Iter-16 G4 — Hunt for a genuinely new mathematical readout of the carrier problem

Mandate. Find a framework not tried in 15 iterations that could either (i) prove the net-naturality theorem (carrier no-go) or (ii) exhibit a generative carrier. Screen test (binding, from the iter-15 dossier §6): the candidate must have a plausible mechanism for making EITHER the causal index-set order n₁ OR the Lorentzian signature η an output of (algebra, state) data — not an index of the source category, not a datum installed on the carrier space, not a hypothesis of the reconstruction. For no-go-direction candidates the dual screen applies: the framework must be able to pose the no-go without presupposing its answer (i.e., without installing n₁/η in its own language).

Default posture honored: DOWNGRADE. Prior probability of a pass was set low; the outcome below is consistent with that prior.

Headline outcome. Eight candidate frameworks swept; seven grade REPRODUCES-SCREEN (one of them, infinite-index subfactor invariants, borderline IRRELEVANT), one grades PASS-CANDIDATE-conditional — continuous model theory of W*-probability spaces, in the no-go direction only. The deep dive (§3) locates its two obstructions precisely: the localized half of the carrier problem is not even posable in the model-theoretic language without installing n₁ as a sort (the smuggle dichotomy reappears as a choice of logical signature), but the non-localized half — the commutant-to-generated gap of Lemma N1 — is posable and yields a genuinely new, well-formed partial theorem target: the definable-carrier no-go (DCNG). Verdict unchanged; one new named lever proposed for the watch-list; one dossier-hardening consequence (Morinelli–Neeb 2024 backs the §6 "interchangeable smuggle items" claim at theorem level).

---

§1. Method

1. Inputs. Iter-15 dossier (esp. §4 routes, §5 already-tried frameworks, §6 smuggle items), iter-12 Krein closure (three collapse channels A/B/C), iter-6 generation-construction survey (four readout families; installation/selection trichotomy; smuggle checklist g₁–g₇ + the two iter-6 extensions). EPISTEMICS.md tag discipline throughout.
2. Candidate pool. The eight classes (a)–(h) from the track mandate; (h) executed as a live 2024–2026 web sweep (math.OA / math-ph / hep-th search terms as specified) whose best find is folded in as candidate (h) = Morinelli–Neeb net-to-Euler-element.
3. Grading instrument. For each candidate: what it outputs from (algebra, state); where n₁/η would have to enter; grade ∈ {PASS-CANDIDATE, REPRODUCES-SCREEN, IRRELEVANT}; one key load-bearing citation, web-verified this session (§6 ledger). REPRODUCES-SCREEN = the framework re-expresses the carrier problem with the geometric input relocated but still installed; IRRELEVANT = no mechanism even in principle.
4. Verification discipline. Eleven load-bearing citations resolved live (2026-07-03) against arXiv/publisher listings; abstract-level phrases quoted where captured. No fabricated sources; paraphrase-level verifications are marked as such in §6. Nothing below is claimed at higher tag than its verification supports.

---

§2. The graded candidate table

Per-candidate paragraphs

(a) Noncommutative Dirichlet forms / quantum Markov semigroups / first-order calculus. This is the one framework in the pool where differential structure is genuinely an output: Cipriani–Sauvageot showed a tracially symmetric completely Dirichlet form on a C*-algebra decomposes through a closable derivation valued in a Hilbert bimodule [ESTABLISHED — JFA 201 (2003)], and — decisively for our III₁ setting — Vernooij–Wirth (CMP 2023) removed the trace: "the generator of the L² implementation of a KMS-symmetric quantum Markov semigroup can be expressed as the square of a derivation with values in a Hilbert bimodule" (per the arXiv:2303.15949 abstract), so the machinery does run on the physical vacuum state of a type III₁ wedge algebra [ESTABLISHED]. Two structural facts kill the carrier reading. First, the Dirichlet/Markov property IS complete positivity: Vernooij–Wirth's proof "hinges on … a new completely positive map … essential to obtain the required inner product on the Hilbert bimodule" — the emitted tangent bimodule is positive-definite by the same axiom that makes the theory exist. An "indefinite Dirichlet form" is a contradiction in terms (loss of Markovianity/contractivity), so the only signature this calculus can ever emit is Riemannian: η would have to be installed on the bimodule by hand, which is smuggle-item η verbatim. This relocates the smuggle with unusual precision — in Dirichlet-form vocabulary, positivity (g₆) is not a hypothesis but the definition of the theory. Second, the semigroup itself is a choice (extra datum beyond (M, φ)); nothing canonical selects one, so even the Riemannian output is not (algebra, state)-canonical. No localization concept exists (no net, no n₁ mechanism). Grade: REPRODUCES-SCREEN [INFERENCE, high — the positivity-constitutive point is structural, not a survey gap].

