OP-48 point (c): a state-/collar-free composition surrogate on Type III₁?

Status: SETTLED as a conditional no-go for OP-48(c) (medium-high confidence); headline verdict encodes-not-generates STANDS and is tightened, not flipped. Last updated: 2026-06-08 Iteration: 5

Goal

Resolve OP-48 point (c): exhibit, or prove impossible, a state-independent, collar-free composition surrogate that singles out quantum theory from split-style composition axioms on a Type III $_1$ factor — converting the partial $A+B$ merge into a full merge, or establishing a clean no-go.

Iteration 4 left (c) OPEN: the split property gives only a state-/collar-dependent factorization (canonical only relative to the vacuum modular conjugation $J$ , failing at zero collar). The closing question this note attacks: is there a canonical, choice-free composition surrogate — Haagerup standard form / natural cone, Connes spatial theory, Connes fusion, the funnel property, or an operational commuting-pair axiomatization — and does it single out QM, or does any QM-selecting surrogate necessarily reintroduce a state/collar dependence?

See ../open-problems.md (OP-48), the predecessor note 2026-05-xx-iter4-op48-split-composition.md for the iter-4 framing, and the domain page ../domains/operator-algebras.md.

Setup (real math)

Two background facts fix the arena.

(i) The local algebra is hyperfinite Type III $_1$ . In relativistic QFT a single local algebra $\mathcal{M}=\mathcal{A}(O)$ is the hyperfinite Type III $_1$ factor (Connes–Haag–Araki) ESTABLISHED. Consequently there is no trace, no minimal projection, no density matrix, and no factorization $\mathcal{H}\neq\mathcal{H}_{\mathcal{M}}\otimes\mathcal{H}_{\mathcal{M}'}$ . So the GPT local-tomography axiom $\dim V_{AB}=\dim V_A\cdot\dim V_B$ is not statable verbatim on $\mathcal{M}$ ESTABLISHED.

(ii) Iter-4 settled the split route. The split property restores an exact $\mathcal{H}\cong\mathcal{H}_1\otimes\mathcal{H}_2$ for a nested inclusion $O_1\Subset O_2$ , but only with a nonzero collar and only canonically relative to a cyclic–separating $\Omega$ via the intermediate Type I factor associated with $N\vee JNJ$ . Hence it is both state-dependent (the canonical intermediate factor is fixed by $\Omega$ 's modular $J$ ; Doplicher–Longo 1984 give infinitely many intermediate factors otherwise) and collar-dependent (Fewster: the split inclusion fails for touching/adjacent regions) ESTABLISHED.

Point (c) asks for a surrogate avoiding both dependencies. The structural candidates, with their honest state/collar status:

1. Haagerup standard form $(\mathcal{M},\mathcal{H},J,\mathfrak{P})$ . Every von Neumann algebra has a canonical self-dual natural cone $\mathfrak{P}$ and conjugation $J$ with $J\mathcal{M}J=\mathcal{M}'$ ; the quadruple is unique up to unitary equivalence (Haagerup 1975; Araki, and Connes for the $\sigma$ -finite case) ESTABLISHED, web-confirmed. This is genuinely canonical and state-independent: the datum is the cone $\mathfrak{P}$ , not a preferred vector. But it furnishes only the commutant pairing $(\mathcal{M},\mathcal{M}')$ — not a tensor factorization.

2. Connes fusion / Sauvageot relative tensor product. The Connes spatial derivative $d\psi/d\varphi$ and Sauvageot's relative tensor product $\mathcal{H}\otimes_N\mathcal{K}$ of bimodules (correspondences) over a common $N$ give a composition operation that "adjusts the naive algebraic tensor product," defined intrinsically from the bimodule structure, requiring no Hilbert-space tensor factorization and no collar (Connes; Sauvageot 1983; Thom) ESTABLISHED, web-confirmed. It is independent of the chosen faithful normal weight up to unique isomorphism, and is defined for any type including III $_1$ .

3. The van Luijk operational program. The van Luijk–Stottmeister–Werner–Wilming work (arXiv:2401.07299; PRL 133, 261602 (2024) = arXiv:2401.07292; review arXiv:2510.07563, Dec 2025) derives the von Neumann–algebraic description of a subsystem from operational axioms (Limits, Ancillas, Dilations, Reversibility, Postselection). Its core theorem: the implementable operations are exactly the $\mathcal{M}$ -inner ones for a unique von Neumann algebra $\mathcal{M}$ , and a bipartite system is a commuting pair $(\mathcal{M}_A,\mathcal{M}_B)$ on a shared $\mathcal{H}$ — state-independent, collar-free, no tensor product assumed ESTABLISHED, web-confirmed.

