Track G1, iteration 16: the net-naturality assault — Lemma G, the multi-wedge closure, and the (E_O) residual

Status: ATTEMPT executed on the single open analytical gate (the carrier problem / commutant-to-generated gap, dossier). Outcome: PARTIAL — a major hardening, not a full closure. Three sub-results: (G-a) the localized horn of Lemma G is proved by a short, self-contained, operator-level argument (Borchers scaling + unique vacuum + Reeh–Schlieder), upgrading the iter-10 Prong-1 [INFERENCE] to a stated-and-proved lemma; (G-b) the multi-wedge naturality lever — verified genuinely unused in iterations 9–10, which worked with a single wedge's modular data — closes the non-localized sliver the iter-10 A2 referee flagged as "not fully treated" (algebra-compatible carriers in $\{\Delta^{it}\}'$ but not in $M$): under naturality with respect to all wedge flows, every such carrier is a canonical implementer of an internal (gauge) symmetry, its signature a charge/statistics grading, $\Delta_{\rm geo}=0$; the sliver is non-empty (the free-field parity $(-1)^N$ certifies it) but exhaustively geometry-void; (G-c) the surviving residual of the whole net-naturality conjecture is located exactly: vacuum ergodicity on double-cone algebras in massive theories (hypothesis $(E_O)$ below), plus the standing $n_1$ localization-template hypothesis (unchanged, irreducible per iter-9). No verdict flip is claimed; the standing verdict is hardened on the negative side. All hedge-moving claims are flagged VERDICT-RELEVANT and are submitted to the adversarial referee, default posture DOWNGRADE. Last updated: 2026-07-03 Iteration: 16

Boundary inherited and NOT relitigated: $\eta_0=\mathrm{sgn}(\ln\Delta)$ exists smuggle-free but is geometry-void, algebra-incompatible, non-localizable (iter-7); Lemma N2, referee-upheld (iter-8/9); the Krein route CLOSED-NEGATIVE and the (relative) bicentralizer route NON-load-bearing (iter-12); the iter-10 A1 centerpiece STRUCK (ergodicity empties $\mathcal M_\omega$, not the $B(H)$ gap) and the A2 referee's residual-sliver flag — this note's target.

Scope honesty: §3 is this session's own mathematics, derived stepwise; every load-bearing step is individually graded. §4's modern-technology citations were web-verified live this session (arXiv abstract pages fetched; for arXiv:2606.23636 the PDF full text was fetched and its introduction read directly). Pre-arXiv classics (Bisognano–Wichmann, Borchers 1992, Doplicher–Longo 1984, DHR, Reeh–Schlieder, Takesaki) are cited at [ESTABLISHED — canonical, pre-arXiv, not web-verifiable] per EPISTEMICS §4 rule 2. No numerics were run this session; nothing below depends on a numerical demonstration.

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1. The question (the exact open object)

The dossier's residual, verbatim in structure: modular naturality forces a carrier $\eta$ into the commutant $\{\Delta_O^{it}\}'$ in $B(H)$, not into the abelian $W^*(\Delta_O)$; for type III$_1$ the modular spectrum is degenerate so these differ. The open question: is every algebra-compatible ($\mathrm{Ad}\,\eta\in\mathrm{Aut}(M)$), $\mathrm{Aut}(\mathcal N,\Omega)$-natural, localized carrier in that commutant forced to be a function of $\Delta$ (geometry-void, $\eta_0$-class) or signature-inherited from the pre-installed causal order — or can a genuinely new generated object live there?

Two housekeeping facts sharpen the target before any new mathematics:

(Q1) The gap is a multiplicity statement. [INFERENCE, high — standard] For a BW wedge, the Borchers commutation relation $\Delta_W^{it}U(a)\Delta_W^{-it}=U(e^{\mp2\pi t}a)$ (lightlike translations, positive generators) plus Mackey imprimitivity for the $ax{+}b$ group forces $\ln\Delta_W$ to have homogeneous Lebesgue spectrum on $\Omega^\perp$ — absolutely continuous, spectrum $\mathbb R$, and (for any theory with particle content, e.g. the free scalar field: rapidity line $\otimes$ transverse momenta $\otimes$ Fock layers) infinite uniform multiplicity. Hence $\{\Delta_W^{it}\}' \;\cong\; L^\infty(\mathbb R)\,\bar\otimes\,B(K),\qquad \dim K=\infty,$ and $W^*(\Delta_W)\cong L^\infty(\mathbb R)\otimes\mathbb C$ is very far from maximal abelian in $\{\Delta_W^{it}\}'$. The "MASA hope" (multiplicity one $\Rightarrow$ commutant $=$ functions of $\Delta$ $\Rightarrow$ gap closes identically) fails, definitively. Any closure must control the multiplicity space $K$ by additional structure, not by spectral non-degeneracy.

(Q2) The prior iterations used one wedge. Verified against the iter-9 and both iter-10 notes: Lemma N1/N2, the centralizer analysis (A2), and the struck ergodicity closure (A1) all work with a single modular operator $\Delta$ (one state, one algebra). The naturality definition (dossier §1) quantifies over all of $\mathrm{Aut}(\mathcal N,\Omega)$ — which by BW contains the modular flows of every wedge — but no prior track ever intersected the operator-level constraints of two or more wedges. The multi-wedge lever is genuinely new. [ESTABLISHED — checked against the repo this session]

One definitional fork governs everything below. The dossier defines naturality by "$\eta_{\alpha x}=\alpha_*\eta_x$ (equivalently $\mathrm{Ad}\,U_\alpha$ commutes with the family $\{\eta_x\}$)". This has two honest readings:

• (R-global) the carrier is a single form/fundamental symmetry $\eta$ on $H$ (one fiber). Equivariance then degenerates to invariance: $[\eta,U_\alpha]=0$ for every $\alpha\in\mathrm{Aut}(\mathcal N,\Omega)$ — in particular $[\eta,\Delta_W^{it}]=0$ for all wedges $W$ simultaneously.
• (R-field) the carrier is a genuine field $\{\eta_x\}_{x\in\mathcal I}$; equivariance constrains each fiber only through the stabilizer of its index ($\mathrm{Stab}(W)\supset$ own boost $+$ edge translations $+$ edge rotations; $\mathrm{Stab}(O)\supset$ rotations for a double cone), plus the consistency web across fibers (which is what Lemma N2 exploits).

Both readings are treated; neither is silently favored.

2. Lemma G — the formalization

Stated so an operator algebraist can referee it cold.

