Iteration 10 synthesis — the commutant-to-generated gap, attacked head-on (verdict unchanged, ninth consecutive confirmation)

One-line outcome. The designated verdict-bearing lever — close the iteration-9 commutant-to-generated gap and push the principal hedge to unconditional HIGH — was executed and did not fire. Two of three submitted outcome labels were corrected by the referee (A1's condition-shedding centerpiece struck as a major error; A3's RESOLVED-NEGATIVE refuted to SHARPENED-OPEN). The standing verdict is UNCHANGED, ninth consecutive confirmation; the principal hedge moves in neither grade nor condition; the carrier-convergence count stays five; no carrier lives in the gap.

Per-track verdict table.

The decisive correction (A1). The attack's verdict-bearing tail is sound and uncontested — PARTIAL unchanged, count FIVE, no localized indefinite centralizer element, the reverse-WW ⟺ carrier-no-go identification and the encoding-screen inheriting unchanged. But the centerpiece is a category error on which gap iteration 9 located. Ergodicity of the Bisognano–Wichmann vacuum (Borchers' Lebesgue boost spectrum; Longo 1978) makes the modular centralizer $\mathcal M_\omega=\mathcal M\cap\{\Delta^{it}\}'$ trivial. But the iteration-9 gap is the representation-space multiplicity gap $\langle\Delta\rangle''\subsetneq\{\Delta^{it}\}'$, driven by spectral degeneracy of $\Delta$ on $\mathcal H$. Because $\Delta\notin\mathcal M$, triviality of the centralizer says nothing about this gap, which stays non-empty even in the BW case and contains the canonical indefinite $\eta_0=\mathrm{sgn}(\ln\Delta)$. So the gap does not close, the Lemma-N1 deficiency is not repaired, and HYP-CKV-VACUITY stays at -R6 — conditional HIGH on $n_1$ AND the located commutant-to-generated gap.

What is genuinely kept (re-establishing, not advancing). Two ESTABLISHED sub-results survive A1: (1) the III₁ modular centralizer is genuinely large for degenerate spectrum — integrable weight ⟹ Type II$_\infty$ centralizer (Connes–Takesaki flow of weights), and any finite $(N,\tau)$ is realizable as a centralizer (Popa, math/0011084) — so the gap is real and wide, not closable on signature alone; (2) the state restricts to a faithful normal trace on its centralizer (Takesaki), so the centralizer is semifinite and the indefiniteness it admits is trace-balanced — signed dimension, no localization. The geometric-vs-degenerate exclusion (a geometric modular flow keeps a non-trivial centralizer only via an internal gauge multiplet, which by DHR commutes with the boost and carries zero spacetime localization) re-establishes the inherited Lemma-N2/$n_1$ obstruction.

No carrier in the gap (A2), with the vacuity surviving two corrected sub-arguments. Where geometry lives (Lebesgue modular spectrum), the centralizer is trivial — no algebra-internal carrier (Shlyakhtenko free-quasi-free structure theory; Houdayer 0809.3827 / Boutonnet–Houdayer 1602.01741 corroborate but are not the biconditional). Where the gap is non-empty (almost-periodic sector), the strongest carrier candidate the program has produced — $\mathrm{sgn}(h)$ for a centralizer $h\notin W^*(\Delta)$, indefinite signature $(8,8,0)$, and algebra-compatible by an inner automorphism since $\mathrm{sgn}(h)\in\mathcal M$ — clears the two bars ((i) indefinite, (iv) algebra-compatible) that iteration-7's $\eta_0$ failed, yet remains geometry-void ($\Delta_{\rm geo}=0$). The two corrections: its complete $\mathrm{Aut}(\mathcal M,\omega)$-invariant is the finer per-modular-eigenspace signature (not the global $(p,q)$ pair — 4 toy conjugacy classes, Skolem–Noether), still discrete/input-indexed/geometry-void; and the "boost-orbit-constant ⟹ wedge-filling" localization argument is struck as sector-inconsistent, the localization-failure surviving via the inherited $n_1$ presupposition.

The one available positive movement, and why it is only partial. Combining A1 and A2, the commutant-to-generated residual is substantially closed for algebra-internal (centralizer, Ad-inner) carriers. But a residual sliver — algebra-compatible carriers with $\mathrm{Ad}\,\eta\in\mathrm{Aut}(\mathcal M)$ yet $\eta\notin\mathcal M$, living in $\{\Delta^{it}\}'\setminus\mathcal M$ — is not discharged. So the hedge narrows substantially toward but not provably to $n_1$ alone; grade and condition both unchanged.

A3 (OP-44 R2) is verdict-neutral. The decidable D=4 cubic-coefficient sub-computation does not resolve negative: under the radial-subtracting TT projector the coincidence limit selects, the leading boost-charge coefficient vanishes at the physical $D=4$ ($m=2$), and even the submission's own $\Pi^{ST}$ Hessian has leading $s^2$-coefficient $(m-1)(m-2)/2 = 0$ at $m=2$. R2 stays SHARPENED-OPEN; the $\ell_P$ floor in 4D is not re-established; the foothold stays INFERENCE, medium.

Why the verdict stands a ninth time. The verdict-flipping condition (a referee-upheld generative, localized, algebra-compatible carrier surviving the encoding screen and the localization-is-geometry result) is exactly what A1/A2 sought in the one place it could still live — the commutant-to-generated gap — and it is not there: the gap does not close, and no carrier lives in it. The residual between HIGH and a theorem is unchanged (the net-naturality theorem + the still-open commutant-to-generated gap). Object-level still handed to experiment; the 2027 $w(z)$ decision point remains the nearest-dated external event.