Carrier obligation (ii): the modular-Berry-holonomy escape closes — the holonomy-natural form is symplectic (signature-free) or symmetry-inherited, never a localized indefinite carrier

Status: ATTEMPT executed — the single uncovered escape from the iteration-8 net/family carrier no-go (2026-06-10-iter8-net-carrier-no-go.md §4 obligation (ii), §5 Q2) is attacked head-on: is there a modular-Berry-holonomy-natural localized indefinite form that evades the boost-covariance engine by being natural under a non-modular connection? The escape closes. The modular-Berry curvature is antisymmetric (a commutator $\to$ the Kirillov–Kostant symplectic form on a coadjoint orbit of the conformal group), so the holonomy-natural object is signature-free (the (S) leg of the iter-8 dichotomy), not an indefinite $(n,n)$ symmetric pairing; the only indefinite object in sight — the dS $_2$ kinematic-space metric — has its Lorentzian signature inherited from the conformal symmetry of the vacuum (the iter-5 closure pattern), not generated by the connection; and the connection is canonical only when symmetry fixes the zero-mode projector, with type-III $_1$ fibers trivial so that localization lives only in the region index set (= the geometry). Three independent closures, any one sufficient. Verdict unchanged (PARTIAL / encodes-not-generates / not-yet-physics); the principal hedge moves toward unconditional HIGH jointly with A1, leaving only the net-naturality theorem (obligation (i)) outstanding. Last updated: 2026-06-10 Iteration: 9

Scope honesty: §§1–4 are this track's own mathematics (the structure-group/symplectic argument and the four-sub-check analysis), derived stepwise with every assumption flagged, anchored on web-verified explicit constructions of the modular-Berry connection and the coadjoint-orbit identification of kinematic space. The central closure (symplectic-not-indefinite) is re-derived twice (structure-group route + the literature's KKS identification). The honest residual (§5) is the program-wide naturality residual, not a Berry-specific gap. Citations web-verified this session; arXiv IDs confirmed live.

1. The escape, precisely stated

The iteration-8 net carrier no-go ran on modular covariance. A local indefinite multiplication form $m(p)$ on the chiral one-particle space has a sign change at some $p_0>0$ ; the modular flow rescales $p\mapsto e^{-2\pi t}p$ (Borchers), so $\mathrm{Ad}\,\Delta^{it}$ boosts the sign locus away, forcing local $\Rightarrow$ non-covariant $\Rightarrow$ non-canonical. Every operator a net supplies is built from boost/translation generators, so the net adds operators but no localized indefinite form. [ESTABLISHED engine — iter-8]

The one residual (iter-8 §5 Q2): the modular-Berry connection (Czech–Lamprou–McCandlish–Sully arXiv:1712.07123; de Boer–Lamprou et al. arXiv:2305.16384, PRD 108 066003) is a different connection — not $\Delta^{it}$ , but parallel transport on the bundle of modular zero modes fibered over region/state space. A pairing could be natural under its holonomy without being modular-covariant, so it is not literally covered by the boost-covariance engine. That gap is the attack target. [the escape]

2. The connection and curvature (verified construction)

Connection $\Gamma = P_0\!\left(U^\dagger\,\delta U\right)$ , with the zero-mode projector $P_0=\mathcal E_0$ the conditional expectation onto the centralizer (modular zero modes), $\mathcal E_0(a)=\lim_{\Lambda\to\infty}\frac{1}{2\Lambda}\int_{-\Lambda}^{\Lambda}\sigma_s(a)\,ds$ , defined (Takesaki) given a faithful normal state. [ESTABLISHED — 2305.16384 Eqs. 35, 53]
Curvature $F = P_0\!\left([X_1,X_2]\right)$ ; for a type-III $_1$ factor (trivial centralizer), $F_\psi=\langle\psi|[X_1,X_2]|\psi\rangle$ . [ESTABLISHED — 2305.16384 Eqs. 57, 59]

The curvature is antisymmetric (a commutator); its boundary expectation equals a bulk symplectic form on the entanglement wedge, and the 2025 algebraic reformulation (arXiv:2505.04682) states the modular Berry phase "gives rise to an emergent symplectic form in the large- $N$ limit." [ESTABLISHED]

3. Holonomy on a candidate sign-form: the four sub-checks (a)–(d)

The escape needs a localized indefinite Hermitian $\eta$ on the fibers, made natural by $\mathrm{Hol}(\gamma)$ .