Standing hypotheses (H). $\mathcal N=\{M(O)\}_{O\in\mathcal I}$ is a local net of von Neumann algebras on a separable Hilbert space $H$, indexed by the double cones and wedges of Minkowski space $\mathbb R^{1,d-1}$ ($d\ge2$), satisfying: isotony; locality; Haag duality; the split property; Poincaré covariance by a strongly continuous positive-energy unitary representation $U$ of $\mathcal P_+^\uparrow$ with unique vacuum $\Omega$ ($0$ a simple eigenvalue of the Hamiltonian $H_0$, $U(g)\Omega=\Omega$); Reeh–Schlieder ($\Omega$ cyclic and separating for every $M(O)$, $O$ a double cone or wedge); the Bisognano–Wichmann property ($\Delta_W^{it}=U(\Lambda_W(2\pi t))$ for every wedge $W$, where $\Lambda_W$ is the boost preserving $W$); local algebras the hyperfinite type III$_1$ factor.

Lemma G (net-naturality, operator form). Let $\eta\in B(H)$ be a self-adjoint unitary ($\eta=\eta^*$, $\eta^2=1$; the fundamental symmetry of a Krein form).

(G-loc) If $[\eta,\Delta_W^{it}]=0$ for some wedge $W$ and all $t$, and $\eta\in M(O)$ for some region $O\subset W$ with $\Omega$ separating for $M(O)$ (in particular any double cone strictly inside $W$, or $W$ itself), then $\eta=\pm1$. No localized modular-natural carrier exists — indefinite or otherwise.

(G-glob) If $[\eta,\Delta_W^{it}]=0$ for all wedges $W$ (reading R-global of naturality) and $\mathrm{Ad}\,\eta(M(W_0))=M(W_0)$ for at least one wedge $W_0$ (algebra-compatibility), then: $[\eta,U(g)]=0$ for all $g\in\mathcal P_+^\uparrow$; $\eta\Omega=\pm\Omega$; $\mathrm{Ad}\,\eta$ preserves every $M(O)$ and is an internal (gauge) symmetry of the net; the $\pm$-eigenprojections $P_\pm$ of $\eta$ are Poincaré-invariant, so the signature grading carries no region dependence: $\Delta_{\rm geo}=0$. Under the split property the symmetry lies in the compact gauge group with its Doplicher–Longo canonical implementation.

(G-field) Under reading R-field: every wedge-indexed fiber $\eta_W$ (own-flow naturality $[\eta_W,\Delta_W^{it}]=0$ is forced, since $\Delta_W^{it}$ stabilizes the index $W$; localization puts $\eta_W\in M(O)\subsetneq M(W)$ or $\eta_W \in M(W)$) is $\pm1$ by (G-loc). In a conformal net the same holds for double-cone fibers ($\Delta_O$ geometric by Hislop–Longo; conformal transport of (G-loc)). The only surviving freedom is: double-cone fibers in massive nets, where $\Delta_O$ is non-geometric; such a fiber satisfies $\eta_O\in M(O)$, $[\eta_O,\Delta_O^{it}]=0$, i.e. $\eta_O$ lies in the centralizer of the vacuum on the double-cone algebra, $M(O)_\omega$. Non-scalar such $\eta_O$ exist iff the following fails:

Hypothesis $(E_O)$ OPEN. The vacuum state $\omega\restriction M(O)$ is ergodic for every double cone $O$: $M(O)_\omega=M(O)\cap\{\Delta_O^{it}\}'=\mathbb C1$.

If $(E_O)$ holds, Lemma G closes completely: the net-naturality conjecture (dossier §2) is a theorem on the stated net class.

Grading: (G-loc) [INFERENCE, high — full proof in §3.1; each step individually ESTABLISHED; awaiting adversarial referee]. (G-glob) [INFERENCE, high on the invariance/gauge chain (§3.2); the DHR "$n_1$-inheritance" framing of the conclusion is INFERENCE, medium-high]. (G-field) reduction [INFERENCE, high]; $(E_O)$ [OPEN].

3. The attack, in full detail

3.1 The localized horn (G-loc): a three-line theorem

Step 1 (Fix of the wedge flow is $\mathbb C\Omega$). Let $\Psi\in H$ with $\Delta_W^{it}\Psi=\Psi$ for all $t$. Let $U_\pm(a)$ be the two lightlike translation groups along the null directions bounding $W$; by BW and Poincaré group theory (pure computation in $\mathcal P_+^\uparrow$), $\Delta_W^{it}\,U_\pm(a)\,\Delta_W^{-it}=U_\pm(e^{\mp2\pi t}a).$ Set $f_\pm(a)=\langle\Psi,U_\pm(a)\Psi\rangle$. Invariance of $\Psi$ gives $f_\pm(a)=f_\pm(e^{\mp2\pi t}a)$ for all $t$: $f_\pm$ is constant on each open half-line. Strong continuity at $a=0$ gives $f_\pm\equiv f_\pm(0)=\|\Psi\|^2$. Equality in Cauchy–Schwarz ($|\langle\Psi,U_\pm(a)\Psi\rangle|=\|\Psi\|\,\|U_\pm(a)\Psi\|$) forces $U_\pm(a)\Psi=\lambda\Psi$ with $\lambda\|\Psi\|^2=\|\Psi\|^2$, so $U_\pm(a)\Psi=\Psi$ for all $a$. The generators satisfy $P_++P_-=2H_0$; hence $H_0\Psi=0$; by uniqueness of the vacuum ($0$ simple), $\Psi\in\mathbb C\Omega$. $\blacksquare$ [Each ingredient ESTABLISHED: the scaling relation is the geometric BW statement (and holds abstractly by Borchers' theorem, CMP 143 (1992) 315, from the spectrum condition alone); positivity of $P_\pm$ is the spectrum condition; simplicity of $\ker H_0$ is the uniqueness-of-vacuum axiom.]

Step 2 (vacuum pinning). $[\eta,\Delta_W^{it}]=0$ and $\Delta_W^{it}\Omega=\Omega$ give $\Delta_W^{it}(\eta\Omega)=\eta\Omega$; by Step 1, $\eta\Omega=c\Omega$. Since $\|\eta\Omega\|=1$ and $c=\langle\Omega,\eta\Omega\rangle=\overline{\langle\eta\Omega,\Omega\rangle}=\bar c$ (self-adjointness), $c=\pm1$.

