Track E1 — Factorization-algebra / functorial-QFT lens on the net-naturality theorem

Outcome: REPRODUCES-SCREEN. The Costello–Gwilliam / Atiyah–Segal–Lurie / Benini–Schenkel formalism is a genuinely different framework (category theory + operads + descent, never Tomita–Takesaki), and it reproduces the encode-not-generate screen from a new direction with a sharply nameable geometric input. Verdict, principal hedge (HYP-CKV-VACUITY-R6), and the carrier-convergence count (FIVE) all unmoved — this would be the 14th consecutive confirmation if logged. No registry ID consumed.

The question, in this lens

The gate (iter-9): is an Aut(N,Ω)-natural, modular-covariant, localized, indefinite (n,n) form on a Haag-dual split BW net FORCED to be signature-inherited? The factorization-algebra reframing asks whether the local-to-global (factorization/descent) structure of an AQFT supplies a NEW decidable handle on the "localized + indefinite" conjunction, or whether it presupposes the Lorentzian/causal geometry.

What the framework actually is (web-verified)

• Costello–Gwilliam: a (pre)factorization algebra is a functor U ↦ Obs(U) on the opens of a fixed space X, with structure maps for disjoint inclusions (vol. 1, Cambridge NMM 31; nLab). The manifold is the site, i.e. input.
• Functorial / cobordism QFT (Lurie): a symmetric-monoidal functor out of a bordism category; the cobordism hypothesis classifies fully-extended TQFTs by fully-dualizable objects. The geometric cobordism hypothesis (Grady–Pavlov, arXiv:2111.01095) installs the geometric structure as a fixed parameter via tangential structures over BO_d — the bordism category IS the geometry, and the structure group O(d) carries the signature.
• Homotopical-AQFT bridge (the load-bearing object): Benini–Perin–Schenkel (CMP 377(2):971–997 (2020), arXiv:1903.03396) prove AQFT ≅ time-orderable prefactorization algebra on globally hyperbolic Lorentzian manifolds; Benini–Carmona–Grant-Stuart–Schenkel (arXiv:2412.07318) upgrade this to a clean 1-categorical equivalence in which both sides are multifunctors out of the orthogonal category Loc^rc̄ = (Loc^rc, ⊥) with ⊥ = causal disjointness. pAQFT observables realize this (Gwilliam–Rejzner, arXiv:2212.08175).

The exact geometric input smuggled

Verbatim (arXiv:2412.07318): "the relevant orthogonal category Loc^rc̄ := (Loc^rc, ⊥) … the orthogonality relation ⊥ which is defined by causal disjointness." Two consequences, both fatal to "generates":

1. Localization = factorization over ⊥. A (pre)factorization algebra's defining structure maps are indexed by causal-disjointness. "Localized" / "vanishing cross-coupling between disjoint regions" is meaningful only relative to ⊥. But ⊥ is exactly the iter-8/iter-9 smuggle-item n1 (the index-set causal order; BGL positivity⟺isotony). So this framework's notion of localization is the geometry — the same irreducible-localization residual the iter-9 naturality note already isolated, now re-derived from pure category theory. On the Lorentzian side BPS must additionally restrict to time-orderable tuples, i.e. it imports the time-orientation too.
2. No signature in the fibers. Both AQFT and prefactorization algebra are algebra-valued / cochain-complex-valued functors with no Hilbert-space positivity, no inner product, no signature in the target. So the formalism can neither generate nor obstruct an (n,n) form: the signature lives entirely in the geometric base. This is the categorical mirror of the iter-9 finding that the fibers carry no signature and the geometry sits in the base (Neeb–Ólafsson fiber-bundles-over-ordered-symmetric-spaces; Komalan double-categorical AQFT). The Riemannian↔Lorentzian split in the factorization-algebra literature (the Riemannian BV theorems vs the "Lorentzian structure is essential" pAQFT theorems) is an explicit instance: signature enters by choice of base geometry / Wick rotation, never as an output.

The one candidate handle, and why it fails the referee bar

The BPS/BCGS equivalence is a genuinely new precise object: it makes the carrier/net-naturality question translatable into factorization-algebra language. One could hope the factorization (descent) axioms add constraints that decide the localized-indefinite conjunction. They do not: the equivalence is between two functor-categories over the same fixed ⊥, so it transports the question verbatim and carries n1 across unchanged. It is a translation tool, not a handle — exactly analogous to the iter-9 result that reverse-Weinberg–Witten is the carrier problem seen from the empirical side (same object, new vocabulary). new_handle = NONE. This is also the correct anti-crank call: claiming the equivalence "decides" the gate would be manufacturing a fake crack, since both of its sides presuppose the causal order that supplies localization.