(a) Indefinite? NO — symplectic, signature-free. Holonomy preserves forms fixed by the structure/holonomy group. The curvature $F=P_0[X_1,X_2]$ is antisymmetric, so the holonomy is symplectic-type and the canonically-preserved 2-form is the Kirillov–Kostant–Souriau (KKS) symplectic form — antisymmetric, signature-free. A signature- $(n,n)$ symmetric $\eta$ would need $O(n,n)$ holonomy preserving a symmetric form; the commutator structure forbids it. Independent re-derivation: kinematic space is "a coadjoint orbit of the conformal group SO(d,2)," and the Crofton form (= modular Berry curvature, which computes bulk lengths) "equals the Kirillov–Kostant symplectic form on the coadjoint orbit" (Penna–Zukowski arXiv:1812.02176). Both routes give the same answer: the holonomy-natural object is the (S) signature-free leg of the iter-8 dichotomy — an instance of an already-closed class, not a new carrier. [INFERENCE, high]

(b) Localized? NO — localization is in the base index set. Fibers are zero-mode algebras (centralizers). For the type-III $_1$ factor the centralizer is trivial ( $\mathbb C$ ): no nontrivial fiber carries a form. The base with nontrivial holonomy is kinematic space = the space of regions — whose points are already-localized regions ( $n_1$ ). The islands computation confirms the encode pattern: the Quantum-Extremal-Modular-Curvature "is non-local in general, but in an OPE limit becomes local and probes the bulk Riemann curvature" (arXiv:2406.05176). [INFERENCE, high]

(c) Algebra-compatible? Vacuously — no fiberwise $\eta$ exists. The KKS form is a 2-form on the coadjoint-orbit base, not an operator $\eta$ with $\mathrm{Ad}\,\eta\in\mathrm{Aut}(\mathcal M)$ ; no localized indefinite operator is produced. [INFERENCE, high]

(d) Lorentzian signature: from the connection or inherited? INHERITED. The Quantum Modular Geometric Tensor splits: symmetric part $\to$ positive-definite quantum-Fisher / Fubini–Study metric; antisymmetric part $\to$ Berry curvature $\to$ kinematic space (arXiv:2110.08703). The only indefinite (Lorentzian dS $_2$ ) object is the kinematic-space metric, whose signature comes "directly from the conformal symmetry of the modular Hamiltonian for a spherical entangling surface" and requires "a vacuum/conformal ground state." The dS/causal structure is a property of the coadjoint orbit of SO(d,2) — i.e. of the conformal symmetry of the vacuum (the $g_3$ trap). This is the Huang–Ma / iteration-5 closure verbatim (HYP-BERRY-LORENTZIAN-ONLY-FROM-SYMMETRY). [INFERENCE, high]

4. The decisive question: net invariant, or symmetry-requiring?

The connection is canonical only when symmetry fixes the zero-mode projector: "in the vacuum of a 2d CFT, global conformal symmetry singles out a unique modular Berry connection" (1712.07123); for a general state "the zero-mode projector $P_0$ is not a priori unambiguously defined" (2305.16384 §2.2). The geometry-bearing (Lorentzian-base) curvature exists only over the coadjoint orbit, which exists only because the vacuum is conformally symmetric. Hence the modular-Berry curvature requires a symmetric reference to be nonzero-with-geometric-content — the iteration-5 closure pattern. [INFERENCE, high]