Step 3 (Reeh–Schlieder kill). $\eta\in M(O)$ and $(\eta-c)\Omega=0$ with $\Omega$ separating for $M(O)$ give $\eta=c1=\pm1$. $\blacksquare$

Remarks (honesty). (i) This subsumes and slightly extends the known wedge-centralizer triviality $M(W)_\omega=\mathbb C1$ (the iter-10 "F3 physics fact", referee-upheld): the point missed in iterations 9–10 is that the localization condition itself confines the carrier to $M(O)\subset M(W)$, where the same two-line ergodicity argument applies — i.e. $M(O)\cap\{\Delta_W^{it}\}'\subseteq M(W)\cap\{\Delta_W^{it}\}'=M(W)_\omega=\mathbb C1$. The iter-10 A2 referee's corrected Prong-1 conclusion ("no localized indefinite carrier exists in the geometric sector") is hereby given a self-contained proof with all hypotheses displayed, and strengthened from "no indefinite localized carrier" to "no non-scalar localized natural carrier at all." (ii) The argument nowhere touches the $B(H)$-multiplicity gap of (Q1) — it does not need to. Localization + one wedge's naturality pins the carrier through the vacuum vector, bypassing the multiplicity space entirely. This is why the gap (huge as an operator-algebra object) is inaccessible to localized carriers. (iii) Consistency check against standing objects: $\eta_0=\mathrm{sgn}(\ln\Delta_W)$ evades (G-loc) because it is in no $M(O)$ (non-localizable — exactly its known defect); the iter-10 centralizer candidates $\mathrm{sgn}(h)$, $h\in M_\omega$, exist only for almost-periodic states, where the BW hypotheses fail — consistent with iter-10 Prong 2. The theorem's boundary matches the known landscape exactly. No standing result is contradicted.

3.2 The multi-wedge lever (G-glob): the non-localized sliver is exactly the gauge gradings

This is the genuinely new structural result, and the direct answer to the iter-10 A2 referee's flag ("algebra-compatible carriers ($\mathrm{Ad}\,\eta\in\mathrm{Aut}(M)$) that lie in $\{\Delta\}'$ but not in $M$ — not fully treated").

Step 1 (wedge boosts generate the Poincaré group). [ESTABLISHED — elementary Lie theory] Boosts of all orientations generate $L_+^\uparrow$ (products of two non-parallel boosts contain Wigner rotations; polar decomposition closes the argument). Translations arise from boosts of translated wedges: with $\Lambda_{W+a}(t)=T(a)\Lambda_W(t)T(-a)$, $\Lambda_{W+a}(t)\,\Lambda_W(t)^{-1}=T\big((1-\Lambda_W(t))a\big),$ and $(1-\Lambda_W(t))$ is invertible on the boost 2-plane (eigenvalues $1-e^{\pm2\pi t}\neq0$), so all translations in the boost plane are reached; varying the wedge orientation reaches all of $\mathbb R^d$. Hence the subgroup of $\mathcal P_+^\uparrow$ generated by $\{\Lambda_W(t): W \text{ wedge},\, t\in\mathbb R\}$ is all of $\mathcal P_+^\uparrow$. Under BW, $\Delta_W^{it}=U(\Lambda_W(2\pi t))$, so $[\eta,\Delta_W^{it}]=0\ \forall W,t \quad\Longrightarrow\quad [\eta,U(g)]=0\ \forall g\in\mathcal P_+^\uparrow.$ (Corroborated structurally by Brunetti–Guido–Longo modular localization, arXiv:math-ph/0203021, where the Poincaré representation is constructed from the wedge modular data — see §7.)

Step 2 (vacuum pinning). By Step 1 of §3.1 applied to any one wedge, $\eta\Omega=\pm\Omega$; WLOG $\eta\Omega=\Omega$ (else pass to $-\eta$, same form up to overall sign).

Step 3 (net-compatibility from one wedge). Assume $\mathrm{Ad}\,\eta(M(W_0))=M(W_0)$. Since $[\eta,U(g)]=0$, for every $g$: $\mathrm{Ad}\,\eta(M(gW_0))=\mathrm{Ad}\,\eta\,\mathrm{Ad}\,U(g)(M(W_0))=\mathrm{Ad}\,U(g)\,\mathrm{Ad}\,\eta(M(W_0))=M(gW_0)$ — all wedges are preserved; by Haag duality $M(O)=\bigcap_{W\supseteq O}M(W)$, all double cones are preserved. So $\gamma:=\mathrm{Ad}\,\eta$ is a net automorphism commuting with the covariance representation and fixing $\Omega$: an internal (gauge) symmetry in the standard sense. Under the split property the group of such symmetries is the compact gauge group with the Doplicher–Longo canonical (split/type-I-interpolation) implementation [ESTABLISHED — Doplicher–Longo, Invent. Math. 75 (1984) 493; pre-arXiv, canonical]; this identifies $\eta$ (given Step 2's vacuum fixing) with the canonical implementer of an involutive gauge symmetry.

Step 4 (geometry audit: $\Delta_{\rm geo}=0$). The spectral projections $P_\pm=(1\pm\eta)/2$ commute with all $U(g)$: the Krein decomposition $H=H_+\oplus H_-$ is Poincaré-invariant, hence assigns nothing to any region; every $\mathrm{Aut}(\mathcal N,\Omega)$-invariant extractable from $(\eta,\mathcal N,\Omega)$ is a global charge/grading datum. In DHR terms, involutive gauge gradings of a Haag-dual split net are superselection/statistics data (Bose–Fermi-type gradings), and the DHR analysis derives that structure from the causal-disjointness relation of the index set — i.e. the sliver's signature content is inherited from the pre-installed causal order $n_1$, in precisely the dossier-§2 sense of "signature-inherited". [INFERENCE, medium-high — the DHR framing; the Poincaré-invariance of $P_\pm$ itself is immediate from Step 1]

Step 5 (the sliver is non-empty — the certifying example). In the free scalar net, the Fock parity $\eta=(-1)^N$ is: self-adjoint unitary; commutes with all $U(g)$, hence with all $\Delta_W^{it}$ and all $J_W$; fixes $\Omega$; implements the internal $\mathbb Z_2$ ($\phi\mapsto-\phi$), preserving every $M(O)$; is indefinite with signature $(\infty,\infty)$ (even/odd Fock grading); is not a function of any $\Delta_W$ (it acts within the multiplicity space $K$ of (Q1)); and is in no $M(W)$ and no $M(W)'$ (it acts nontrivially on both). So the commutant-to-generated gap does host objects beyond $W^*(\Delta)$, even fully natural ones — the honest answer to "can a genuinely new object live there?" is yes, objects live there — but Steps 1–4 classify them exhaustively: they are gauge gradings, geometry-void, their indefiniteness indexed by internal charge/statistics, never by a subregion. [ESTABLISHED — direct computation in Fock space]

Conclusion (G-glob). Under full (R-global) naturality the trichotomy of the conjecture is proved: localized $\Rightarrow$ scalar (G-loc, using any one wedge — a fortiori under all wedges); non-localized + algebra-compatible $\Rightarrow$ gauge grading, geometry-void, $n_1$-inherited signature. Never indefinite and localized. The iter-10 referee's sliver is treated.