(b) Free probability / free Araki–Woods factors. Shlyakhtenko's functor takes as input an orthogonal representation U: ℝ ↷ H_ℝ on a real Hilbert space and outputs Γ(H_ℝ, U)'' with the free quasi-free state; the factor is III₁ when U is nontrivial and non-periodic [ESTABLISHED — Pacific J. Math. 177 (1997)]. But for the BW wedge, the input U is the boost — i.e., the modular flow, equivalently the standard-subspace (BGL) datum, equivalently n₁. Free Araki–Woods is thus a free-probability second quantization of the smuggle item itself: everything geometric about the output was fed in as the one-parameter group. Its genuine value is computational, not generative: e.g., Houdayer proved free Araki–Woods factors have trivial bicentralizer (Proc. AMS 137 (2009)) — which per dossier §5 is the carrier-friendly condition and therefore non-load-bearing for the no-go. A free-probabilistic model could make the commutant-gap computable in specific examples (worth remembering as a computation platform for §3's DCNG target), but it has no mechanism to output n₁ or η. Grade: REPRODUCES-SCREEN [INFERENCE, high].

(c) Subfactor standard invariants / planar algebras / Q-systems for M(O) ⊂ M(W). The standard invariant (λ-lattice / planar algebra / Q-system) is an object of a rigid C-tensor category* — a unitary world: dagger structure, positive-definite inner products on hom-spaces, positive Markov trace. Signature-(n,n) data cannot live there; dropping unitarity lands exactly in the non-unitary/dagger-installed categorical territory already closed as iter-15 E3 (installed reflection data). For the physical inclusion M(O) ⊂ M(W) the index is infinite and there are additional degradations: the full planar-algebra structure is obstructed at infinite index (Penneys, arXiv:1110.3504, defines only a restricted rotation/planar calculus), and Longo Q-systems classify finite-index extensions (Jones–Penneys arXiv:1707.02155 give the compact W*-algebra-object generalization — still unitary). Finally the inclusion itself presupposes selecting O ⊂ W from the ordered index set: n₁ enters before the invariant is even defined. Grade: REPRODUCES-SCREEN, borderline IRRELEVANT (at infinite index the invariant that would need to carry the signature does not exist in usable form) [INFERENCE, high].

(d) Model theory / continuous logic of W-algebras.* The only candidate whose mechanism survives first contact — because it is a candidate for the no-go direction, where the screen demands not an output of n₁/η but a smuggle-free way to pose and prove their non-derivability. The infrastructure exists and is recent: σ-finite W*-probability spaces (M, φ) are axiomatizable as metric structures, with a 2025 simplification via totally bounded elements + the Rieffel–Van Daele bounded approach to modular theory (Arulseelan–Goldbring–Hart–Sinclair, arXiv:2501.14153), extended in 2025 to a "language and axiomatization of full left Hilbert algebras" with well-behaved (Ocneanu-type) ultraproducts (Arulseelan, arXiv:2508.17241) [ESTABLISHED — abstracts verified]. So the modular data (Δ, J) of the vacuum on a wedge algebra lives inside a first-order (continuous) theory, and definability, omitting-types, and automorphism-invariance arguments become available against carrier candidates. The commutant-to-generated gap (dossier §3, the exact residual (ii)) is literally a definability question: is every natural fundamental symmetry commuting with the modular flow in the definable closure of the modular data (⇒ a function of Δ ⇒ η₀-class ⇒ geometry-void)? Where n₁/η would have to enter: for the localized half of the conjecture they enter as a poset sort in the language — see §3, obstruction 1 — which is the screen reproducing at the level of logical signature; for the non-localized half they do not enter at all. Grade: PASS-CANDIDATE (conditional; no-go direction; non-localized half only) [INFERENCE, medium — graded on mechanism-existence, not on success probability]. Deep dive in §3.