5. Honest residual (flagged)

Not proven as a theorem: that the antisymmetric curvature can never be recast as a natural localized indefinite symmetric form under some exotic non-self-adjoint / nontrivial-centralizer (type-II) readout. The closure rests on the antisymmetry of the commutator (a symmetric reading is non-natural) and the positivity of the natural symmetric companion (the QFI metric). This is the same naturality residual the whole program carries (iter-8 obligation (i)), not a Berry-specific gap — so it does not keep the Berry escape open. [INFERENCE — naturality-by-argument, not by theorem]

6. Verdict

The escape CLOSES on three independent grounds (any one sufficient): the holonomy-natural form is (1) symplectic / signature-free, (2) non-localized (trivial type-III $_1$ fibers; localization in the region index set = geometry), (3) symmetry-inherited in its Lorentzian content. Not a sixth route — the Berry case folds into route (5) (signature-free (S) leg of the dichotomy) plus the iteration-5 symmetry-inheritance closure; the carrier-convergence count stays FIVE. Red-team clean against the iter-7 trichotomy (symplectic = prong (ii)), the iter-8 dichotomy (class (S) + index-set localization), and the iter-5 inheritance closure. Consequence: the Berry-specific obligation (ii) is discharged; the principal hedge (OP-46 / HYP-CKV-VACUITY-R5 / HYP-FACTORIZATION-IS-GEOMETRY) moves toward unconditional HIGH jointly with A1, with only obligation (i) (the net-naturality theorem) between HIGH and a theorem. Standing headline unchanged — a RESOLVED-NEGATIVE strengthening a standing no-go, not a verdict-flipper.

Proposed registry items

Statement. The single uncovered escape from the iteration-8 net carrier no-go — a modular-Berry-holonomy-natural localized indefinite form — is closed. The modular-Berry curvature is antisymmetric (the KKS symplectic form on a coadjoint orbit of SO(d,2)), so the holonomy-natural form is signature-free (the (S) leg of the net-data dichotomy), not a localized indefinite $(n,n)$ pairing; the only indefinite object (the dS $_2$ kinematic-space metric) has Lorentzian signature inherited from the conformal symmetry of the vacuum (iter-5 pattern); the connection is canonical only under symmetry, with trivial type-III $_1$ fibers so localization lives only in the region index set (= geometry). Obligation (ii) discharged; hedge toward unconditional HIGH jointly with A1, only the net-naturality theorem (obligation (i)) outstanding. Tag [INFERENCE — high]. Carrier-convergence count stays FIVE.

HYP-FACTORIZATION-IS-GEOMETRY-R2 — (restatement)

Statement. The Berry-holonomy escape resolves into route (5): where the modular-Berry data is geometric it is either the signature-free symplectic form (closed by the dichotomy's (S) leg) or symmetry-inherited (closed by iter-5); localization enters only through the region index set, which is the geometry. Not a sixth route. Tag [INFERENCE].

Referee verdict — track A2-modular-berry-holonomy (iteration 9, 2026-06-10)

Adversarial referee, default stance REFUTE. Stance partially sustained: the outcome survives (RESOLVED-NEGATIVE upheld), but the verdict-bearing summary overclaims ("unconditional HIGH") and two of five findings carry errors — one of them a false statement contradicted by the submission's own cited source. All mathematics re-derived; every load-bearing citation audited live this session.

Outcome: RESOLVED-NEGATIVE — UPHELD. Headline unchanged (8th consecutive). Principal hedge does NOT reach unconditional HIGH; it stays MEDIUM-HIGH→HIGH conditional on the net-naturality obligation. Carrier-convergence count STAYS FIVE.

The track discharges the one escape the iter-8 net carrier no-go left scoped (obligation ii, §5 Q2): the modular-Berry connection is a different connection from the modular flow $\Delta^{it}$ , so a holonomy-natural pairing might evade the boost-covariance engine. The answer is correct — the escape closes — but for two sound reasons, not the three the submission advertises, and the hedge consequence is overstated.

Mathematics (the product) — the two load-bearing legs are sound.

