 T} (k+2)) <= 0 for EVERY threshold T (only the shift index
  Q = -1 survives), so 1 = K_T contradicts (smul_eq_zero + iotaVL_ne_zero).
* => FockResidualHonestCoeff (batch 219-A's corrected package) is REFUTED on its second conjunct; the ledger must
  use HonestNegPairingCoeff.
* The corrected coefficient GENERALISED: honestBlockCoeffJ a b j m Q = binomCoeff (blockFirstIdx j 0 m <a,b> - Q)
  (j + (-<a,b>).toNat), with honestBlockCoeffJ_zero (at j = 0 it IS batch 219-A's binomial, rfl-level), the
  corrected face HonestNegPairingCoeff (free support + DETERMINED binomial) with its j = 0 / j >= 1 slices, and
  rankOne_coeffJ_attained (attained at the falsifying datum: single block Q = -1 with binomCoeff(-1,2) = 1).
* The general-N/general-j coefficient is DERIVED (through the landed zAlignedGeneratorFlat: expand flatMul,
  reindex p = i+n-l, apply the landed Chu-Vandermonde MM.FLM86ZM.binomCoeff_add_eq_sum) and numerically verified,
  and its formal proof reduces to ONE NAMED VERTEX-ALGEBRA-FREE identity --- **FlatBinomialExpansion**: for a
  commutative Q-algebra A, X Y : Z -> A, s : Z, n : N, TOT : Z,
    sum_{l in range (n+1)} binomCoeff s l . ((n-l)! : Q)^-1 . flatMul (iterDeriv (n-l) X) Y (l + TOT - n)
      = sum_{p in range (TOT.toNat+1)} binomCoeff (p + s) n . (X p * Y (TOT - p)),
  whose only gap is the (l,i) |-> p double-sum reindexing in Finset.sum_range.  Numerically verified at N = 1
  (j = 0..3, c = 1..3, m = 1..6, all blocks, 0 mismatches).  HonestNegPairingCoeff at general (a,b,w), the
  off-generator support and the l >= 1 second slot remain open.
* Brief errors: target (1) as spelled was a definitional triviality (honestBlockCoeff IS that binomial by def) ---
  the content is the FIELD identity; powerFam_nestedExp needs 0 <= <a,b_j> (positive pairing) and is NOT the
  route (zAlignedGeneratorFlat + binomCoeff_add_eq_sum suffice at generators); the "same cardinality object"
  diagnosis is inaccurate at j >= 1 (there the pairBlock coefficient is a genuine binomial PRODUCT, yet the face
  is still false --- wrong index attribution / coupled window); the batch-211-B checked instance does not extend
  to m > 0.

**B — the finiteness half proved; the matching is open** (Definitions/MMFLM86ResidueInsertion.lean 217 +
...SpanningLift.lean 326).

* **ResidueInsertion8625 IS PROVED --- but it is only the FINITENESS side** (its own docstring says so):
  iterateOf (ZAligned E) (alpha_c(-r)u) v k = 0 for k >= N, a corollary of (8.10.5) at the honest field
  (componentTruncationAligned kills the whole j-family: exists_iterateOf_eq_zero_of_truncation,
  residueInsertion8625), with non-vacuity machine-checked (iterateOf_zAligned_lattice_ne_zero: the iterate is
  NONZERO at the (8.5.11) boundary index -<a,b>-1).  => **the (8.6.25)/(8.6.27) MATCHING (the normal-ordered
  insertion Y(Y(A,z_0)B,z_2) = 8Y(A,z_2+z_0)Y(B,z_2)8 . prod (z_0+w_i-x_j)^{<a_i,b_j>}) is UNNAMED and OPEN**, and
  proving the finiteness half does NOT advance item (i).  Also: batch 217-A/219's "obstruction (b)" is WRONG ---
  vectorwise (8.10.5) DOES give state-level thresholds at arbitrary pairs; the real obstruction is WINDOW
  SYNCHRONIZATION (their (a)).
* **Corollary851SpanningToAllStates <-> Corollary851SpanningEventualWindows Y** (per-pair eventual windows; both
  spellings equivalent, spanningGrounded_iff_eventual), with the reduction corollary851Lift_of_eventual PROVED for
  arbitrary Y (finite expansion exists_stateFinset_mem_span + a common window exists_corollary851_of_mem_stateSpan
  + MM.FLM85Fock.corollary851_of_mem_span); the input is NOT implied by the landed hypothesis and the gap is
  machine-checked (not_corollary851_succ_of_first_ne_zero); the lift itself is neither proved nor refuted.
  Falsification: two scalar pairs with windows [3,4,5] and [0] --- each has a window, none common.

**Accounting after batch 220**: Fock = HonestNegPairingCoeff (both slices) <-- FlatBinomialExpansion;
(i) = the unnamed (8.6.25)/(8.6.27) matching + Corollary851SpanningEventualWindows (+ the mode-field input is
PROVED); the six boundary inputs.

---

### Batch 221 — 🎯 FlatBinomialExpansion PROVED (with an honest correction), HonestNegPairingCoeff at l = 0 PROVED AT THE LATTICE GENERATORS, and the (8.6.25)/(8.6.27) matching NAMED with half of it a THEOREM plus the mode-step bridge PROVED

**A — the flat identity and its instantiation** (Definitions/MMFLM86FlatBinomial.lean 554 lines +
...FlatBinomialApply.lean 285; 25 guards).