3.3 The field reading (G-field): where the residual truly lives

Under (R-field), equivariance constrains the fiber $\eta_x$ only through $\mathrm{Stab}(x)$:

• Wedge fibers. $\mathrm{Stab}(W)$ contains the boost $\Lambda_W$; own-flow naturality $[\eta_W,\Delta_W^{it}]=0$ is therefore forced by naturality itself (not an extra assumption). Localization ((loc): $\eta_W$ affiliated to $M(O)\subsetneq M(W)$, or even just $\eta_W\in M(W)$) triggers (G-loc): every wedge fiber is $\pm1$. The equivariant field $W\mapsto\mathrm{sgn}(\ln\Delta_W)$ ($\eta_0$ as a field) survives exactly because it is localized in no $M(O)$ — reconfirming iter-7/9 from a new direction.
• Conformal double cones. In a conformal net $\Delta_O$ is geometric (Hislop–Longo / Longo's conformal-net axioms); a double cone is conformally a wedge, and the (G-loc) argument transports (the conformal group supplies one-parameter subgroups with the same scaling relation inside $O$, and the conformal vacuum is unique): double-cone fibers are $\pm1$ too. For conformal nets, Lemma G closes entirely. [INFERENCE, high — transport argument; not spelled out at BW-level rigor this session]
• Massive double cones — the residual. $\mathrm{Stab}(O)\cap\mathcal P_+^\uparrow=SO(d-1)$ (rotations); naturality forces only $[\eta_O,\Delta_O^{it}]=0$ (own, non-geometric flow) and rotation invariance. Then $\eta_O\in M(O)\cap\{\Delta_O^{it}\}'=M(O)_\omega$, the centralizer of the vacuum on the double-cone algebra — and here the (G-loc) engine has no fuel: no Borchers translation semigroup maps a bounded $O$ into itself, so $\mathrm{Fix}(\Delta_O^{it})=\mathbb C\Omega$ is not provable by scaling. The existence of a non-scalar fiber is exactly the failure of hypothesis $(E_O)$ (vacuum ergodicity on double cones). This is the entire analytic residual of the carrier problem after this track.

Counterexample analysis (what a gap-living carrier must now look like). [INFERENCE, high] Any surviving generative carrier must be: (i) a field over double cones only (all wedge fibers scalar — so it has no wedge-level signature, already in tension with reading any Lorentzian metric off it); (ii) each fiber a non-scalar involution in $M(O)_\omega$ — requiring the massive vacuum to have an exact modular symmetry of a bounded region, i.e. $(E_O)$ false; (iii) even then, its per-fiber invariant is (by the iter-10 A2 referee-corrected classification, inherited) a sign pattern resolved per modular eigenspace of $\Delta_O$ — discrete data indexed by the input spectrum, choice-dependent, non-metric: $\Delta_{\rm geo}=0$ again; and (iv) assembling fibers into a localized indefinite field re-triggers Lemma N2's sign-locus obstruction and the $n_1$ template. The residual is thus double-walled: $(E_O)$ (existence wall, [OPEN]) and the centralizer-sign vacuity (generativity wall, [INFERENCE, high], inherited-with-correction from iter-10). A counterexample must breach both.

4. What modern technology gives (tool-by-tool, verified)

1. Marrakchi–Vaes, arXiv:2305.14217 (Crelle 809 (2024) 247–260) — verified live; abstract quoted verbatim in §7. Ergodic states form a dense $G_\delta$ among faithful normal states on any III$_1$ factor with separable predual. Yield for the gap: informative-negative. Genericity cannot pin the specific vacuum state on $M(O)$: $(E_O)$ is a specific-state question structurally outside the theorem's reach. It does make $(E_O)$ plausible (ergodicity is generic and the vacuum has no known reason to be exceptional on a bounded region) — grade of that plausibility: [SPECULATIVE]. It also confirms the landscape: nothing about $B(H)$-level multiplicity, consistent with the iter-10 referee.

2. Marrakchi, arXiv:2606.23636 (22 Jun 2026, 12pp) — verified live; PDF full text fetched; introduction read directly; Theorems A/B quoted in §7. Proof method, from the paper's own introduction: the weak relative Dixmier property for the inclusion + construction of a binormal state $\Phi\in B(L^2(M))_*$ encoding it + the ultrapower implementation of binormal states technique of [Ma25], with the key novelty the approximate eigenstate technique of [Ma18] used to make $\Phi$ well-behaved with respect to the modular operator ($\Phi(f(\log\Delta_\varphi))=f(0)$) — explicitly not a spectral-gap argument, and easier than Haagerup's maximality argument. Yield: FAILED for the gap, informative. The technique operates on $B(L^2(M))$ but its conclusions target the relative bicentralizer $B(N\subset M,\varphi)\subseteq M$ and modular intertwiners in $M$ ($xy=\sigma_t^\varphi(y)x$, Theorem B) — all algebra-internal objects; the carrier-gap object lives in $\{\Delta^{it}\}'\setminus M$. Consistent with the standing iter-12 non-load-bearing status of the bicentralizer route. One genuinely suggestive transferable: the discipline of pinning a state at the modular-spectral point 0 is the same move as our vacuum pinning (§3.1 Step 2) — the parallel is structural, not logical. Its Theorem E (ergodic $\sigma^\varphi$ with globally invariant $N$ $\Leftrightarrow$ irreducibility of the core inclusion $c(N)\subset c(M)$) is the closest published statement to $(E_O)$-type questions and is flagged to the watchlist.