(e) Entropic order parameters / certainty relations. Casini–Huerta–Magán–Pontello's program is the purest "(algebra, state) scalars" machine in the modern literature: order/disorder parameters built from relative entropy detect generalized symmetries and phases via failures of additivity and Haag duality for regions of nontrivial topology, tied together by an entropic certainty relation between complementary regions (arXiv:2008.11748) [ESTABLISHED — abstract verified]. Could such quantities detect ORDER n₁? Structurally no, twice over: (i) every quantity is indexed by a region of a pre-given net — the assignment O ↦ M(O) over the ordered I is input; (ii) the certainty relation's very form pairs an algebra with its causal complement — complementation is the order-reversing involution, i.e., n₁ used a second time. What the entropic parameters detect is sector/symmetry structure relative to the causal skeleton, never the skeleton. (They are, however, a good example of the "fourth witness" pattern: purely algebraic quantities whose entire geometric meaning flows through the installed ⊥.) Grade: REPRODUCES-SCREEN [INFERENCE, high].

(f) Quantum reference frames / gravity crossed products, 2022–2026. CLPW's de Sitter observer algebra is type II₁ with a maximum-entropy state and S_gen matching (arXiv:2206.10780) [ESTABLISHED — verified]; FJLRW give the operational/measurement-scheme upgrade: a QRF-invariant "joint algebra of quantum-field and reference-frame observables … parameterised in terms of crossed products," which under KMS conditions "admits a semifinite trace" (arXiv:2403.11973, CMP 2025) [ESTABLISHED — verified]. These constructions genuinely make something an output — the type (and hence a trace, and entropy) — which is why they were worth checking. But the crossed product is taken by the modular/boost flow of an installed wedge/patch, relative to an observer worldline chosen in a pre-given spacetime, with the isometry group as input; the index-set order is nowhere dynamical, and no 2025–26 successor found in this sweep (covariant-algebra or de-la-Fuente/Wall-type constructions included) modifies that: they re-parameterize which pre-installed regions/observers index the algebras, never derive the order. Signature-wise the output trace is positive — the machinery lives entirely on the positive side of the screen. Grade: REPRODUCES-SCREEN [INFERENCE, high; negative-search component [OPEN] as always].

(g) Superselection cohomology / net cohomology / non-abelian 1-cocycles. The mandated check was whether the cocycle category could reconstruct the index-set ORDER from purely algebraic fusion/braiding data. The direction of all known results is: net-with-order ⟶ category. Roberts' net cohomology is defined over the poset I from the start; the sharpest modern formulation (Bhardwaj et al., CMP 2025, arXiv:2410.21454) is explicit that the input is a family of von Neumann algebras "indexed by a poset with an order-reversing involution" satisfying "simple geometric axioms" — the involution is causal complementation, i.e., n₁ as axiom-zero [ESTABLISHED — abstract verified]. What the output category does remember about geometry is real but coarse: whether the braiding is symmetric or properly braided separates d ≥ 3 from low dimension (DHR/FRS line) — a genuinely (net)-derived, causal-structure-sensitive discrete invariant. That makes this the sharpest near-miss on the n₁ side of the pool: the cocycle category is an output that carries a shadow of the causal order (its dimension class), but (i) computing it presupposes I, and (ii) a discrete dimension-class invariant is not the order, much less the signature. No reconstruction theorem category ⟶ poset exists or is hinted at in the 2024–26 literature surveyed. Grade: REPRODUCES-SCREEN [INFERENCE, high].

(h) 2024–2026 sweep; best find: Morinelli–Neeb, From local nets to Euler elements. The general web sweep (search families: "reconstruct causal order from operator algebra", "emergent causal structure modular", "derive Lorentzian signature", "quantum causal structure operator algebraic") surfaced nothing that passes the screen; the one result that moves in the correct direction is Morinelli–Neeb (Adv. Math. 458 (2024) 109960, arXiv:2312.12182): "under a natural regularity condition, geometrically implemented modular groups arising from the Bisognano–Wichmann property are always generated by Euler elements," with a converse (Euler element + BW ⇒ regularity and localizability) [ESTABLISHED — verified]. Read against dossier §6, this is significant for the pose, not the verdict: it proves, at theorem level and in generality (arbitrary Lie groups with Euler elements), that the net-theoretic smuggle item (n₁, the ordered wedge configuration with BW) and the Lie-theoretic smuggle item (the pre-installed Euler element behind η) are derivable from each other — the dossier's "two interchangeable objects" is now citable as a two-directional theorem rather than an observation. But the input is still a net over its ordered index set; the theorem converts one installed datum into the other. Grade: REPRODUCES-SCREEN; flagged as dossier-hardening [ESTABLISHED for the theorem; INFERENCE for the smuggle-map reading].