* **HONEST CORRECTION (machine-checked)**: the batch-220 spelling sum_{p in Finset.range (TOT.toNat+1)} is FALSE at
  TOT < 0 (rangeSpelling_refuted at (1_{z^0}, 1_{z^{-1}}, 0, 0, -1): LHS 0 vs RHS 1, because every flatMul index
  l+TOT-n <= TOT vanishes while range(TOT.toNat+1) = {0} leaves the spurious term C(s,n).X 0 * Y TOT).  The honest
  window is **Finset.Icc 0 TOT** (empty for TOT < 0); flatBinomialExpansion_range recovers the recorded spelling
  exactly where it is true (0 <= TOT), so no content is lost.  Also corrected: commutativity of A is unnecessary
  and WRONG for the carrier (Module.End is noncommutative), and the landed flatMul_apply_Icc [Algebra Q A] scalar
  does not apply there --- hence the general-ring F-scalar forms flatMul_eq_sum_Icc / iterDeriv_apply_smul_F.
* **flatBinomialExpansion PROVED** for [Ring A] [Algebra F A] (F char 0, A not assumed commutative):
    sum_{l<=n} C(s,l)((n-l)!)^{-1} . flatMul (iterDeriv (n-l) X) Y (l+TOT-n)
      = sum_{p in Icc 0 TOT} C(p+s,n) . (X p * Y (TOT-p)).
  Proof content: iterDeriv rising-factorial closed form -> flatMul expansion over Icc k TOT (landed
  sum_Icc_shift) -> the Icc 0 TOT form -> l |-> n-l reflection -> Finset.sum_comm -> per-p collapse
  ((k!)^{-1}.rising = C(p,k), prod_range_reflect) + Chu-Vandermonde.  (The suggested
  MM.FLM86GAI.sum_factorial_flatMul_iterDeriv is a NESTED-flatMul Leibniz rule and does not apply.)
* **INSTANTIATION**: flatGeneratorValueZ_eq_sum_binomCoeff / zAlignedGeneratorFlat_eq_sum_binomCoeff --- the block of
  a-index P = p + <a,c> carries exactly C(P,j), CONFIRMING batch 220-A's honest coefficient --- and
  **honestNegPairingCoeff_at_generators: HonestNegPairingCoeff at l = 0 is a THEOREM at the lattice generators**
  (witness s = (Icc 0 TOT).image (fun p => blockFirstIdx j 0 m <a,b> - p - <a,c>); only landed material, NO new
  assumption), with non-vacuity rankOne_sum_binomCoeff_eq_iotaVL.
* Residual: HonestNegPairingCoeff at ARBITRARY w (zAlignedGeneratorFlat is a generator statement; the general
  vector needs the mode action of ZAligned, i.e. its Fock second half); the l >= 1 second slot; the matching at
  arbitrary vectors.  Brief errors: the spelled identity is false at TOT < 0; 'commutative Q-algebra' is
  unnecessary and wrong; the reduction needs the general-ring rising-factorial closed form (the landed
  iterDeriv_apply_smul is Module.End-only).

**B — the matching named, half proved, and the bridge proved** (Definitions/MMFLM86Matching8625Aux.lean 314 +
...Matching8625.lean 387; 24 guards).

* **ResidueMatching8625 E := Nonneg /\ Neg /\ NegSecondSlot** --- FLM (8.6.25)/(8.6.27) at the sharpest landed
  slice (A = iota(a), B = [x^l](E^-_b(x) iota(b)), every j, every slot l, BOTH pairing signs);
  iterateOf_iotaVL_inserted8625 proves the LHS IS the (8.6.27) iterate at the (8.5.24) index -(j)-1.
* **residueMatching8625Nonneg is a THEOREM** (from the landed MM.FLM86MI.ZAligned_createdState) --- HALF THE
  MATCHING IS NO LONGER A RESIDUAL --- and the BOTH-WAY ledger
  **residueMatching8625_iff_coeff_and_slot : ResidueMatching8625 E <-> HonestNegPairingCoeff E /\
  ResidueMatching8625NegSecondSlot E**.
* **THE BRIDGE IS PROVED**: IterateInsertion8625 / CommutatorInsertion8625 -> modeStepLeft_of_insertion /
  modeStepRight_of_insertion / **modeStepExists_of_insertion : both halves -> Corollary851ModeStepExists
  (ZAligned E)**, with the window NOT assumed (landed (8.10.5) exists_iterateOf_zAligned_eq_zero);
  modeFieldCommInput8625 re-exports the PROVED batch-219-B input, and the remaining cross-contraction gap is named
  **ModeFieldCommInsertionStep8625** (the (8.4.25) cross-contraction producing the (8.6.25) pairing factor on the
  commutator side).
* The NEW l >= 1 slot: honestBlockCoeffJL a b j l m Q = binomCoeff (blockFirstIdx j l m <a,b> - Q) (j + N); it is
  NOT implied by HonestNegPairingCoeff (at the 220-A datum the coefficient is 0 vs the landed l = 0 value 1:
  honestBlockCoeffJL_one_rankOne).
* Corrections: deltaExpCoeff lives in MM.FLM85Ex (not FLM85Br); corollary851ModeStepExists_iff_forall_existsAt
  does not exist (the both-way slices are ...Left/RightExists_iff_forall_existsAt); batch 219-B's "the bridge =
  ResidueInsertion8625" is misleading (the bridge is the pair of halves above).

**Accounting after batch 221**: Fock = HonestNegPairingCoeff at arbitrary w (the generator slice is PROVED) +
ResidueMatching8625NegSecondSlot; (i) = ModeFieldCommInsertionStep8625 + Corollary851SpanningEventualWindows; the
six boundary inputs.  The boundary theorem must be re-stated on the corrected (non-vacuous) package.

---

### Batch 222 — 🎯 THE DELIVERY IS RE-HUNG ON A NON-VACUOUS CORRECTED PACKAGE; the Fock face reduces to the Fock half with the LATTICE FACTOR PROVED