3. Houdayer–Marrakchi, arXiv:2511.11409 (selflessness) — verified live; abstract quoted in §7. Trivial bicentralizer $\Rightarrow$ $(M,\varphi)$ selfless for every faithful normal state. Since the hyperfinite III$_1$ factor has trivial bicentralizer (Haagerup 1985), the wedge/double-cone vacua are selfless W$^*$-probability spaces. Yield: FAILED-INFORMATIVE. Abstract-level content is again $M$-internal; no statement about $\{\Delta^{it}\}'$ in $B(H)$ or about centralizers of specific states was extractable from the abstract, and the body was not load-bearing for any step above. Flagged as a candidate future tool for $(E_O)$ (selflessness is a strong absence-of-inner-symmetry property; whether it constrains $M(O)_\omega$ for the vacuum specifically is a question for the watchlist).

4. Continuous core / Takesaki duality. The core $c(M)=M\rtimes_\sigma\mathbb R$ with its dual trace re-packages the algebra side (centralizer, flow of weights) — precisely the side iterations 10 closed. The $B(H)$-multiplicity gap is invisible to the core ($B(H)$ is type I; the crossed-product dualities do not act on the representation-space multiplicity). Yield: nothing beyond iter-10; the §3 route bypasses it. [INFERENCE, high]

5. Longo canonical endomorphism / Doplicher–Longo standard split inclusion. Load-bearing, positively: supplies the compactness and canonical implementation of the gauge group in (G-glob) Step 3 [ESTABLISHED — DL, Invent. Math. 75 (1984); pre-arXiv]. The infinite-index subfactor theory of $M(O)\subset M(W)$ was not needed: the vacuum-vector pinning replaces index technology.

6. Almost-periodic weights / Sd–T invariants. The BW wedge $\Delta_W$ is purely absolutely continuous on $\Omega^\perp$ (Q1) — no almost-periodic escape exists for the vacuum on wedges; an almost-periodic weight is a different state, and iter-10 Prong 2 (referee-corrected) already graded that sector geometry-void. The residual $(E_O)$ is exactly the question of whether the vacuum on a double cone secretly has a point-spectrum/centralizer component — connecting the carrier problem to the (famously unknown) modular spectral theory of massive double-cone algebras. Yield: reframing of the residual as a concrete spectral question.

5. Outcome

PARTIAL — major negative-side hardening; no closure; no flip.

Severity. The dossier's "commutant-to-generated gap" residual, as posed over $\{\Delta_O^{it}\}'$ in $B(H)$, is now resolved in the negative on every branch except one: the localized branch dies by (G-loc); the non-localized algebra-compatible branch is exhaustively the gauge-grading class (G-glob), which is geometry-void and — via DHR — signature-inherited from the causal order, i.e. the conjecture's own "signature-inherited" horn; the field branch survives only on massive double-cone fibers conditional on $\neg(E_O)$, and even there generativity is separately walled off by the inherited centralizer-sign classification. Honest failure location: a full theorem is blocked exactly at $(E_O)$ — the ergodicity of the vacuum on a bounded region in a massive theory — a clean, externally attackable operator-algebra/AQFT question that current technology (genericity results, bicentralizer flow, selflessness) provably does not decide (§4.1–4.3).

VERDICT-RELEVANT claims (to face the adversarial referee):

1. (G-loc)+(G-glob) restructure the principal hedge's condition: from "the net-naturality theorem AND the located commutant-to-generated gap" (R6) to "$n_1$ AND the $(E_O)$ sliver" — a strict narrowing of the second condition, not its discharge. The verdict itself ("encodes, does not generate") is unchanged; this hardens it. Referee checkpoints: (a) the Fix-lemma chain (§3.1 Step 1 — scaling relation scope, C–S equality step, simple $\ker H_0$); (b) the naturality-reading fork (§1) — that (R-global) invariance is the honest degeneration of the dossier's equivariance for a single form; (c) the wedge-boost generation computation (§3.2 Step 1); (d) whether "affiliated" could smuggle unbounded forms evading $\eta^2=1$ boundedness (we claim not: a fundamental symmetry is bounded by definition); (e) the Doplicher–Longo compactness usage (structural only; the $\Delta_{\rm geo}=0$ conclusion needs only Poincaré-invariance of $P_\pm$).
2. No RESOLVED-POSITIVE anywhere; no new route — this is route (5)'s operator-level completion; the carrier-convergence count stays FIVE.

6. Consequences proposed for the wiki (all referee-gated)

1. OPEN_PROBLEMS / carrier dossier §3: replace the residual item "(ii) the commutant-to-generated gap" by the two-branch statement: (ii-a) closed for localized carriers (Lemma G-loc) and for the non-localized natural sliver (Lemma G-glob: gauge gradings, geometry-void); (ii-b) surviving sliver: massive double-cone fibers, equivalent to $\neg(E_O)$. Register $(E_O)$ as a named open hypothesis with its own entry.
2. HYP-CKV-VACUITY-R6 → R7 (proposed, referee-gated): condition narrows from $\{n_1,\text{commutant gap}\}$ to $\{n_1, (E_O)\text{-sliver}\}$; grade unchanged (conditional HIGH). Note the house history: two prior R7 mints were struck; this one differs by carrying a written proof for the closure branches and by NOT claiming the gap "empty" — the gap is non-empty ($(-1)^N$, $\eta_0$) and classified.
3. New registry lemma LEM-NET-NATURALITY-G: statement = §2 Lemma G verbatim, grade [INFERENCE, high] until refereed.
4. EXPERIMENT/EXTERNAL WATCHLIST: add (a) $(E_O)$ — modular spectral theory / centralizer of the vacuum on massive double-cone algebras (adjacent published tech: Marrakchi 2606.23636 Theorem E on core-irreducibility $\Leftrightarrow$ invariant ergodic states; selflessness 2511.11409); (b) any conformal-to-massive transport result for double-cone modular flows.
5. GAPS_AND_CONTRADICTIONS: the carrier problem's classification is "unsolved-but-consistent," now with the unsolved part reduced to $(E_O)$ + $n_1$; no logical inconsistency introduced.

7. Verified-citation ledger (this session, live)

Pre-arXiv canonical sources used without live verification (per EPISTEMICS §4.2, real landmark results, no fabrication): Bisognano–Wichmann, J. Math. Phys. 16 (1975) 985 & 17 (1976) 303; H.-J. Borchers, The CPT-theorem in two-dimensional theories of local observables, CMP 143 (1992) 315; S. Doplicher, R. Longo, Standard and split inclusions of von Neumann algebras, Invent. Math. 75 (1984) 493; Doplicher–Haag–Roberts, CMP 23 (1971) 199 & 35 (1974) 49; Reeh–Schlieder (1961); Takesaki, Theory of Operator Algebras II (trace on the centralizer); Hislop–Longo, CMP 84 (1982) 71 (double-cone modular flow, conformal case). Marked [ESTABLISHED — canonical, pre-arXiv].