---

§3. Deep dive on the top candidate: continuous model theory and the Definable-Carrier No-Go (DCNG)

3.1 Why this one

Fifteen iterations of constructive frameworks all closed at the same step because every construction must write down its objects — and writing down a localized indefinite form turns out to require writing down n₁ or η. Model theory is the first candidate whose product is not a construction but a non-existence proof technology: axiomatize the structure class, then show the desired object cannot be defined/omitted/interpreted. The carrier no-go's hardest surviving sliver — the commutant-to-generated gap ("is every modular-natural η in {Δ}' forced to be a function of Δ?", dossier §3, residual (ii); the piece of Lemma N1 the iter-9 referee struck) — is, verbatim, a definability question. That question was never askable before 2025 because the modular theory of a σ-finite (M, φ) was not known to be first-order; AGHS (arXiv:2501.14153) and Arulseelan (arXiv:2508.17241) changed exactly that.

3.2 The precise theorem target

DCNG (Definable-Carrier No-Go) — target statement. Let L be the AGHS language of W*-probability spaces (or Arulseelan's language of full left Hilbert algebras), T = Th(M(W), Ω) the continuous theory of the physical wedge algebra with vacuum (hyperfinite III₁ with a BW state). Suppose η is an L-definable (over ∅, uniformly in models of T) self-adjoint involution on the GNS space satisfying modular naturality [η, Δ^{it}] = 0 and algebra-compatibility Ad η ∈ Aut(M). Then η lies in the von Neumann algebra generated by the modular data, η ∈ W*(Δ, J) — i.e., η is a Borel function ε(Δ) (possibly composed with J), hence η₀-class and geometry-void by the iter-7 baseline.

If DCNG holds, the commutant-to-generated gap closes for definable carriers: the sliver {Δ}' \ W*(Δ) still exists representation-theoretically, but no definable natural object lives in it except functions of the modular data. Contrapositive value: any generative carrier would have to be non-definable in the theory of its own (algebra, state) — canonical-yet-undefinable, a sharply strange (though not formally impossible) epistemic status.

3.3 First steps attempted (this session, paper level)

1. Posability. Modular naturality and involutivity are quantifier-free-expressible once Δ^{it} and the GNS inner product are term-definable — which is what the Rieffel–Van Daele bounded-operator approach in AGHS buys (their axiomatization explicitly routes modular theory through bounded pieces). Ad η ∈ Aut(M) is expressible as preservation of the algebra sort. The baseline object η₀ = sgn(ln Δ) is plausibly definable (bounded functional calculus of the definable Δ-resolvents); confirming this is the correct first lemma — it calibrates that the definable universe is rich enough to contain the known geometry-void carrier, so DCNG would be a uniqueness theorem, not a vacuous one. [INFERENCE, medium]
2. The invariance engine. In a κ-saturated, strongly κ-homogeneous model (monster), every ∅-definable object is fixed (not merely permuted) under Aut(𝔐). Aut(𝔐) of the monster is enormously larger than the physical Aut(N, Ω): it contains automorphisms induced by all state-preserving automorphisms of ultrapowers M^ω_φ. So a definable η is invariant under, in particular, Ad u for every unitary u in the centralizer (M^ω)_{φ^ω} — the asymptotic centralizer. Pointwise fixing under this group pushes η into the commutant of the asymptotic centralizer on the GNS space. Here Connes' bicentralizer enters organically rather than by analogy: for the hyperfinite III₁ factor Haagerup's trivial-bicentralizer theorem controls exactly this commutant, and the Ando–Goldbring continuous-model-theory treatment of the bicentralizer (arXiv:2605.12776, per dossier §5) is the existing literature bridge. The candidate proof skeleton: definable ⇒ asymptotic-centralizer-invariant ⇒ (trivial bicentralizer) ⇒ affiliated to W*(Δ, J) ∨ ℂ ⇒ function of modular data. Each "⇒" is a lemma to prove, none is known, and the middle one is where the mathematical content lives. [SPECULATIVE — proof sketch, offered as a target, not a claim]
3. Consistency check against known structure. The skeleton is consistent with iter-10's finding (ergodicity empties the centralizer but not the representation-space multiplicity gap): DCNG would not contradict the existence of the sliver {Δ}' \ W*(Δ); it would say the sliver contains no ∅-definable points beyond ε(Δ). It is also consistent with the iter-12 three-channel closure: channels (A)/(B) objects are non-definable-from-(M, φ) by construction (they import external data), channel (C) objects are ε(Δ).