**A — the corrected boundary** (Definitions/MMFLM810BoundaryCorrected.lean, 473 lines, MM.FLM810BC, 14 guards).

* EvenCocycleResidualsCorrected C h2 := ResidueMatching8625 E /\ Corollary851SpanningEventualWindows
  (ZAligned E) /\ exists Ins, IterateInsertion8625 E Ins /\ ModeFieldCommInsertionStep8625 E Ins, at
  E = C.toCentralExtension h2.  The FOUR PROVED conjuncts (MM.FLM86HAP.negTwoFirstDerivZero_of_symmetricForm,
  MM.FLM86NTP.negThreeExpandedResidualAligned_evenCocycle, MM.FLM86YFD.coherentCreatedStepPosNonneg,
  MM.FLM86M8625.residueMatching8625Nonneg) are DISCHARGED and appear only in EvenCocycleResidualsCorrectedLedger.
* Normal form, BOTH WAYS: evenCocycleResidualsCorrected_iff_normal (via the landed both-way ledger
  residueMatching8625_iff_coeff_and_slot), with the right side the honest residual list
  (HonestNegPairingCoeff /\ ResidueMatching8625NegSecondSlot /\ Corollary851SpanningEventualWindows /\
  exists Ins(...)); non-vacuity on the 0 <= <a,b> slice is machine-checked.
* **Boundary re-landed**: theorem8102_of_correctedResiduals / theorem8102_of_correctedNormal (composing
  MM.FLM810TE.theorem8102_of_evenJacobi), and **corollary851AllStatesExists_of_corrected: the package implies
  the ALL-STATES (8.5.13)** --- so the only residual passage is EvenCocycleJacobiStepCorrected, i.e. the
  ASSEMBLY from all-states (8.5.13) + the landed faces to (8.10.8) (Theorem 8.8.9's argument).
* **FINDING**: MM.FLM810TE.theorem8102_of_residuals CANNOT be composed --- its EvenCocycleResiduals contains
  the refuted CoherentCreatedStep, which is UNSATISFIABLE at the boundary's own hpd
  (MM.FLM810HA.not_evenCocycleResiduals_of_posDef).  The old path is formally DEAD; the corrected one replaces it.
* Audit: Probe/b222a_audit.py (comment-stripped, word-boundary) checks 26 refuted names --- 0 consumed, exit 0,
  negative-controlled; the Lean header carries the refutation table.
* Brief errors: MM.FLM86NTP.negTwoFirstDerivZero_* does not exist (it is
  MM.FLM86HAP.negTwoFirstDerivZero_of_symmetricForm); the old boundary cannot be composed; the batch-221
  accounting OMITTED the all-pairs iterate half IterateInsertion8625 E Ins (the bridge needs both halves, and the
  named matching covers only the k = l = 1 generator slice) --- the package carries an explicit exists Ins with
  both clauses; Corollary851ModeStepExists is MM.FLM86C851P; residueMatching8625_iff_coeff_and_slot needs hs.

**B — the Fock half** (Definitions/MMFLM86NegPairingCoeffAllAux.lean 464 lines, MM.FLM86NPCA +
...CoeffAll.lean 413, MM.FLM86NPCL; 24 guards).

* honestNegPairingCoeff_at_generators_window: 221-A's generator face with the support exhibited as the
  CANONICAL WINDOW Icc (<b,c>) (blockFirstIdx j 0 m <a,b> - <a,c>), the unique maximal support at iota(c).
* **honestNegPairingCoeff_on_latticeSpan / _on_latticeCombination: THE FACE IS A THEOREM AT EVERY w IN THE
  F-SPAN OF THE LATTICE GENERATORS (the lattice factor F{L} <= V_L)**, with the explicit union-of-windows
  support --- strictly extends 221-A; it works because blocks outside a generator's window VANISH there.
* **honestNegPairingCoeff_iff_atStates (BOTH WAYS)**: the face <-> HonestNegPairingCoeffAtStates (the Fock half
  at the (8.5.4) states, ONE uniform support per finite family).  Forward = per-vector block finiteness
  (exists_diag_support of the landed one-sided finiteness = FLM (8.10.5) per state) + **"an infinite field is not
  a finite union of proper subspaces"** (Submodule.exists_forall_notMem_of_forall_ne_top); backward = spanning
  MM.FLM85Un.spanningStatement + F-linearity of both sides (the coefficients do not depend on w).
* **honestNegPairingCoeff_iff_finiteDimUniform**: the uniformity in the named input is FORCED by the target ---
  the per-state reading is strictly weaker and cannot imply it.
* Falsification: the face at the NON-VACUUM, NON-GENERATOR w = iota(1) + iota(2) (rank-one model).
* **EXACT RESIDUAL: MM.FLM86NPCL.HonestNegPairingCoeffAtStates E** (the Fock half: the field at a
  creation-mode-descended state, i.e. the (8.6.9)/(8.6.25) re-expansion there).  The l >= 1 slot was not
  attempted (its generator LHS is only available as a ratio sum of coherentFieldNat self-recursions).
* Corrections/refinements: the F-linearity + spanning route closes the LATTICE factor only (spanning consumes the
  Fock half); batch 215-A's "basis-state restriction is a non-reduction" applies to the OLD refuted (lo,hi,Tfun)
  window spelling --- in the current per-vector Finset currency the lattice reduction IS valid (a theorem now);
  **modeFieldCommInput does NOT give the Fock half directly** (it is the commutator in the STATE slot, whereas the
  Fock half is the field at a state evaluated at a creation-mode VECTOR --- the tool is MM.FLM86MLA.zLineModeAt_all
  / AnchorModeComm plus the mode action on the normal-ordered blocks); the CANONICAL WINDOW was the missing piece
  in the brief's route.

**Accounting after batch 222**: the boundary is NON-VACUOUS; residual list = HonestNegPairingCoeffAtStates
(Fock half) + ResidueMatching8625NegSecondSlot + Corollary851SpanningEventualWindows +
ModeFieldCommInsertionStep8625 + the ASSEMBLY EvenCocycleJacobiStepCorrected; the six boundary inputs.

---

### Batch 223 — 🎯 THE ASSEMBLY IS BUILT: the step reduces to TWO named statements (the windows conjunct + the l < 0 faces at non-generator pairs), and the l >= 0 faces are PROVED not residual

**A — the assembly** (Definitions/MMFLM810AssemblyAux.lean 389 + MMFLM810Assembly.lean 370 + AssemblySlices.lean
203 + AssemblyGrounded.lean 168).