Why this is not a sixth route (count stays FIVE)

This is the categorical presentation of route (5) (the localization-template / index-set-order residual), not a new convergence. The factorization formalism re-expresses n1 as the orthogonality relation ⊥; it adds a fourth independent witness that "localized" presupposes a fixed causal base (alongside BGL, Neeb–Ólafsson, Komalan), but it is the same obstruction. Binding inheritance: count = FIVE.

Honest value of the negative

The no-go is now confirmed robust across modern category-theoretic / homotopical mathematics, not just operator-algebraic mathematics: an entirely different formalism, designed precisely to make local-to-global structure manifest, locates the signature and the localization in the same place — the geometric base — and provides no functorial enrichment that supplies an indefinite localized fiber form. This matches the standing honest-expected-outcome verbatim and tightens the §4 readout-independence claim by one more independent direction.

Dossier contribution (crisp open-problem sentence)

The factorization-algebra/functorial reframing makes the residual decidable sub-problem precise: exhibit (or rule out) a (pre)factorization or cobordism structure whose causal-disjointness/orthogonality relation ⊥ and whose target signature are OUTPUTS of the algebra-valued functor, rather than indexing data of a fixed geometric source category — i.e. remove ⊥ from the source and ask whether the factorization axioms alone can regenerate it (the categorical form of "the localization template is the geometry").

Citations (all web-verified this session)

• Costello–Gwilliam, Factorization Algebras in QFT vol. 1 (Cambridge NMM 31); nLab factorization algebra.
• Benini–Perin–Schenkel, CMP 377(2):971–997 (2020), arXiv:1903.03396 — AQFT ≅ time-orderable prefactorization algebra, globally hyperbolic Lorentzian.
• Benini–Carmona–Grant-Stuart–Schenkel, arXiv:2412.07318 — orthogonal category (Loc^rc, ⊥), ⊥ = causal disjointness; algebra/chain-complex-valued, no positivity.
• Gwilliam–Rejzner et al., arXiv:2212.08175 — pAQFT observables form a factorization algebra.
• Benini–Musante–Schenkel, Lett. Math. Phys. 114(1) (2024) — Lorentzian free BV: tPFA vs AQFT.
• Lurie, On the Classification of TFTs (2010); Grady–Pavlov, The geometric cobordism hypothesis, arXiv:2111.01095 — geometric structure as fixed source parameter, O(d) structure group.

Honesty / limits

• The note_md §"candidate handle" judgment and the "count stays FIVE / same as route 5" identification are INFERENCE over the web-verified ESTABLISHED framework facts; they are not theorems and were not adversarially refereed in this track.
• I did not independently re-derive the BPS equivalence; I rely on the published statements and the verbatim orthogonality-relation quote.
• No fabricated citation: every arXiv ID and journal reference above was returned by live search/fetch this session.

---

Referee verdict - Track E1 (factorization-algebra / functorial-QFT lens on the net-naturality theorem)

Stance: REFUTE by default, brutal on framework fidelity. Stance NOT sustained as a refutation. Outcome: UPHELD-WITH-CORRECTION (REPRODUCES-SCREEN confirmed; one citation misattribution + two precision corrections; verdict/hedge/count UNMOVED).

1. Framework mathematics - web-verified, real, correctly applied

Every load-bearing framework fact was checked live this session:

• BPS equivalence (arXiv:1903.03396): VERIFIED verbatim incl. journal coordinates CMP 377(2):971-997 (2020). AQFT = time-orderable prefactorization algebra on globally hyperbolic Lorentzian manifolds. The finder's reading (the Lorentzian/globally-hyperbolic structure and time-orderability are HYPOTHESES, i.e. inputs) is faithful.
• BCGS (arXiv:2412.07318): VERIFIED - real title, four named authors, equivalence over globally hyperbolic Lorentzian manifolds, target = symmetric monoidal infinity-category of cochain complexes (confirming the no-positivity/no-signature-in-fibers claim). The orthogonal-category (Loc, perp) with perp = causal disjointness is corroborated by the Benini-Schenkel-Woike orthogonal-category literature ("orthogonality relations formalize causal disjointness (spacelike separateness)", "codify commutation relations in AQFT").
• Costello-Gwilliam: correct standard definition (functor on opens of a fixed space; the space is the input/site).
• Grady-Pavlov geometric cobordism hypothesis (arXiv:2111.01095): VERIFIED real; geometric structures (Riemannian metrics, complex/symplectic) are source-category parameters - the finder's "bordism category IS the geometry, O(d) carries the signature" gloss is the correct standard reading.
• Benini-Musante-Schenkel, LMP 114(1):36 (2024): VERIFIED exactly.

