Track B2 (iteration 12): the 2023-2026 external-math sweep finds no lever on the net-naturality residual - CONFIRMS-SATURATION

Status: ATTEMPT executed - a web-verified 2023-2026 sweep of the operator-algebra / AQFT literature for any result closing, refuting, or directly addressing the net-naturality theorem or the located commutant-to-generated gap (the actual standing residual, iters 9/10). Outcome: CONFIRMS-SATURATION. No such result exists: every new paper presupposes geometry as input, is orthogonal, or re-confirms an open external problem. The residual stays OPEN exactly as iterations 10 and 11 left it. Verdict unchanged (11th consecutive confirmation); principal hedge HYP-CKV-VACUITY stays at -R6; carrier-convergence count stays FIVE; registry implication NONE. Last updated: 2026-06-19 Iteration: 12

Scope honesty: every arXiv identifier below was web-verified live this session against its actual abstract/listing (all eight real, correctly attributed). Binding referee correction applied: the iter-7 "J carries reflection positivity" gloss is NOT attached to Naaijkens-Penneys-Wallick (arXiv:2605.10693) - that paper's reflection-positivity axiom is internal to its local-topological-order (spin-system) framework and makes no claim about the modular conjugation J carrying a Euclidean datum; it is recorded here as orthogonal.

1. The sweep, channel by channel `[INFERENCE, high]`

Morinelli-Neeb-Olafsson Euler-element / geometric-BW program (arXiv:2508.10960 Part I; arXiv:2603.26390 Part II "Geometric Bisognano-Wichmann and Spin-Statistics"; arXiv:2412.20410 [single-author Morinelli, "A geometric perspective on AQFT"]; arXiv:2511.09360 [nets of real subspaces on homogeneous spaces; per-paper authorship to confirm before any bibliography use]). Each presupposes the Euler element and the causal homogeneous space - geometry as INPUT - and derives BW / Spin-Statistics in the forward direction. This is the smuggle the iter-11 A3 audit named ("forced to be signature-inherited from a pre-installed Euler-element / causal index set"): a cleaner statement of the encode-not-generate screen, not a net-naturality theorem, not a carrier, not a sixth route.
Weiner algebraic Haag theorem (arXiv:1006.4726, the iter-9 conditional complete-invariant theorem) stays conditional on split + geometric modular action - the geometric inputs; no 2023-2026 result makes it unconditional.
Hong Liu lecture notes (arXiv:2510.07017, "entanglement, von Neumann algebras, and emergence of spacetime") presuppose bulk AdS/CFT geometry; subregion-subalgebra duality recovers, does not generate the metric - fails the encode-not-generate screen.
Naaijkens-Penneys-Wallick (arXiv:2605.10693, "Local topological order, Haag duality, and reflection positivity") place a reflection-positivity axiom on the Euclidean side of a spin-system LTO framework and use Tomita-Takesaki for Haag duality on cone regions. It constructs no Lorentzian metric, no indefinite pairing, no Krein structure, no carrier witness, and makes no J-as-reflection-positivity-carrier claim (binding correction). Orthogonal.
Ando-Goldbring (arXiv:2605.12776, "Model theory and Connes' bicentralizer problem") re-confirms Connes' bicentralizer problem is OPEN in general (model-theory reformulation; not AQFT nets). It re-validates the binding iter-11 correction and explicitly does NOT make the carrier no-go equivalent to the bicentralizer problem.
Emergent-Lorentzian working manuscripts (e.g. the non-peer-reviewed rgemergence.com manuscript, which presupposes a Fisher metric + a posited monotone flow) are inherited, not generated - they install the geometry and fail the screen. Tagged [SPECULATIVE] / non-peer-reviewed.

2. Adjudication - no lever, residual stays OPEN `[INFERENCE, high]`

No 2023-2026 result closes the net-naturality theorem (negatively => no-go unconditional), refutes it (=> a carrier exists), or even directly addresses the located commutant-to-generated gap. The new mathematics is confirmatory (geometry-as-input: the MNO Euler program, Weiner, Liu) or orthogonal (Naaijkens-Penneys-Wallick) or re-confirms an open external problem (Ando-Goldbring, Connes bicentralizer). The reverse-Weinberg-Witten witness stays a literature-absence OPEN, as of 2026-06, the empirical-side shadow of the carrier problem (iter-9 identity), un-registered, NOT OP-50. The residual is reasoning-undecidable from inside the wiki, exactly as iterations 10 and 11 left it - external-math-gated.

3. Registry implication

NONE. No new numeric ID; no grade/condition change to any hedge; HYP-CKV-VACUITY stays -R6; carrier-convergence count stays FIVE; verdict unchanged (11th consecutive). The MNO Euler-pair theorems are flagged under HYP-ENCODING-SCREEN as a 2026 encode-pattern instance, explicitly NOT a sixth route. Iter-11 A1 open subquestion #1 (relative-bicentralizer triviality for physics nets) is addressed separately by track B1 (retired as a non-load-bearing lever).

Referee verdict — Track B2 external-math sweep (iteration 12, 2026-06-19)

Adversarial referee, default stance REFUTE. Every cited arXiv ID web-verified live this session against the actual abstract/listing; every load-bearing characterization re-derived against the binding iteration-11 corrections, the iter-9 reverse-WW identification, and the standing carrier no-go. Stance NOT sustained as a refutation — the submission survives with one supporting-gloss correction and one citation-form note, neither verdict-bearing.

Outcome: CONFIRMS-SATURATION — UPHELD-WITH-CORRECTION. Headline unchanged (11th consecutive confirmation); carrier-convergence count FIVE; registry implication NONE.