* **NEW MATHEMATICS**: crossBracket_eq_crossIterate_of_nonneg_of_grounded --- every `l >= 0` face of (8.8.41) at
  EVERY state/pairing is the `(z_1-z_2)^l` translate of the grounded `Res_{z_0}` face
  ((8.8.12)+(8.8.30); finite commutator sum + the landed binomial finite difference binomCoeffFinDiff +
  reindexing).  So `l > 0` is NOT a residual --- batch 202's `l = -c` faces cover only the GENERATOR chamber with
  `0 <= <a,b>` (MMFLM810NegCFace.lean:422), and `l > 0` is a different sign (crossBracket_swap flips the states and
  variables, not `l`).
* the named skeleton: FacesAtStates (= the target, `JacobiEvenCocycle <-> FacesAtStates` Iff.rfl),
  `faces_iff_split`, `corollary851_iff_zeroFace` (both ways), `FaceWindows`/`faceWindows_zAligned`
  ((8.10.8)'s finiteness = the landed (8.10.5), a theorem), ZeroFaces/PositiveFaces/NegativeFaces,
  NegativeOnGenerators, **NegativeResidual E** (the `l < 0` faces at pairs NOT lattice generators with
  `0 <= <a,b>`) and GroundedZeroFace.
* **THE COMPOSITION**: faces_of_groundedZeroFace, stepCorrected_of_groundedZeroFace, and
  stepCorrected_of_spanningEventual: the package's own Corollary851SpanningEventualWindows + NegativeResidual imply
  EvenCocycleJacobiStepCorrected.  Also groundedZeroFace_of_spanningEventual (spanning machinery + bilinearity of
  iterateOf; iterateOf_eq_zero_of_mem_span).
* **FINDINGS**: (1) batch 202's "every l = -c face" is only the generator chamber --- the residual is STRICTLY
  larger; (2) the step's stated hypothesis Corollary851AllStatesExists (an exists-K form) is STRICTLY WEAKER than
  the grounded form the assembly consumes (a window below the (8.10.5) threshold cannot be raised: the campaign's
  own not_corollary851_succ_of_first_ne_zero); the missing passage is named WindowSyncAtStates and is closed by the
  package for the campaign; (3) the section 8.10 route is NOT shorter (its Props are unproved `def ... : Prop` and
  its landed calculus is the `l = 0` coefficient alone); (4) the `l >= 0` generator chambers (batch 187) and the
  finiteness are theorems, so the `l >= 0` content costs nothing; (5) `ZAligned` is defined twice (FLM810JF2 and
  FLM810JZF, identical bodies) --- the files set maxHeartbeats 600000 for that unification.
* The step is NOT proved outright: the `l < 0` normal-ordering engine at non-generator states / negative pairings
  remains open (`NegativeResidual E`).

**B — the commutator half is the windows statement** (Definitions/MMFLM86Insertion8625ProveAux.lean 352 +
...ProveFalsify.lean 149 + ...Prove.lean 351).

* honestIns8625 E a b c r i j = iterateOf (ZAligned E) a b i j (the honest (8.6.24) insertion; the (8.6.4) labels
  c, r are inert), so `IterateInsertion8625 E (honestIns8625 E)` is rfl --- **the iterate half constrains nothing**;
* modeFieldCommInsertionStep8625_iff_commutator: the (8.6.9)/(8.6.10) guard carries no content (batch 219-B's
  modeFieldCommInput8625 is a theorem);
* **HEADLINE (BOTH WAYS)**: modeFieldCommInsertionStep8625_iff_spanningEventual :
  `ModeFieldCommInsertionStep8625 E (honestIns8625 E) <-> Corollary851SpanningEventualWindows (ZAligned E)` --- the
  target IS the batch-220-B window-synchronization residual of (8.5.13); plus insertionEventualWindows_iff_spanningEventual
  (all-states vs spanning spellings), the window-lowering lemmas (corollary851Residue_eq_of_iterate_eq_zero_of_le,
  corollary851_of_grounded_of_le --- the converse of the landed Corollary851.of_cutoff) and the grounded/eventual
  insertion predicates;
* **REDUNDANCY FINDING**: insertionConjunct8625_of_spanningEventual shows the package conjunct
  `exists Ins, Iterate /\ Step` is IMPLIED by the package's own windows conjunct, hence
  **evenCocycleResidualsCorrected_iff_matching_and_windows : EvenCocycleResidualsCorrected C h2 <->
  ResidueMatching8625 E /\ Corollary851SpanningEventualWindows (ZAligned E)** (and the normal form drops the
  insertion conjunct); the boundary is re-landed on the reduced hypothesis (corollary851AllStatesExists_of_matching_and_windows,
  theorem8102_of_matching_and_windows);
* falsification (non-vacuum, non-generator): honestIns8625_ne_zero (nonzero at the (8.5.11) boundary index) and
  **not_commutatorInsertion8625_zeroIns_rankOne** (the zero insertion FAILS the commutator half at the landed rank-one
  datum) --- the target is not a tautology; the probe re-checks the (8.6.21)--(8.6.24) reassembly
  (deltaExpCoeff i (i-m-1) = C(m,i)) and reproduces the landed trivial-cocycle refutation on
  `alpha_1(-1) iota(3)`;
* corrections: the ledger DOUBLE-COUNTED ModeFieldCommInsertionStep8625 and Corollary851SpanningEventualWindows
  (they are equivalent, so item (i) has ONE residual); batch 222-A's "two insertion halves (open)" is wrong for the
  iterate half (rfl); the brief's route via (8.4.25)/(8.4.36) was mis-attributed (the proof consumes only
  modeFieldCommInput8625, (8.10.5), Corollary851.of_cutoff, iterateOf_eq_zero_of_corollary851_succ,
  exists_stateFinset_mem_span, corollary851_of_mem_span).

**EXACT RESIDUAL after batch 223**: (1) `Corollary851SpanningEventualWindows (ZAligned E)` (the package's own
conjunct); (2) `MM.FLM810ASM.NegativeResidual E` (the `l < 0` faces at non-generator/negative-pairing pairs --- the
normal-ordering engine at non-generator states); (3) the Fock named residuals (`HonestNegPairingCoeffAtStates`,
`ResidueMatching8625NegSecondSlot`); (4) the six boundary inputs.


**HANDOVER**: the state of the campaign as of batch 223 (delivered reductions, the remaining book formulas, the refuted-spelling blacklist, the module map and the verification discipline) is summarised in `HANDOVER.md` at the repository root.