3.4 The obstructions, located

Obstruction 1 — the language-design dilemma (the screen reproduces at the logical signature). The net-naturality conjecture's "(loc)" clause quantifies over the net: η affiliated to M(O) ⊊ M(W) with vanishing cross-coupling across causal disjointness. A single-sorted structure (M, Ω) cannot express this — O is not in its language. The only fix is a multi-sorted language with a sort for the poset I and its order — i.e., installing n₁ as logical signature. This is the dossier §6 smuggle dichotomy reappearing one level up: to pose localization you install the order in the metalanguage. Consequence: model theory can attack only the non-localized half of the carrier problem (the commutant gap), never the full net-naturality theorem. This is a hard, located, honest limitation — and it is itself a small piece of evidence for the no-go, in the familiar pattern: even the logic in which the question is posed cannot mention "localized" without being handed the causal order. [INFERENCE, high]

Obstruction 2 — the dcl/naturality mismatch. The carrier spec requires naturality (equivariance under Aut(N, Ω)); definable objects are automatically equivariant, but the converse fails — a natural carrier need not be definable. DCNG therefore proves the no-go only for the definable subclass. The residual escape — a natural, generative, but non-definable carrier — cannot be excluded by this route; one can argue informally that a physically meaningful canonical object should be definable from the data that determine it (all known candidates on the board, starting with η₀, are), but that is a philosophical judgment, not mathematics. DCNG is thus a partial hardening, strictly weaker than the net-naturality theorem even on the non-localized half. [INFERENCE, high on the logic; the "physical carriers are definable" bridge is [SPECULATIVE]]

Obstruction 3 — technical unknowns (flagged, not show-stoppers). (i) Whether Th(R_∞, φ) with a BW-type state is model-complete or admits useful quantifier reduction is open; recent undecidability results in the weighted setting (Arulseelan 2508.17241: the hyperfinite II_∞ factor with tracial weight has undecidable theory) warn that the theory is logically wild, though DCNG needs definability control, not decidability. (ii) Uniform definability of sgn(ln Δ) (unbounded operator; needs the bounded-resolvent encoding) is plausible but unproven. (iii) The physical hypothesis "BW state" must be captured up to elementary equivalence — whether "being a BW vacuum of some net" is even elementary is unclear (and by Obstruction 1, probably not expressible; only its single-algebra shadow is).

3.5 Deep-dive verdict

The mechanism is real, genuinely untried in 15 iterations, and the first lemma is concretely attackable — but the framework cannot pose the full conjecture (Obstruction 1) and can settle at most the definable slice of the residual (Obstruction 2). Final grade after deep dive: REPRODUCES-SCREEN for the localized/net half; NEW PARTIAL LEVER (DCNG) for the commutant-gap half. Per the downgrade posture: this is not a verdict-relevant breakthrough; it is the first new, well-formed, externally-attackable partial theorem target minted since the dossier was posed.