See also

• 2026-06-19-iter15-carrier-problem-dossier.md — the posed problem this note attacks
• 2026-06-10-iter9-carrier-naturality.md — Lemma N1/N2; the commutant-to-generated gap (referee correction)
• 2026-06-10-iter10-carrier-in-the-gap.md · 2026-06-10-iter10-naturality-commutant-gap.md — the two iter-10 tracks; the struck closure; the sliver flag this note answers
• ../EPISTEMICS.md — tag discipline governing every grade above

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Referee verdict — R1 (binding)

Adversarial referee R1, iteration 16. Default stance REFUTE. Prior: two previous "gap closes / hedge advances" submissions (iter-9 Lemma N1; iter-10 A1 ergodicity closure) were STRUCK, and the iter-10 A2 R7 proposal was struck-in-part; that history was applied as the operative bar. Every mathematical step of §§3.1–3.3 was independently re-derived from scratch; claims (1) and (4) were checked against the binding iter-9/10 records and the dossier; the G3 draft §2 was cross-checked for consistency; the two load-bearing pre-arXiv citations were web-verified live this session.

STANCE: UPHELD-WITH-CORRECTIONS. The two closure lemmas (G-loc, G-glob core) survive full adversarial re-derivation at proof grade — the first hedge-moving submission in this program's carrier track to do so. Seven binding corrections, all precision-level, none load-bearing. HYP-CKV-VACUITY-R7 IS MINTED, with the corrected condition string below. Verdict unchanged; no flip; count stays FIVE.

A. What is sound and verified (re-derivations)

A1. Fix(Δ_W^{it}) = ℂΩ (§3.1 Step 1) — re-derived, correct, proof-grade. I reproduced the argument independently: (i) the Borchers/BW commutation relation Δ_W^{it}U_±(a)Δ_W^{-it} = U_±(e^{∓2πt}a) is the geometric BW statement for the two lightlike translation groups along the wedge's null bounding directions, and holds abstractly (spectrum condition + half-sided translations) by Borchers' theorem, CMP 143 (1992) 315 — web-verified live this session (Springer/Project Euclid, vol. 143, pp. 315–332). (ii) Invariance of Ψ gives f_±(a) = f_±(e^{∓2πt}a), so f_± is constant on each open half-line {a>0}, {a<0}; strong continuity at a=0 pins both constants to ‖Ψ‖². (iii) |⟨Ψ,U_±(a)Ψ⟩| = ‖Ψ‖² = ‖Ψ‖‖U_±(a)Ψ‖ is the equality case of Cauchy–Schwarz, forcing U_±(a)Ψ = λΨ, and ⟨Ψ,λΨ⟩ = ‖Ψ‖² forces λ = 1. (iv) With P_± = H₀ ± P¹ ≥ 0 and P_±Ψ = 0, H₀Ψ = 0; simplicity of ker H₀ (unique-vacuum axiom, explicitly in (H)) gives Ψ ∈ ℂΩ. Every step checks. The argument is real mathematics of the standard wedge-ergodicity type, here written self-contained with all hypotheses displayed. Positive energy enters exactly twice (Borchers' abstract route; the meaning of "unique vacuum") — as the submission says.

A2. Vacuum pinning and the Reeh–Schlieder kill (§3.1 Steps 2–3) — re-derived, correct; the domain trap is avoided. The submission commutes η only with the bounded unitaries Δ_W^{it}, never with Δ^{1/2}; for bounded η this is domain-free. ηΩ is Δ_W^{it}-fixed ⟹ ηΩ = cΩ by A1; ‖ηΩ‖ = 1 (η unitary) and c = ⟨Ω,ηΩ⟩ = c̄ (η self-adjoint) give c = ±1. No J-commutation is needed anywhere. Step 3 is exact: η − c1 ∈ M(O), (η − c1)Ω = 0, Ω separating for M(O) ⟹ η = c1. The remark-(i) inclusion chain M(O) ∩ {Δ_W^{it}}′ ⊆ M(W)_ω = ℂ1 (isotony + wedge ergodicity) is also correct and is the cleanest statement of why the B(H)-multiplicity gap (Q1) is inaccessible to localized carriers. (G-loc) is a theorem on the stated hypotheses.

A3. The group-generation identity (§3.2 Step 1) — verified by direct computation in 𝒫₊^↑. With Λ_{W+a}(t) = T(a)Λ_W(t)T(−a) and the semidirect-product relation Λ T(b) Λ^{-1} = T(Λb): Λ_{W+a}(t)Λ_W(t)^{-1} = T(a)·Λ_W(t)T(−a)Λ_W(t)^{-1} = T(a)T(−Λ_W(t)a) = T((1−Λ_W(t))a). ✓ On the boost 2-plane Λ_W(t) has eigenvalues e^{±2πt} on the null eigendirections, so 1−Λ_W(t) is invertible there for t ≠ 0 (eigenvalues 1−e^{±2πt} ≠ 0) — all boost-plane translations are reached; every direction of ℝ^d (including e₀) lies in some wedge's boost plane, so varying W reaches all translations. Boosts of all orientations generate L₊^↑ (Wigner rotations from non-parallel boost products; trivial in d = 2 where L₊^↑ is the boosts). Hence ⟨{Λ_W(t)}⟩ = 𝒫₊^↑, and under BW (transported to all wedges by covariance) [η,Δ_W^{it}] = 0 ∀W ⟹ [η,U(g)] = 0 ∀g ∈ 𝒫₊^↑. Correct.

A4. The gauge-symmetry chain (§3.2 Steps 2–3) — proof-grade with one hypothesis-hygiene note (C5). Poincaré transitivity on wedges + [η,U(g)] = 0 propagates Ad η-compatibility from W₀ to all wedges. The duality step M(O) = ∩{W⊇O}M(W) I re-derived: O = ∩{W ⊇ O} as regions; O′ = ∪{W⊇O}W′ (spacelike-separated double cones admit a separating wedge); wedge duality M(W′) = M(W)′ follows from BW (geometric J_W) within (H); then ∩{W⊇O}M(W) = (∨{W⊇O}M(W′))′ ⊆ M(O′)′ = M(O) by Haag duality, and ⊇ is isotony. ✓ So γ = Ad η is a net automorphism commuting with U(𝒫₊^↑) and fixing the vacuum state — an internal symmetry in the standard sense. The Doplicher–Longo compactness/canonical-implementation usage is structural only, exactly as the submission's own checkpoint (e) says; the citation is correct — web-verified live: Invent. Math. 75 (1984) 493–536. Uniqueness of the vacuum-fixing implementer up to phase holds because ∨(net) = B(H) (wedge duality + factoriality, wedges in the index set).