---

### Batch 224 — TARGET: residual #1 SHARPENED: the (8.5.13) window synchronization IS the grounded one-mode step, and the (8.5.4) lattice corner is a THEOREM

**A — the sharpening** (Definitions/MMFLM86WindowSync.lean 348 + MMFLM86WindowSyncDelivery.lean 159).

* the three predicates: **GroundedPairWindow** (a window at which (8.5.13) holds AND at which the (8.5.24)
  iterate family already vanishes — FLM's finiteness (8.10.5) at that pair, so the truncated residue is the
  complete sum (8.5.26)); **GroundedLatticeCorner**; and the sharpened residual **GroundedModeStepAtStates**
  (“groundedness at a (8.5.4) pair is inherited by each of the pair's two one-creation-mode descendants
  `alpha_c(-r)u`, `r >= 1`”, the step CONSUMING its own grounded input — exactly the form FLM's induction
  (8.6.4)--(8.6.27) has to supply);
* **THE (8.5.4) LATTICE CORNER IS A THEOREM, UNCONDITIONALLY AT THE EVEN COCYCLE**:
  groundedLatticeCorner_evenCocycle — the (8.5.13) half is the landed batch-204-A
  corollary851_lattice_of_canonical_le, whose negative-pairing input is the PROVED
  MM.FLM86NTP.residueNegativeAligned_evenCocycle (so no negative-pairing hypothesis appears), and the
  vanishing half is (8.5.11) at the honest field (MM.FLM810JF2.stateVanishesBelow_ZAligned) read through the
  (8.5.24) iterate.  The cement of the (8.5.4) induction is therefore already in place;
* **THE EQUIVALENCE, BOTH WAYS AND UNCONDITIONALLY**:
  **spanningEventual_zAligned_iff_groundedStep : Corollary851SpanningEventualWindows (ZAligned E) <->
  GroundedModeStepAtStates (ZAligned E)**, with no hypothesis beyond the even cocycle, `(2:F) != 0` and the
  symmetry of the form.  Forward (groundedModeStep_of_spanningEventual): the one-mode successor of a (8.5.4)
  pair is again a (8.5.4) pair (`stateOfVL (⟨(r,c) :: l, a⟩)` is `alpha_c(-r) stateOfVL (⟨l,a⟩)` by the rfl
  unfolding stateOfVL_cons), so the residual applies to it directly — no induction.  Backward
  (spanningEventual_of_groundedModeStep): the (8.5.4) list induction of the landed twin
  corollary851_spanningStates_exists_of_lattice_and_step, re-run in the grounded currency, each step
  consuming the grounded window its own induction hypothesis produced for the input pair.  The intermediate
  form is spanningEventual_iff_groundedStep_of_corner, which keeps the corner as a hypothesis;
* **THE WINDOW-FREE CURRENCY IS STRICTLY WEAKER, ON RECORD**: groundedStep_imp_windowFree drops the vanishing
  conjunct (the converse is NOT claimed and is exactly the batch-223-A WindowSyncAtStates gap: a window below
  the (8.10.5) threshold cannot be raised, the machine-checked
  MM.FLM86SL.not_corollary851_succ_of_first_ne_zero); groundedPairWindow_lattice_nonempty exhibits the
  corner as a witness that the step's hypothesis is satisfiable — the step is not a statement about an empty
  hypothesis, and its CONCLUSION at the descendants is what stays open;
* **THE DELIVERY IS RE-HUNG ON THE SHARPENED SPELLING** (MMFLM86WindowSyncDelivery.lean, by composition of
  landed theorems only): groundedZeroFace_of_groundedStep (the assembly's grounded `Res_{z_0}` face, the
  currency whose `l >= 0` translates are the landed crossBracket_eq_crossIterate_of_nonneg_of_grounded);
  stepCorrected_of_groundedStep (the reduced Jacobi step `EvenCocycleJacobiStepCorrected` from the grounded
  step + MM.FLM810ASM.NegativeResidual); allStates_of_matching_and_groundedStep; and **
  theorem8102_of_matching_and_groundedStep** — FLM Theorem 8.10.2 with the window hypothesis replaced by the
  grounded one-mode step (the Fock matching conjunct `ResidueMatching8625` and the six boundary inputs pass
  through unchanged).
* FINALCHECK: the two modules are imported by FinalCheck.lean; the batch is accepted with the driver green.

**EXACT RESIDUAL after batch 224**: (1) **`GroundedModeStepAtStates (ZAligned E)`** — residual #1 in its
sharpened form, whose proof has to supply the (8.6.23)/(8.6.24) cross contraction at the descendant pair
(equivalently, by batch 223-B, the honest commutator half of the (8.6.25)/(8.6.27) matching);
(2) `MM.FLM810ASM.NegativeResidual E`; (3) the Fock named residuals (`HonestNegPairingCoeffAtStates`,
`ResidueMatching8625NegSecondSlot`); (4) the six boundary inputs.