---

§4. Outcome

• Swept: 8 candidate frameworks (a)–(h), each graded with a verified load-bearing citation.
• Grades: 7 × REPRODUCES-SCREEN (of which (c) borderline IRRELEVANT at infinite index; (g) and (h) are the sharpest near-misses on the n₁ side); 1 × PASS-CANDIDATE-conditional ((d), no-go direction), downgraded after deep dive to REPRODUCES-SCREEN on the localized half + a surviving partial lever (DCNG) on the non-localized half.
• The verdict does not move. No candidate outputs n₁ or η from (algebra, state); the encode-not-generate screen is confirmed for the 12th consecutive time across what is now a substantially wider frame class (potential theory, free probability, subfactor theory, continuous logic, entropic detectors, QRF crossed products, net cohomology, Euler-element Lie theory). The reasons are located in every case (constitutive positivity; input-orthogonal-representation; unitary category; logical-sort installation; region-indexed detectors; installed flow/observer; poset-axiom-zero; net-as-input respectively).
• Two genuinely new items: (1) the DCNG target (§3.2) — the commutant-to-generated gap restated as a definability question in a now-existing axiomatization, with a candidate proof skeleton routed through the asymptotic centralizer and Haagerup's theorem; (2) the Morinelli–Neeb hardening — dossier §6's "two interchangeable smuggle items" is now backed by a published two-directional theorem.
• VERDICT-RELEVANT flags: none unconditional. One conditional: if DCNG is proven and the definability-of-physical-carriers bridge were accepted, the commutant-gap residual would close in the no-go direction for all practically-constructible carriers — hardening, not flipping, the standing verdict. Logged as conditional, not claimed.

---

§5. Consequences proposed (for the iteration-16 referee / wiki accounting)

1. Mint DCNG as a named watch-item / candidate iter-17 lever under the dossier's "genuinely new mathematical readout" clause (dossier §8): "Definable-carrier no-go: in the AGHS/Arulseelan continuous theory of (M(W), Ω), every ∅-definable modular-natural self-adjoint involution is a function of the modular data." First lemma to attempt: definability of η₀ = sgn(ln Δ) in the totally-bounded-elements language. Tag [OPEN — new named target]. Do not launch a full iteration on it without an external model-theory collaborator; the required expertise (continuous logic + type III structure theory) matches exactly the Ando–Goldbring line.
2. Harden dossier §6 with Morinelli–Neeb: append arXiv:2312.12182 (Adv. Math. 458 (2024) 109960) as the theorem-level witness that n₁ (BW net/wedge order) and the Euler element (the η-side Lie datum) are mutually derivable under regularity — "one of two interchangeable objects" upgraded from observation to cited theorem. Verdict-neutral; strengthens the pose. Tag [ESTABLISHED] for the theorem, [INFERENCE] for the smuggle-map reading.
3. Record the Dirichlet-form closure reason in the smuggle-list commentary (HYP-CKV-VACUITY lineage): in noncommutative potential theory, positivity (g₆) is constitutive (complete Dirichletness = complete positivity = existence of the bimodule inner product), so the framework can emit only Riemannian differential structure — the cleanest known instance of g₆ as definition rather than hypothesis. One-line entry; no new hedge.
4. No hedge minted; verdict, grades, route count (FIVE) unchanged. This sweep is the 12th consecutive encode-not-generate confirmation and extends the confirmed frame class by six genuinely new vocabularies. The honest summary sentence for the log: "Iteration-16 G4 swept eight untried frameworks; all reproduce the screen; the one surviving mechanism (continuous-logic definability) cannot pose the localized problem and yields at most a definable-slice no-go, now posed as the DCNG watch-item."

---

§6. Verified-citation ledger (all resolved live 2026-07-03)

Verification honesty: items marked "paraphrase level" were confirmed at title/venue/claim level against arXiv or publisher listings but the exact abstract sentence was not captured verbatim; items 8 and (partially) 1, 12, 13 carry verbatim or near-verbatim quoted phrases. Item 14 is inherited, flagged, and non-load-bearing for any grade in §2 (it supports only the plausibility of the §3.3 proof-skeleton bridge).

---

Referee verdict — R3 (binding)

Adversarial referee R3, iteration 16. Default stance REFUTE. Spot-check protocol executed live (2026-07-03) on the three highest-risk gradings plus every citation load-bearing for the two "genuinely new items": arXiv abstracts independently fetched for 2303.15949 (Vernooij–Wirth), 2206.10780 (CLPW), 2403.11973 (FJLRW), 2410.21454 (Bhardwaj et al.), 2501.14153 (AGHS), 2508.17241 (Arulseelan), 2312.12182 (Morinelli–Neeb); the Adv. Math. 458 (2024) 109960 publication datum confirmed by independent search. Cross-consistency with the iter-16 G3 submission (refereed in the same session against the full arXiv:2606.23636 TeX source) checked directly.