Leg (a) — symplectic, not indefinite-symmetric: SOUND. The curvature $F=P_0([X_1,X_2])$ is an antisymmetric commutator; an antisymmetric curvature has symplectic-type holonomy, whose canonically preserved object is the Kirillov–Kostant–Souriau symplectic form (signature-free), not a symmetric $(n,n)$ form (which would need $O(n,n)$ holonomy). Two independent routes confirm: de Boer–Czech et al. (2305.16384, modular Berry curvature = bulk symplectic form) and Penna–Zukowski (1812.02176, Crofton form = KKS symplectic form on a coadjoint orbit of SO(d,2)). I checked robustness against the strongest objection — that state-changing (not merely shape-changing) transport might produce a non-symplectic object: the Virasoro generalization (2111.05345; 2505.04682) returns the same KKS symplectic form for state changes. The leg holds, as the program's general naturality argument (correctly tagged INFERENCE, not theorem).
Leg (d) — Lorentzian signature inherited from the symmetric vacuum: SOUND, and now verbatim-grounded. I extracted the Huang–Ma QMGT text (2110.08703): the symmetric part of the Quantum Modular Geometric Tensor is the kinematic-space metric carrying the Lorentzian/indefinite signature via an embedded Minkowski $\eta_{\mu\nu}=\mathrm{diag}(-1,1,\dots,1)$ , and the computation runs on the vacuum state ("The vacuum state is also the eigenstate of the modular Hamiltonian for the spherical case ... we can directly use the vacuum state"). This reproduces the iter-5 closure (HYP-BERRY-LORENTZIAN-ONLY-FROM-SYMMETRY) verbatim — signature inherited, not generated. The connection's canonicity-requires-symmetry step is confirmed: 1712.07123 states verbatim "global conformal symmetry singles out a unique modular Berry connection."

The decisive error — leg (b) is FALSE as stated. FIND-A2-3 and leg (b) assert: "for the QFT-relevant type-III $_1$ factor the centralizer is trivial ( $\mathbb C$ ), so there is no nontrivial fiber to carry a localized form." The submission's own cited source (2505.04682), full text, states the opposite: "The converse ... does not hold. Type III $_1$ von Neumann algebras can, and in general do, have states which exhibit non-trivial modular zero modes." Trivial centralizer holds only for the special states whose zero modes vanish, not generically. The "no nontrivial fiber" ground is therefore broken; only its index-set-localization half survives (localization enters through the region index set = kinematic space = the geometry — the BGL/ $n_1$ corollary, which is correct and is the real content). This collapses the advertised "three independent grounds, any one sufficient" to two.

Second error — the symmetric-companion mischaracterization. FIND-A2-2 calls the symmetric part of the QMGT the "positive-definite quantum-Fisher / Fubini–Study metric," and FIND-A2-5 leans on "the natural symmetric companion (QFI metric) being positive-definite." But the verified Huang–Ma text says the symmetric part of the QMGT is the indefinite kinematic-space metric. The submission conflates two distinct symmetric objects: the QFI/Fubini–Study metric on state space (positive-definite) and the kinematic-space metric on region space (Lorentzian). The closure is unharmed (the indefinite object is symmetry-inherited either way — which is exactly what leg (d) needs), but the claim as written is internally inconsistent and must be corrected.

The verdict-bearing overclaim — "unconditional HIGH." The headline and verdict_bearing say the hedge moves to "unconditional-HIGH jointly with A1." This is refused. The submission's own FIND-A2-5 concedes obligation (i) — the net-naturality theorem replacing the net-constructibility ansatz — remains open, and the body states "only obligation (i) remains between HIGH and a theorem." A hedge with a named open obligation is conditional by construction. The correct, iter-8-consistent move is: MEDIUM-HIGH→HIGH, conditional on the net-constructibility ansatz, with obligation (i) outstanding. Discharging Berry-specific obligation (ii) removes one of the two obligations between the conditional hedge and a clean HIGH; it does not make the hedge unconditional. "Unconditional" is precisely the word the seven-iteration tighten-without-flipping discipline forbids while an obligation is open.