---

### Batch 225 — TARGET: residual #1 IS FLM's display — the window synchronization is ABSORBED

**A — the absorption** (Definitions/MMFLM86WindowSyncAbsorbed.lean 354).

* **THE GENERAL BRIDGE BETWEEN THE CURRENCIES**: groundedPairWindow_iff_aboveThreshold: for any `N`
  above which the (8.5.24) iterate family of a pair vanishes, a grounded window of that pair IS
  (8.5.13) at EVERY window at or above `N` (raising: Corollary851.of_cutoff; lowering:
  MM.FLM86M8625P.corollary851_of_grounded_of_le --- both windows see the same complete residue),
  plus pairWindowAboveThreshold_mono.  This is the precise sense of FLM's parenthetical in (8.5.13)
  (“the window is the iterate's vanishing bound”);
* **THE NEW MATHEMATICS**: groundedPairWindow_modeVL_left_of_commutatorInsertion / ..._right_...: the
  (8.10.5) threshold of the DESCENDANT pair is ITSELF a grounded window there, and the identity at it
  is the display (8.6.23)/(8.6.24) read at the honest insertion
  MM.FLM86M8625P.honestIns8625 (whose (8.6.27) half is rfl).  No window is RAISED — so the batch-223-A
  WindowSyncAtStates obstruction (“a window below the (8.10.5) threshold cannot be raised”) is
  bypassed, not solved — and no input window is consumed;
* **HEADLINE 1**: **groundedModeStep_of_commutatorInsertion : CommutatorInsertion8625 E
  (honestIns8625 E) -> GroundedModeStepAtStates (ZAligned E)**, with NO hypothesis at all: no symmetry
  `hs` of (7.1.1), no even cocycle, no `(2:F) != 0`, no boundary input.  So residual #1's own
  hypothesis (the grounded window of the input pair) is REDUNDANT, and the whole
  window-synchronization layer of residual #1 lies INSIDE the display;
* DescendantsGroundedWindows (the hypothesis-free spelling of the conclusion) and
  descendantsGrounded_of_commutatorInsertion (the displays give it outright).  The direction is
  recorded: the step is derived from this spelling by dropping the hypothesis, NOT the other way
  round (the step consumes its input pair's window);
* **HEADLINE 2 (the accounting, both ways, at an even cocycle)**:
  **groundedModeStep_iff_commutatorInsertion_of_evenCocycle : CommutatorInsertion8625 E
  (honestIns8625 E) <-> GroundedModeStepAtStates (ZAligned E)** — composition of batch 223-B's
  commutatorInsertion8625_iff_spanningEventual with batch 224's
  spanningEventual_zAligned_iff_groundedStep (the even cocycle enters only through the lattice
  corner); and its spanning corollary spanningEventual_of_commutatorInsertion_of_evenCocycle.  **So
  residual #1 is EXACTLY FLM's (8.6.23)/(8.6.24) cross contraction, with no window bookkeeping left
  outside it**;
* the delivery re-hung on the display: groundedZeroFace_of_commutatorHalf (the assembly's grounded
  `Res_{z_0}` face), stepCorrected_of_commutatorHalf (the reduced Jacobi step from the display +
  NegativeResidual), theorem8102_of_matching_and_commutatorHalf (**FLM Theorem 8.10.2 on the display
  list**: ResidueMatching8625 + the honest commutator half + the reduced Jacobi step + the six
  boundary inputs);
* non-vacuity on record (batch 223-B's falsifications): the zero insertion FAILS the commutator half
  at the rank-one datum (MM.FLM86M8625P.not_commutatorInsertion8625_zeroIns_rankOne) while the honest
  insertion is nonzero at the (8.5.11) boundary index (honestIns8625_ne_zero) — the honestness of the
  insertion is genuinely consumed by the passage above;
* verification: tools/build.ps1 OK; forbidden_tokens all zeros; FinalCheck.lean green; every public
  declaration guarded with footprint exactly [propext, Classical.choice, Quot.sound].

**EXACT RESIDUAL after batch 225**: (1) **`CommutatorInsertion8625 E (honestIns8625 E)`** — FLM's
DISPLAYED (8.6.23)/(8.6.24): the cross contraction of the normally-ordered (8.6.9) object of
`alpha_c(-r)u` with `Y(v,z_2)`, i.e. the (8.4.25) contraction producing the (8.6.25) pairing factor on
the commutator side.  No window synchronization, no groundedness quantifier and no finiteness input is
left outside it; (2) `MM.FLM810ASM.NegativeResidual E`; (3) the Fock named residuals
(`HonestNegPairingCoeffAtStates`, `ResidueMatching8625NegSecondSlot`); (4) the six boundary inputs.

---

### Batch 226 — TARGET: residual #1 in the per-pair DISPLAY currency of (8.6.23)/(8.6.24); the per-pair bridge; the step IS the residual

**A — the display currency** (Definitions/MMFLM86DisplayPair.lean 336).

* **DisplayAtPair / DisplayAllStates**: FLM's (8.6.23)/(8.6.24) at ONE pair of states — for every window
  `K` above which the (8.5.24) iterate family of the pair vanishes, the commutator family (8.5.15) IS the
  (8.5.13) residue of the iterate at `K` — with the inert (8.6.4) labels `c, r` dropped (at the honest
  insertion they select nothing beyond the pair).  **displayAllStates_iff_commutatorInsertion8625** (both
  ways): the label-free display IS the landed `CommutatorInsertion8625 E (honestIns8625 E)`, so batch
  225's reduction is consumable and readable in the label-free currency;
* **THE PER-PAIR BRIDGE, BOTH WAYS**: **displayAtPair_iff_groundedPairWindow_of_threshold** — given any
  `N` above which the iterate family vanishes, the display at a pair IS the existence of a grounded
  window at that pair (forward: `N` itself is one; backward: raise by `Corollary851.of_cutoff`, lower by
  `MM.FLM86M8625P.corollary851_of_grounded_of_le`).  At the honest field the (8.10.5) threshold exists,
  so **displayAtPair_zAligned_iff_groundedPairWindow** and **displayAllStates_iff_groundedAllStates** hold
  unconditionally.  Per pair, the two currencies of the campaign coincide: “the window of Corollary 8.5.1
  is the iterate's threshold”, on the nose;
* **displayAtPair_lattice_evenCocycle**: the display at the LATTICE pairs is a THEOREM at the even cocycle
  (the batch-204-A corner read through the bridge).  So the display step's hypothesis is satisfiable, and
  the corner is the induction's base case in the display currency;
* **THE STEP**: **DisplayModeStep** — the display at a pair is inherited by each of the pair's two
  one-creation-mode descendants, the step consuming the display at its own input pair (no groundedness
  conjunct, no mode labels, no (8.6.4) data) — with:
  * displayModeStep_of_groundedModeStep_of_evenCocycle (grounded step ⇒ display step, via residual #1's
    honest commutator half and batch 225's descendant windows at ARBITRARY pairs);
  * groundedModeStep_of_displayModeStep_of_evenCocycle (display step ⇒ grounded step at the (8.5.4)
    pairs, via the per-pair bridge);
  * **displayModeStep_iff_groundedModeStep_of_evenCocycle** (the two steps are the same statement);
  * **HEADLINE: honestDisplay_iff_displayModeStep_of_evenCocycle :
    DisplayAllStates (ZAligned E) <-> DisplayModeStep (ZAligned E)**.  Forward: restriction.  Backward: the
    display step at the spanning pairs gives the batch-224 grounded step, hence the spanning eventual
    windows (`MM.FLM86WS.spanningEventual_of_groundedModeStep` with the landed corner), hence
    `GroundedZeroFace` at ALL states (`MM.FLM810ASG.groundedZeroFace_of_spanningEventual`), hence the
    display at every pair (the per-pair bridge, with `MM.FLM810ASM.corollary851_iff_zeroFace`);
  * displayAllStates_iff_spanningEventual_of_evenCocycle (the composition with batches 223-B/224/225);
* verification: tools/build.ps1 OK; forbidden_tokens all zeros; FinalCheck.lean green; every public
  declaration guarded with footprint exactly [propext, Classical.choice, Quot.sound].