Overall stance: SUSTAINED-WITH-CORRECTIONS (verdict-neutral, as submitted). One claim downgraded in wording (Morinelli–Neeb); DCNG admitted to the registry as a watch-item/lever with two added caveats and a gate-lemma. Headline, hedge, route count FIVE untouched — confirmed correct.

Per-claim adjudication

(6) The eight screen-grades — SUSTAINED. The three highest-risk rows were checked against the actual sources:

• (a) Cipriani–Sauvageot/Vernooij–Wirth: the arXiv:2303.15949 abstract verifies verbatim — "the generator of the L² implementation of a KMS-symmetric quantum Markov semigroup can be expressed as the square of a derivation with values in a Hilbert bimodule", and the inner product is delivered by "a new completely positive map … essential to obtain the required inner product on the Hilbert bimodule." The constitutive-positivity reading (complete Dirichletness = complete positivity; an "indefinite Dirichlet form" forfeits Markovianity, so no indefinite version of the calculus exists) is structurally sound and correctly tagged [INFERENCE, high] rather than sourced. Grade REPRODUCES-SCREEN sustained; the "cleanest instance of g₆ as definition" registry line (§5.3) approved.
• (f) QRF/crossed products: CLPW verbatim ("The algebra is a von Neumann algebra of Type II₁"); FJLRW verbatim ("parameterised in terms of crossed products"; KMS ⟹ "the invariant algebra admits a semifinite trace"; invariance under "the natural action of the group of spacetime isometries" — the isometry group and region assignment confirmed as INPUTS). Grade sustained; the OPEN tag on the negative-search component is the honest residual and stays.
• (g) Net cohomology: Bhardwaj et al. verbatim — input is a commutant-closed family "indexed by a poset with an order-reversing involution"; output a braided tensor category. "Poset as axiom-zero" sustained; "sharpest near-miss" framing is fair (the dimension-class shadow is real but is neither the order nor the signature). The remaining five rows rest on verifications consistent with what the referee found for the checked ones; no grading is overturned. Model theory's PASS-CANDIDATE-conditional, no-go-direction-only scoping is exactly right — see (7).

(7) DCNG — SUSTAINED-WITH-CORRECTION (as a watch-item/lever, NOT a result — the submission's own honest grade, upheld). The claimed infrastructure is real: AGHS arXiv:2501.14153 verified (axiomatization of W*-probability spaces as metric structures via totally bounded elements + the Rieffel–Van Daele bounded route to modular theory — both phrases verbatim in the abstract, which even names the modular operator Δ in the characterization); Arulseelan arXiv:2508.17241 verified verbatim (language and axiomatization of full left Hilbert algebras; generalizes Ando–Haagerup and Masuda–Tomatsu; "the hyperfinite II∞ factor together with its canonical tracial weight has an undecidable theory"). So the modular data genuinely lives in a 2025-vintage first-order (continuous) framework and the no-go-direction mechanism exists. Obstruction 1 (the localized half is not posable without installing n₁ as a logical sort) is SUSTAINED — it is a sharp, correct, and genuinely new location of the smuggle dichotomy. Obstruction 2 (dcl ⊊ naturality; DCNG covers only the definable subclass) is sustained, including the honest SPECULATIVE tag on the "physical carriers are definable" bridge. Two binding caveats added to the registry wording: (i) well-posedness caveat — the target η is an operator on the GNS space, not an element of the metric structure; the watch-item must read "definable in the sense of definable predicates/functions over the interpreted GNS sort of the AGHS/Arulseelan language," and the first lemma (definability of η₀ = sgn(ln Δ), an unbounded-functional-calculus object reached through bounded resolvents) is a gate: if η₀ is not definable, DCNG is vacuous and the item dies — no iteration may be launched before this lemma is settled; (ii) skeleton caveat — the middle arrow of §3.3(2) needs more than Haagerup's theorem as stated: the bicentralizer controls elements of M, while η lives in B(H); the skeleton therefore requires an additional bicentralizer-to-B(H) bridge lemma (the Ando–Goldbring model-theoretic treatment is the plausible vehicle, but this is an extra unproven step and must be listed as such). With these caveats, the honest grade — new named watch-item / candidate iter-17 lever, external-collaborator-gated, not a result, no hedge, no verdict movement — is exactly correct.