Citation audit (live web) — clean, no fabrication

All eight arXiv identifiers resolve to real papers with correct authorship and titles, and each is accurately characterized:

2508.10960 (MNO, "Orthogonal pairs of Euler elements I") — verified; classifies orthogonal Euler pairs, relates wedge causal complements to twisted complements group-theoretically. Euler element presupposed — geometry as input, forward direction. Does not close the commutant-to-generated gap.
2412.20410 (Morinelli, "A geometric perspective on AQFT") — verified, single-author; overview of the Neeb-Olafsson geometric AQFT program; wedge localization built on Euler elements. Geometry-as-input.
2511.09360 ("Nets of real subspaces on homogeneous spaces and AQFT") — verified; nets over a presupposed causal homogeneous space. Confirmatory.
2603.26390 (MNO II, "Geometric Bisognano-Wichmann and Spin-Statistics") — verified (already vetted iter-9); geometric BW presupposes the Euler/wedge germ. Forward direction.
1006.4726 (Weiner, algebraic Haag's theorem) — verified verbatim at iter-9; complete-invariant statement is conditional on split + geometric modular action, the geometric inputs. No 2025-2026 result makes it unconditional. Finder's negative claim survives the sweep.
2510.07017 (Hong Liu, "Lectures on entanglement, von Neumann algebras, and emergence of spacetime") — verified; lecture notes presupposing bulk AdS/CFT geometry; subregion-subalgebra duality recovers (does not generate) geometry. Fails the encode screen. Correct attribution ("Liu") and genre.
2605.12776 (Ando-Goldbring, "Model theory and Connes' bicentralizer problem") — verified; continuous-model-theory result (selfless class forall-exists axiomatizable; Houdayer-Marrakchi equivalence extended to non-separable diffuse W*-probability spaces). Treats Connes' bicentralizer as a prominent OPEN problem — independently re-validates binding iter-11 correction (4). Not AQFT nets; carrier no-go is NOT equivalent to it.
2605.10693 (Naaijkens-Penneys-Wallick, "Local topological order, Haag duality, and reflection positivity") — verified; LTO reflection-positivity axiom + Tomita-Takesaki Haag duality on cone regions in spin systems. No Lorentzian metric, no indefinite pairing, no Krein structure, no carrier witness. Orthogonal.
Vaillant (rgemergence.com working manuscript) — verified real, non-peer-reviewed; starts from a presupposed Fisher metric + posited monotone flow from which a Lorentzian cone "emerges." Inherited, not generated — fails the encode screen. Correctly tagged non-peer-reviewed.

Binding iteration-11 corrections — all respected

The submission does not lean on the bicentralizer route to close the no-go. It cites Ando-Goldbring 2605.12776 only to re-confirm Connes' bicentralizer is open in general (binding correction 4) and explicitly states "carrier no-go not equivalent to it" (binding corrections 1, 2). It does not resurrect the false HMT-resolution premise, does not conflate trivial centralizer with trivial bicentralizer, and does not assert the inner-horn closure reduces to Connes' problem. Clean.

No-go red-team — clean; no flip, no smuggle

No cited result constructs a Lorentzian/causal metric from bare (algebra, state) data, exhibits an indefinite (n,n) / Krein carrier, or supplies a reverse-Weinberg-Witten witness. Every emergent-geometry candidate (Liu, Vaillant) installs the geometry first; every AQFT-net advance (the MNO Euler program, Weiner) presupposes the causal/Euler datum or split+GMA — i.e. they reproduce, not refute, the iter-9 located gap (GMA needs the Euler element; the index set is smuggle-item n1). The extraordinary-claim gate does not fire. Carrier-convergence count correctly stays FIVE (the sweep adds no route; it is a literature-survey confirmation of the existing five). No conflict with dS-cardinality, the iter-7 trichotomy, the iter-8 net no-go, the iter-9 naturality / reverse-WW identification, or the iter-10 commutant-gap null.

Corrections (neither verdict-bearing)

Trim one supporting gloss. The finder's "Naaijkens Penneys Wallick ... consistent with J carrying reflection positivity" attributes the iter-7 J-as-RP-carrier mechanism to 2605.10693. The paper's RP axiom is internal to the LTO framework and makes no such J claim. Record the paper as orthogonal (Euclidean-side RP, no carrier) without the J-attribution. Conclusion (orthogonality) unaffected.
Citation-form precision. 2412.20410 is single-author (Morinelli); not every Euler-program ID is three-author. Attribute per-paper if the bibliography is edited. No theorem misstated.

Net

A verdict-neutral, citation-clean external sweep. The honest core — no 2023-2026 result closes, refutes, or directly addresses the net-naturality theorem or the located commutant-to-generated gap; new mathematics is confirmatory (geometry-as-input) or orthogonal; Connes' bicentralizer re-confirmed open and carrier no-go not equivalent to it; reverse-WW still OPEN with no witness — is fully sustained. The principal hedge HYP-CKV-VACUITY stays at R6 in grade and condition; HYP-FACTORIZATION-IS-GEOMETRY and HYP-ENCODING-SCREEN unchanged. No new numeric ID consumed. Iteration 11 open subquestion 1 (relative-bicentralizer triviality for physics-relevant Bisognano-Wichmann / conformal nets) remains the sharpest next lever and is correctly reported as unanswered in 2025-2026. CONFIRMS-SATURATION upheld with correction; eleventh consecutive confirmation; object-level forecast unchanged; wager remains not-yet-physics.