**EXACT RESIDUAL after batch 226** (content unchanged, currency sharpened): (1) **`DisplayModeStep (ZAligned E)`
<-> `DisplayAllStates (ZAligned E)` <-> `CommutatorInsertion8625 E (honestIns8625 E)`** — FLM's displayed
(8.6.23)/(8.6.24) cross contraction, now with no groundedness conjunct, no mode labels and no inert data;
(2) `MM.FLM810ASM.NegativeResidual E`; (3) the Fock named residuals (`HonestNegPairingCoeffAtStates`,
`ResidueMatching8625NegSecondSlot`); (4) the six boundary inputs.

---

### Batch 227 — TARGET: residual #1 IS the source's Theorem 8.6.1 display (8.6.3), in the source's own carrier

**A — the bridge to the spine** (Definitions/MMFLM86SourceBridge.lean 226).

* **corollary851_iff_sourceUnshifted / corollary851_iff_sourceStatement**: at ONE pair and window, the
  factor-free source carrier `MM.FLM86SG.sourceResidue861Unshifted` IS the landed (8.5.13) (the landed
  `sourceResidue861Unshifted_eq_corollary851Residue` is `rfl`), and the carrier WITH the source's
  operator-valued grading factor `(z_1/z_2)^{a}` (7.1.34)--(7.1.35) is the same statement because the
  factor's finite-support sector sum is the identity (`sourceResidue861_eq_unshifted`).  So the spine's
  carriers introduce nothing: the two right-hand sides are the same family;
* **corollary851_iff_theorem861SourceStatement**: at a labelled (8.6.2) pair `(u' ⊗ ι(a), v' ⊗ ι(b))` and
  under the source's hypothesis (8.6.1), `MM.FLM86SG.Theorem861SourceStatement` at window `K` IS the landed
  `Corollary851` at that window (the `Prop` is the `(m,n)`-pointwise reshaping of the same operator
  identity);
* **displayAtPair_iff_forall_theorem861SourceStatement**: **residual #1 at a labelled pair** (batch 226's
  per-pair display) IS the source's (8.6.3) at EVERY window above which the (8.5.24) iterate family
  vanishes — the honest reading of the source's window `K` (its "finite sum" for the states of (8.6.2));
* **HEADLINE: displayAllStates_iff_sourceStatements_of_evenCocycle** (BOTH WAYS, at an even cocycle with
  the (8.6.1) hypothesis discharged by `MM.FLM86.hypothesis861_of_evenCocycle`):
  **`DisplayAllStates (ZAligned E)` ↔ the source's (8.6.3) at every labelled (8.6.2) state pair and every
  window above the vanishing bound**.  Forward: the per-pair bridge of batch 226 with the (8.10.5)
  threshold.  Backward: the labelled pairs include the (8.5.4) states (`MM.FLM85Un.stateOfVL_eq_tmul` with
  `u' = MM.FLM85Un.fockWord`), the display there gives the spanning grounded windows, hence the spanning
  eventual windows (`MM.FLM86SL.spanningGrounded_iff_eventual`), hence the grounded `Res_{z_0}` face at
  ALL states (`MM.FLM810ASG.groundedZeroFace_of_spanningEventual`), hence the display at every pair (the
  per-pair bridge, `MM.FLM810ASM.corollary851_iff_zeroFace`);
* **CONSEQUENCE (the point of the batch)**: the campaign's residual #1 and the spine's "single named next
  step" (8.6.3) are ONE statement, machine-checked both ways.  The source's own route to (8.6.3) — its
  (8.6.4)--(8.6.27), i.e. the OPE (8.6.12), the mode--field commutator (8.6.9)/(8.6.10) (PROVED here in
  batch 219-B) and the residue matching (8.6.21)--(8.6.27) — is available as a map, with the spine file's
  per-step status table (`MMFLM86SpineGap.lean`) for every numbered step;
* verification: tools/build.ps1 OK; forbidden_tokens all zeros; FinalCheck.lean green; every public
  declaration guarded with footprint exactly [propext, Classical.choice, Quot.sound].

**EXACT RESIDUAL after batch 227** (unchanged content, now identified with the source): (1) **the source's
(8.6.3) = `DisplayAllStates (ZAligned E)` = `DisplayModeStep (ZAligned E)` = `CommutatorInsertion8625 E
(honestIns8625 E)`** — FLM's displayed cross contraction; (2) `MM.FLM810ASM.NegativeResidual E`; (3) the Fock
named residuals (`HonestNegPairingCoeffAtStates`, `ResidueMatching8625NegSecondSlot`); (4) the six boundary
inputs.

---

### Batch 228 — TARGET: the FOCK half of (8.6.25)/(8.6.27) is ONE one-mode step (the mode action on the blocks), and its lattice base is a THEOREM

**A — the Fock half isolated** (Definitions/MMFLM86NegPairingFockStep.lean 218).

* **FaceAtFamily** (the (8.6.25) face at one finite family of (8.5.4) states, the support common to the
  family) and the definitional identification **honestNegPairingCoeffAtStates_iff_familyFaces** (the
  ledger's Fock half is the conjunction of its family faces);
* **THE ONE-MODE STEP, NAMED**: **FockStep E** — the face at a family is inherited when a LIST of
  (8.5.4) creation modes is appended to every state of the family (the list form is the one the (8.5.4)
  induction consumes, via `List.take 1`/`List.drop 1` on each datum); its one-mode instance is derived
  as **fockStep_imp_oneMode**.  This is the piece the ledger recorded as missing ('the mode action, i.e.
  its Fock second half'): its proof compares the two sides' recursions in the TEST vector — the
  (8.6.9)/(8.6.10) mode--field commutator at the inserted state on the left (PROVED for all states,
  batch 219-B) against the mode action on the normal-ordered blocks `blockOfShift E a b j 0 m Q` on the
  right, with the coefficients compared by the landed `MM.FLM86FB.flatBinomialExpansion` (batch 221-A);
* **THE BASE IS A THEOREM**: **faceAtFamily_lattice** — for a family all of whose states are lattice
  states, the face holds with the union of the canonical windows `canonWindow` of the family as the
  common support (the landed generator window `honestNegPairingCoeff_at_generators_window` + the
  vanishing of a block outside its own canonical window + `blockSum_subset`).  So adding modes to a
  family is the ONLY content left;
* **HEADLINE (both ways, `hs` only)**: **honestNegPairingCoeffAtStates_iff_fockStep** :
  `HonestNegPairingCoeffAtStates E <-> FockStep E`.  Forward: a list-extension of (8.5.4) data is again
  (8.5.4) data, so the face supplies the support.  Backward: strong induction on the total number of
  oscillators `sum_i #(D i).osc`, peeling ONE mode from every state of the family at a time; the base is
  `faceAtFamily_lattice` and each round is `FockStep`;
* **CONSEQUENCE**: the Fock half (residual #3) is now a single one-mode step, exactly as batch 218
  isolated the (8.6.9) content as `ModeFieldCommAnnStep` (proved in batch 219-B).  The lattice factor,
  the generator slice, the windows, the uniform-support bookkeeping and the (8.5.4) induction are all
  out of the residual;
* verification: tools/build.ps1 OK; forbidden_tokens all zeros; the blacklist audit (b222a_audit.py)
  reports 0 refuted names consumed; FinalCheck.lean green; every public declaration guarded with
  footprint exactly [propext, Classical.choice, Quot.sound].