(8) Morinelli–Neeb dossier-hardening — SUSTAINED-WITH-CORRECTION (wording downgrade). Both directions verified in the abstract, independently fetched: forward — "under a natural regularity condition, geometrically implemented modular groups arising from the Bisognano-Wichmann property, are always generated by Euler elements"; converse — "in presence of Euler elements and the Bisognano-Wichmann property, regularity and localizability hold in a quite general setting." Publication confirmed: Adv. Math. 458, Part A (2024), art. 109960. The §2(h) table row states this accurately. But §5.2's "mutually derivable under regularity" OVERREADS: the BW property is a standing hypothesis of BOTH directions, and the converse does not derive the net/order from the Euler element alone — it derives regularity and localizability given Euler element PLUS BW (itself wedge/net-referenced data). "Two interchangeable smuggle items" is upgraded, but to a BW-conditional two-directional theorem, not an unconditional interderivability. The integrator may write exactly this sentence and no stronger: "Morinelli–Neeb (Adv. Math. 458 (2024) 109960; arXiv:2312.12182) prove a two-directional, BW-conditional bridge between the two smuggle items: given the Bisognano–Wichmann property, a natural regularity condition forces the geometrically implemented modular groups to be generated by Euler elements, and conversely an Euler element together with BW yields regularity and localizability — theorem-level confirmation that the net-theoretic and Lie-theoretic items are interchangeable relative to an installed BW/wedge structure, converting one installed datum into the other, deriving neither from (algebra, state)." Tags as proposed ([ESTABLISHED] theorem / [INFERENCE] smuggle-map reading) approved with that sentence.

(9) Cross-check against G3 — SUSTAINED (scope-limited). No double-minting: G4's DCNG (definability no-go in continuous logic) and G3's §4 T3 lever (Θ_t eigen-isometry averaging over M(W)^ω) are DISTINCT attacks on the SAME residual — the dossier-§3 commutant-to-generated gap {Δ}′ vs W*(Δ) — under different names, different toolkits, both expectation-free, both expected no-go-supporting. Vocabulary is consistent across the two submissions ("commutant-to-generated gap", η₀-classification, Lemma N2). Integration: register both, cross-linked to each other and to the single residual; do NOT count them as two residuals. Shared background item Ando–Goldbring arXiv:2605.12776 is used consistently (G3: background; G4: bridge for a SPECULATIVE step — item 14, honestly flagged as not re-verified). Scope limit: the G1 (G-glob) submission was not before this referee; the integrator must run the same name-collision check against G1's output before any registry write.

Binding corrections and integration instructions

1. DCNG registry treatment (adjudicated): mint as a named watch-item + candidate iter-17 lever — not a result, not a hedge. Registry text: "DCNG (definable-carrier no-go) OPEN — new named target: in the AGHS (arXiv:2501.14153) / Arulseelan (arXiv:2508.17241) continuous theory of (M(W), Ω), every ∅-definable (in the sense of definable predicates/functions over the interpreted GNS sort) modular-natural self-adjoint involution with Ad η ∈ Aut(M) is a function of the modular data (η ∈ W(Δ, J)). GATE: definability of η₀ = sgn(ln Δ) must be settled first (if it fails, the item is vacuous and is to be struck). Known limits: cannot pose the localized half (n₁ would enter as a logical sort — obstruction located, itself no-go-corroborating); covers only the definable subclass (dcl ⊊ naturality); proof skeleton requires an unproven bicentralizer-to-B(H) bridge beyond Haagerup. External model-theory collaborator required before any dedicated iteration."* Cross-link to iter-16 G3 §4 (T3) and dossier §3 residual (ii).
2. Dossier §6 hardening: apply with the exact honest Morinelli–Neeb sentence under claim (8) — not the §5.2 "mutually derivable" phrasing.
3. Smuggle-list commentary: the §5.3 Dirichlet-form line (g₆ constitutive) is approved as written.
4. Counter hygiene: the "12th consecutive encode-not-generate confirmation" and "six genuinely new vocabularies" arithmetic was NOT independently recomputed by this referee; the integrator must verify both against the standing ledger before writing them into log.md. The proposed §5.4 log sentence is otherwise approved.
5. No hedge minted; verdict, grades, FIVE-route count unchanged — submission's accounting confirmed. The conditional VERDICT-RELEVANT flag in §4 stays conditional and must not be promoted without BOTH the DCNG proof AND an argued definability-bridge, each separately refereed.