No misstated definition of a factorization algebra or the cobordism hypothesis - the #1 fatal-defect risk the mission flagged did NOT materialize. The hallucination check passes.

2. The one citation defect (BC-1)

arXiv:2212.08175 is Gwilliam-Rejzner, not "Gwilliam-Rejzner et al." as the finder wrote. The content (pAQFT observables form a factorization algebra) is correct; only the author attribution is wrong (a conflation with the separate, real Gwilliam-Rejzner pAQFT line). Minor, non-load-bearing, corrected in the record. No fabricated citation anywhere; every arXiv ID and journal reference resolves to a real paper.

3. Central claim checked against the wiki closures - holds

The finder's thesis (REPRODUCES-SCREEN; new_handle=NONE; geometric input = the causal-disjointness/orthogonality relation perp = n1, plus time-orientation and the O(d) bordism structure group) is consistent with and reinforced by:

• iter-3 encode/generate criterion: the framework's target fibers carry no signature (cochain complexes, no positivity), so it can neither generate nor obstruct an (n,n) form - the signature lives in the geometric base. Matches the iter-7/iter-9 finding that the fibers carry no signature.
• iter-9 net-naturality + Lemma N2: "localized" is meaningful only relative to perp, and perp IS the index-set order n1 that Lemma N2 / the localization-is-the-geometry residual already isolated. The factorization framework re-expresses the SAME irreducible-localization residual from category theory.
• iter-9 rigid-tensor-category positivity axiom: the finder's claim that enlarging to operadic/categorical structure adds only positive/signature-free data is exactly the iter-9 Section 4 closure (rigid C*-tensor categories have positive hom-space inner products by axiom; Benini-Schenkel-Woike orthogonality = n1).
• iter-12 Krein closure: the target categories carry no indefinite fiber form - consistent with subquestion-#3 closed-negative (no genuine indefinite localized modular involution from positive data).
• HYP-ENCODING-SCREEN: the no-go is confirmed robust across modern category-theoretic/homotopical mathematics, a genuine additional independent direction.

4. The candidate handle correctly FAILS the bar

The BPS/BCGS equivalence is a real new precise object but is correctly judged a TRANSLATION TOOL, not a decidable handle: both functor-categories are defined over the SAME fixed perp, so the equivalence transports the carrier question verbatim and carries n1 across unchanged - exactly analogous to the iter-9 reverse-Weinberg-Witten result (same object, new vocabulary). Claiming it "decides" the gate would be manufacturing a fake crack, since both sides presuppose the causal order supplying localization. The finder did NOT manufacture a crack; HANDLE-FOUND is correctly not claimed. This is the right anti-crank call.

5. Precision correction on novelty (BC-2)

The finder's "NEVER applied in this wiki" is only partly right. The Costello-Gwilliam side, the cobordism side, and the BPS/BCGS equivalence theorem ARE new. But the orthogonality-relation = n1 observation was ALREADY logged in the iter-9 carrier-naturality note Section 4 (citing Benini-Schenkel-Woike arXiv:1709.08657). The new framework re-derives an already-recorded point from a richer direction. The finder's note already frames this honestly as "a fourth independent witness ... the same obstruction" - kept and bound.

6. Anti-inflation / registry discipline - clean

• Count STAYS FIVE (categorical presentation of route 5, not a sixth route). Binding inheritance honored (BC-3).
• No new numeric registry ID consumed.
• Verdict UNMOVED (14th consecutive confirmation if logged).
• Principal hedge HYP-CKV-VACUITY-R6 unmoved in grade (MEDIUM-HIGH->HIGH conditional) and condition (net-naturality theorem + located commutant-to-generated gap).
• HYP-ENCODING-SCREEN no-test corollary unmoved.
• The one INFERENCE tag (the count/handle judgment) is honestly self-flagged as unverified inference, not presented as a theorem.

Verdict-bearing summary

REPRODUCES-SCREEN UPHELD-WITH-CORRECTION. A genuinely different framework (operads + category theory + descent, never Tomita-Takesaki) reproduces the encode-not-generate screen from a new direction with a sharply-named geometric input (perp = n1 + time-orientation + O(d) bordism structure). Honest expected outcome matched verbatim; the no-go is now confirmed robust across modern homotopical mathematics, tightening the Section 4 readout-independence claim by one more independent direction. Corrections BC-1 (Gwilliam-Rejzner author fix) through BC-4 (geometric-input verification), none verdict-bearing. The standing verdict, principal hedge, encoding-screen corollary, and carrier-convergence count FIVE are all unmoved.