**EXACT RESIDUAL after batch 228**: (1) residual #1 = the source's (8.6.3) (the display; batches 225--227);
(2) `MM.FLM810ASM.NegativeResidual E`; (3) **`MM.FLM86NPFS.FockStep E`** (the Fock half of
(8.6.25)/(8.6.27), = `HonestNegPairingCoeffAtStates`, = `HonestNegPairingCoeff`), whose proof is the mode
action on the normal-ordered blocks; (4) `MM.FLM86M8625.ResidueMatching8625NegSecondSlot`; (5) the six
boundary inputs.

### Batch 229 — TARGET: FLM Theorem 8.6.1 and Theorem 8.8.9 + Remark 8.8.11 as THEOREMS, with the open input per statement closed by a REGISTERED `sorry`

Owner's directive: "未证明部分使用sorry" --- the two source theorems were to exist as Lean theorems at
the boundary field of Theorem 8.10.2, with the unproved part closed by a registered `sorry` (not
hidden behind a description file and not pretended away).

**Delivered** (`Definitions/MMFLM810Theorem861Stated.lean`, namespace `MM.FLM810T861S`):
* `displayAllStates_evenCocycle (C h2 hs) : DisplayAllStates (ZAligned (C.toCentralExtension h2))`
  --- THE one hole: residual #1 (the display (8.6.23)/(8.6.24) = Corollary 8.5.1 (8.5.13) at the
  (8.6.2) states = the source's (8.6.3)); closed by `sorry`.
* `theorem861_all_pairs` --- FLM Theorem 8.6.1, all `u, v in V_L`, at every window above the
  vanish-bound: Corollary 8.5.1 at all states, composed from the hole through the landed
  `displayAllStates_iff_sourceStatements_of_evenCocycle`.
* `theorem861_source` --- the source-faithful (8.6.2)/(8.6.3) form (`Theorem861SourceStatement`,
  grading factor retained).
* `theorem861_residual` --- the same statement in the residual currency, via
  `source861Aligned_iff_residual`.
Every declaration is guarded; its guard prints `[propext, sorryAx, Classical.choice, Quot.sound]`,
so the hole is visible in the build.

**Delivered** (`Definitions/MMFLM88Theorem889Stated.lean`, namespace `MM.FLM88T889S`):
* `theorem889_jacobi (C h2) : forall u v, exists N, JacobiIdentity (ZAligned E) u v N` --- FLM
  Theorem 8.8.9 ((8.8.29)/(8.8.30)) at the boundary field; THE one hole.  Its `l = 0` face is
  residual #1 (`MM.FLM88.corollary851_iff_jacobiIdentity_zero_face`), its `l >= 0` faces are proved
  (`MM.FLM810ASA.crossBracket_eq_crossIterate_of_nonneg_of_grounded`) and its `l = -c` generator
  chamber is proved (`MM.FLM810NCFC.negCFaceAligned_evenCocycle`), so the hole is exactly residual
  #2, `MM.FLM810ASM.NegativeResidual`; closed by `sorry`.
* `theorem889_delta` --- the equivalent form (8.8.30), from `jacobiIdentity_iff_delta`.
* `remark8811_res_z0` / `remark8811_res_z1` / `remark8812_res_z2` --- Remark 8.8.11's `Res_{z_0}`
  face, its `Res_{z_1}` face (8.8.31) and Remark 8.8.12's `Res_{z_2}` face (8.8.32), as theorems,
  from the landed `res_z0_face`/`res_z1_face`/`res_z2_face`.
* `theorem861_of_theorem889` --- the engine link, PROVED OUTRIGHT (footprint exactly the three
  standard axioms): (8.8.29) at a window gives (8.6.3) at that window
  (`source861Aligned_of_jacobiIdentity` composed with `residual_of_source861Aligned`).  The book's
  "we repeat the proof of Theorem 8.6.1" is now a checked theorem.

**Accounting.**  The residual list is UNCHANGED --- nothing that was open is claimed proved, and no
residual was weakened: the two holes ARE residual #1 and residual #2's `l < 0` part.  What is new is
that the theorems EXIST at named, guard-visible locations: `FinalCheck.lean` imports both files (it
does not guard the sorry-consuming declarations, by design) and the acceptance build is GREEN.
`Probe/forbidden_tokens.py` reads `sorry=1` for exactly these two files --- the registered holes of
`README.md`'s `sorry` map --- and all zeros elsewhere.  One ledger name was corrected on the way:
HANDOVER §2's `MM.FLM810ASM.crossBracket_eq_crossIterate_of_nonneg_of_grounded` is really
`MM.FLM810ASA.…` (file `MMFLM810AssemblyAux.lean`, namespace `FLM810ASA`).

**Residual list after batch 229**: (1) residual #1 = `MM.FLM86DP.DisplayAllStates` (the hole of
`theorem861_source`, registered `sorry`); (2) `MM.FLM810ASM.NegativeResidual` (the hole of
`theorem889_jacobi`); (3) `MM.FLM86NPFS.FockStep`; (4)
`MM.FLM86M8625.ResidueMatching8625NegSecondSlot`; (5) the six boundary inputs.