A5. Δ_geo = 0 (§3.2 Step 4, core) — immediate and proof-grade. P_± = (1±η)/2 commute with all U(g); the Krein decomposition is Poincaré-invariant and assigns nothing to any region. This alone carries the geometry-void conclusion. The DHR framing on top of it is a gloss — see C2.

A6. The certifying example (−1)^N (§3.2 Step 5) — verified by direct Fock-space checks. Γ(u) preserves particle number for every one-particle unitary, so [(−1)^N, U(g)] = 0; it is a self-adjoint unitary fixing Ω, implements φ → −φ preserving every M(O), has signature (∞,∞). Not a function of any Δ_W: it is −1 on the 1-particle space and +1 on the 2-particle space while spec(ln Δ_W) is all of ℝ on both layers — no single function f(Δ_W) can do that. Not in M(W) or M(W)′: if (−1)^N ∈ M(W) it would lie in M(W)_ω = ℂ1 (A2 chain) — contradiction; the M(W)′ case is symmetric via J_W-covariance (or the same centralizer argument for the commutant wedge). The sliver is non-empty and the honest answer "objects DO live in the gap" is correctly on the record — this is exactly what the two struck predecessors failed to say.

A7. The G-field reduction and the residual location (§3.3) — correct with C4. Wedge fibers: own-boost commutation is genuine Aut(𝒩,Ω)-equivariance at the stabilizer (Λ_W(t)W = W), then (G-loc) kills localized fibers. Massive double-cone fibers: η_O ∈ M(O) ∩ {Δ_O^{it}}′ = M(O)_ω, where the (G-loc) engine genuinely has no fuel (no translation semigroup maps a bounded O into itself; the Borchers scaling has no analogue). The residual is correctly located. The consistency boundary (η₀ evades by non-localizability; almost-periodic centralizer candidates evade by non-BW) matches the standing landscape with no contradiction.

A8. Cross-check against Track G3 §2 — CONSISTENT, mutually reinforcing. G3's finding ("no state on M(W) has a modular flow globally preserving M(O); the expectation is Takesaki-blocked because ambient flows are geometric and move O") concerns ambient flows on M(W). G1's (E_O) concerns the intrinsic modular flow of (M(O), ω) — a different object, well-defined regardless (Ω is cyclic-separating for M(O)). No conflict; both tracks independently locate the hard unknown at the non-geometric massive double-cone modular data.

A9. Claim (1) checked against the binding records — TRUE. Iter-9 Lemma N2 works with one flow ("the flow acts as a free translation θ ↦ θ−2πt"); Lemma N1 with one Δ_O commutant. Iter-10 A1's whole apparatus is "one state, one algebra" ($\mathcal M_\omega = \mathcal M \cap \{\Delta^{it}\}'$; the struck closure and the referee's correction are both stated for a single Δ: "$\{\Delta\}''\subsetneq\{\Delta^{it}\}'$ (on H, the actual iter-9 gap)"). Iter-10 A2 §1: "modular equivariance forces [h,Δ^{it}] = 0" — one Δ. The dossier's naturality definition quantifies over Aut(𝒩,Ω) — which contains every wedge flow by BW — but no prior track intersected the operator constraints of two wedges. The multi-wedge lever is genuinely new, and (Q2) is fairly stated.

A10. Claim (4) well-posedness and openness — CONFIRMED. (E_O) is a precise statement (triviality of the centralizer of ω↾M(O), a specific state on a specific algebra). It is not settled by any result on record or known to this referee: Marrakchi–Vaes genericity is structurally about generic states, not the vacuum (the submission itself grades the plausibility transfer SPECULATIVE — correct); Fredenhagen-type III₁-ness of M(O) is a type statement, not ergodicity; the massive double-cone modular flow is non-geometric with unknown spectrum — (E_O) is exactly a question about that unknown, consistent with the massive-theory facts and with G3. For conformal nets the Hislop–Longo geometric double-cone flow makes the (G-loc) transport plausible — but see C3. Registering (E_O) as a named OPEN hypothesis is legitimate and useful.

B. Binding corrections (numbered; precision-level, none load-bearing)

1. (C1 — the (E_O) "iff" direction.) (G-field)'s "Non-scalar such η_O exist iff (E_O) fails" is literally true only for bare centralizer elements (a vN algebra ≠ ℂ1 contains a non-scalar involution 2p−1). For carrier fibers the correct statement is directional: (E_O) ⟹ full closure (proof-grade, given G-loc and the posed modular-naturality); ¬(E_O) is necessary but NOT sufficient for a surviving natural fiber — rotation invariance (Stab(O)-equivariance), the cross-fiber consistency web (Lemma N2), and the inherited centralizer-sign vacuity walls all remain. §3.3's own "double-walled" paragraph already says this; the Lemma-G statement line must carry it too. Reword in the registry entry.
2. (C2 — DHR framing downgraded.) Step 4's "the sliver's signature content is inherited from n₁ in the DHR sense" is a gloss, downgrade INFERENCE, medium-high → INFERENCE, medium — framing only. The DHR analysis lives on observable nets and their sector theory; hypothesis (H) fixes one local (Bose) net and does not adjudicate observable-vs-field status. No equivocation error occurs in the proof (the submission never uses irreducibility-of-the-observable-net where a field-net commutant is in play), but the n₁-inheritance conclusion rests on the DHR reading and must not be recorded as theorem. The load-bearing conclusion Δ_geo = 0 stands independently on the Poincaré-invariance of P_± (A5), which is proof-grade.
3. (C3 — conformal branch is sketch-grade.) "For conformal nets, Lemma G closes entirely" rests on a conformal-transport argument the submission itself flags as "not spelled out at BW-level rigor this session." Downgrade to INFERENCE, medium-high — transport sketch. It is NOT load-bearing for the hedge (the hedge condition (E_O) quantifies over the net class; the conformal claim is only that (E_O) is provable there) — but it must not be written into the wiki as a closed branch. Registry wording: "conformal double-cone fibers: expected closed via Hislop–Longo geometric flow + transport of (G-loc); transport not yet written at proof grade."
4. (C4 — source of the double-cone commutation condition.) §3.3's "naturality forces only [η_O,Δ_O^{it}] = 0 (own, non-geometric flow)" is mis-sourced: for massive double cones Δ_O^{it} is not in Aut(𝒩,Ω) (it is non-geometric; it does not map local algebras to local algebras), so Aut(𝒩,Ω)-equivariance forces nothing about it. The commutation is the posed modular-naturality condition of the dossier's carrier definition ("(modular-natural) [η,Δ_O^{it}] = 0", a separate leg). The constraint set — hence the residual — is unchanged; repair the wording.
5. (C5 — hypothesis hygiene on (G-glob).) Algebra-compatibility is assumed at a wedge (W₀). The dossier's compatibility is stated for local algebras generally; if a sliver object's compatibility is granted only at a double cone, promoting it to a wedge needs weak additivity (M(W) = ∨_{O⊂W}M(O)), which (H) does not list. Add weak additivity to (H) in the registry statement (it is standard, and also silently backs ∨(net) = B(H) in the general-index case). As stated — compatibility at one wedge — (G-glob) is complete.
6. (C6 — Q1 scope.) (Q1)'s "infinite uniform multiplicity" is correct for theories with standard particle content but tacitly excludes vectors with spectral support on the null ray p₊ = 0 (measure-zero for mass-shell measures; conceivable in pathological cases). Q1 is housekeeping-negative (the MASA hope fails) and not load-bearing; keep at INFERENCE, high with this scope note.
7. (C7 — ledger repair, not this submission's error.) The iter-12-inherited ledger (reproduced in the G3 draft §7.6) cites "Doplicher–Longo, Invent. Math. 73 (1984) 493–536"; the correct volume is 75 — verified live this session. G1's own ledger is correct. Integrator: propagate the fix to the iter-12/G3 ledgers.