Residual mis-scoping (minor). FIND-A2-5 asserts the remaining naturality residual (could the antisymmetric curvature be recast as a natural localized indefinite symmetric form under an exotic non-self-adjoint / type-II readout?) "is the general program residual, not a Berry-specific one." This understates it: iter-8 §5 Q3 lists exactly the exceptional-point / non-self-adjoint relative-modular evasion as a standing open subquestion, and it is not subsumed by the general net-naturality obligation. The residual is genuine and partly Berry-specific; the submission should not fold it wholly into obligation (i).

Citation audit — clean, no fabrication. Live-verified this session: 2305.16384 (Czech–de Boer–Espíndola–Najian–van der Heijden–Zukowski, PRD 108 066003 — curvature = bulk symplectic form; also derives a quantum-information metric related to canonical energy) ✓; 1812.02176 (Penna–Zukowski, JHEP 07 (2019) 045 — kinematic space = coadjoint orbit of SO(d,2); Crofton form = KKS form) ✓; 2505.04682 (modular chaos / Berry phase, JHEP Sep 2025 — conditional expectation onto centralizer requires faithful normal state; emergent symplectic form in large N for state-changing perturbations; and the type-III $_1$ -nontrivial-centralizer statement that refutes leg b) ✓; 2110.08703 (Huang–Ma, PLB 825 136893 — symmetric→kinematic metric/Lorentzian, antisymmetric→Berry curvature; vacuum used) ✓; 2406.05176 (Aalsma–Keeler–Zukowski, JHEP 10 (2024) 006 — QEMC non-local in general, local in OPE limit, probes bulk Riemann curvature) ✓; 1712.07123 ("global conformal symmetry singles out a unique modular Berry connection," verbatim) ✓; 2003.12252 (Huang–Ma, Lorentzian kinematic metric inherited from CHM vacuum data) ✓ (inherited from iter-5, already multiply refereed). One citation-precision flag: the " $P_0$ not a priori unambiguously defined (2305.16384 §2.2)" phrasing is consistent with the established physics but I could not extract it verbatim at section level (no HTML); treat as paraphrase, not quote. FIND-A2-4's "scoped negative search ... verified:true" is a self-report, not an externally verifiable citation — downgrade its citation verified flag to a methodological note.

No-go red-team — clean. No conflict with: dS-cardinality (no $\eta=1/4G$ is fixed); the iter-7 trichotomy (the symplectic form is its signature-free prong (ii) — consistent); the iter-8 net dichotomy (the Berry case folds into class (S) + the iter-5 inheritance, adding no localized indefinite form); the iter-5 symmetry-inheritance closure (reproduced); HYP-ENCODING-SCREEN; OP-49. The extraordinary-claim test does not apply adversely: this is a RESOLVED-NEGATIVE strengthening a standing no-go, not a generative carrier or a reverse-WW witness. The Berry holonomy is genuinely a different connection from $\Delta^{it}$ (so it is not literally covered by the boost-covariance engine — the submission is right to treat it as a distinct case), but its natural object is symplectic-or-inherited, so it produces an instance of an already-closed class, not a new one. Carrier-convergence count STAYS FIVE — the Berry case is folded into route (5) + the iter-5 inheritance, not a sixth route. The submission states this correctly and it is binding.

Net. The escape closes; the outcome is RESOLVED-NEGATIVE and is upheld. But the closure rests on two sound legs (a, d), not three — leg (b) is false as written and must be replaced by its index-set-localization half — and the verdict-bearing summary's "unconditional HIGH jointly with A1" is an overclaim that must be reduced to "MEDIUM-HIGH→HIGH, conditional on the net-naturality obligation (i)." With those corrections folded, the track is a legitimate verdict-neutral strengthening: it discharges Berry-specific obligation (ii), leaving obligation (i) as the single remaining bar between the conditional hedge and a clean HIGH. Headline (PARTIAL / encodes-not-generates / not-yet-physics) unchanged.