C. Adjudication of claims (1)–(5)

D. Exact registry/hedge instruction for the integrator

1. HYPOTHESES.md — mint HYP-CKV-VACUITY-R7 (replacing R6), condition string verbatim:

   "Conditional HIGH on: (i) the localization-template hypothesis n₁ (irreducible, per iter-9, unchanged); AND (ii) hypothesis (E_O) — vacuum ergodicity on massive double-cone algebras, M(O)_ω = M(O) ∩ {Δ_O^{it}}′ = ℂ1 for every double cone O. Direction: given the Lemma-G hypotheses (H, + weak additivity) and n₁, (E_O) makes the net-naturality no-go a theorem (Lemma G, referee-verified); ¬(E_O) is necessary for any counterexample fiber but NOT sufficient (rotation equivariance, the Lemma-N2 cross-fiber web, and the inherited centralizer-sign vacuity remain as generativity walls). The commutant-to-generated gap is NOT closed: it is non-empty ((−1)^N, η₀) and classified — localized branch dead (G-loc), fully-natural non-localized branch = gauge gradings with Poincaré-invariant eigenprojections, Δ_geo = 0 (G-glob)." Grade unchanged (conditional HIGH). Note in the entry: two prior R7 mints struck (iter-10); this mint differs by referee-verified proofs and by not claiming gap emptiness.

2. Register LEM-NET-NATURALITY-G: statement = §2 Lemma G with corrections C1, C4, C5 folded in; grade: (G-loc) and (G-glob) Steps 1–3 + Step-4 core [INFERENCE, high — referee-verified proof, iteration 16]; Step-4 DHR framing [INFERENCE, medium — framing]; (G-field) reduction [INFERENCE, high]; conformal transport [INFERENCE, medium-high — sketch].
3. Register (E_O) as a named open hypothesis (own entry, [OPEN]), with the adjacent-technology pointers (Marrakchi 2606.23636 Theorem E; selflessness 2511.11409) as watchlist items per §6.4.
4. OPEN_PROBLEMS / dossier §3: replace residual item (ii) per submission §6.1, but with C1's directional wording, and keep the explicit sentence "the gap is non-empty and classified, not closed."
5. FINDINGS iteration-16 update: outcome PARTIAL — negative-side hardening; verdict unchanged (would be the 15th consecutive confirmation); count stays FIVE; no RESOLVED-POSITIVE; hedge advances R6 → R7 per this verdict.
6. What the integrator must NOT write: do NOT write "the commutant-to-generated gap closes/is empty" in any form; do NOT record the conformal closure as established; do NOT record the DHR n₁-inheritance as a theorem (Δ_geo = 0 is the theorem; the inheritance is framing); do NOT phrase R7 as "condition (ii) discharged" — it is narrowed to (E_O), not discharged; do NOT import the "Invent. Math. 73" volume number (C7).

E. Smuggle / extraordinary-claim red-team

• Multi-wedge family as n₁? The wedge index set IS causal-order data — but the standing conjecture already quantifies over Aut(𝒩,Ω)-naturality on the net, and by BW every wedge flow is in Aut(𝒩,Ω) (the dossier says so explicitly, §1). Using all wedges therefore proves something about the POSED conjecture; it does not change the problem. Clean.
• No new installation. No step installs η: the objects exhibited ((−1)^N, η₀-as-field) are found and classified as geometry-void, not offered as carriers. No g₁–g₇/n₁–n₃ item enters a positive construction — there is no positive construction. Clean.
• Extraordinary-claim gate: no RESOLVED-POSITIVE, no flip, no new route (route-5 completion; count FIVE). The only verdict-relevant move is a hedge-condition narrowing, now referee-verified. Gate does not fire.
• Standing-record conflicts: none found. η₀ evades (G-loc) by non-localizability (consistent with iter-7/9); (−1)^N's existence is consistent with the iter-10 A2 referee's non-empty-sliver flag and answers it; the iter-10 binding "gap does NOT close" is respected verbatim (the gap stays non-empty); iter-12 Krein/bicentralizer closures untouched; G3's expectation-free finding reinforced (A8).

F. Verified-citation ledger additions (R1, live 2026-07-03)

Final adjudication line: UPHELD-WITH-CORRECTIONS. (1) SUSTAINED · (2) SUSTAINED (proof-grade) · (3) SUSTAINED-WITH-CORRECTION · (4) SUSTAINED-WITH-CORRECTION · (5) SUSTAINED-WITH-CORRECTION — HYP-CKV-VACUITY-R7 minted on the corrected condition string {n₁, (E_O)}; verdict unchanged; carrier-convergence count FIVE; the gap is classified, not closed.