Referee correction carried (op48c:claim). The axiom-enumeration label "A1–A5" and the descriptor "Theorem A" are the wiki's paraphrase of arXiv:2510.07563, not verbatim named results in the source, and are used here only as shorthand. Likewise, Haag duality is a property a net may satisfy or violate (it can fail under spontaneous symmetry breaking, for disconnected regions, etc.); what is genuinely type-agnostic is the commuting-pair / standard-form / Connes-fusion structure, not "Haag duality for all types." [FLAGGED]

The attempt

Each candidate was carried to its honest limit, then assembled into the strongest QM-selection argument.

Step 1 — A state-independent, collar-free composition surrogate DOES exist. Combine:

(a) the Haagerup standard form, which canonically (state-independently) furnishes the commuting pair $(\mathcal{M},\mathcal{M}')$ via $J\mathcal{M}J=\mathcal{M}'$ and the self-dual cone $\mathfrak{P}$ ;
(b) the van Luijk operational axiomatization, which derives precisely this commuting-pair structure from operations; and
(c) Connes fusion $\otimes_N$ as the intrinsic, weight-independent composition law over a shared algebra.

Each ingredient is web-confirmed ESTABLISHED. Therefore the iter-4 framing — "any canonical composition re-imports a reference state (BREAK 1) and a collar (BREAK 2)" — is FALSE for this surrogate: the commuting-pair / standard-form / Connes-fusion package is genuinely collar-free (no $O_1\Subset O_2$ ; defined for touching/coincident algebras) and reference-state-free (canonical from the cone, not a vector). This is a real advance on (c): the surrogate exists and is exhibited.

Step 2 — Does it single out QM? Here the attempt breaks. The surrogate is provably type-agnostic: the commuting-pair structure and standard form hold for every von Neumann algebra type (I, II, III) verbatim. They characterize "von-Neumann / quantum-logical composition in general," not Type-I textbook QM. In particular they contain no local-tomography clause that would force the complex field and the tensor product; the commuting-pair / Haag-duality datum is strictly weaker than $\dim V_{AB}=\dim V_A\dim V_B$ .

Step 3 — The operational invariant. The same framework supplies a state-independent operational invariant that separates types: the worst-case embezzlement error $\kappa_{\max}(\mathcal{M}_A,\mathcal{M}_B)$ (van Luijk et al., arXiv:2401.07299, Thm E; companion arXiv:2401.07292). Its values ESTABLISHED, web-confirmed:

\kappa_{\max}=2 \quad\text{(semifinite: Type I \emph{and all} Type II — no embezzling states)},

\kappa_{\max}=\frac{2\,(1-\sqrt{\lambda})}{1+\sqrt{\lambda}} \quad\text{(Type III}_\lambda), \qquad \kappa_{\max}=0 \quad\text{(Type III}_1\text{ — every normal state embezzling: universal embezzler).}

Note the convention: $\kappa$ is a failure measure — $\kappa=0$ means embezzlement succeeds universally, $\kappa=2$ means it maximally fails.

So the collar-free commuting-pair composition, applied to the physically relevant Type III $_1$ local algebra, has a non-classical operational signature — universal entanglement embezzlement — that finite-dimensional QM (Type I, $\kappa_{\max}=2$ ) provably does not share.

Referee correction carried (op48c:claim, op48c:NI1) — load-bearing. $\kappa_{\max}=2$ holds for Type I AND all of Type II (semifinite), not Type I alone. So $\kappa_{\max}=2$ does not identify Type I, and the value of the embezzlement invariant alone does not force a tensor factorization: the wiki's own CLPW construction yields a Type II $_1$ factor, which has $\kappa_{\max}=2$ yet has no local-tomography subsystem factorization (it has a trace, not subsystem tensor factors). The Type-I interposition in Step 4 below therefore comes from the split property (an independent structural input), not from " $\kappa_{\max}=2$ " per se. [FLAGGED — corrects the iter-5 first-pass framing.]

Step 4 — The assembled selection argument (and where it must be hedged). The remaining inference is: to force QM-like composition (a genuine tensor factorization with bounded embezzlement) you must leave Type III $_1$ — i.e. interpose a Type I factor, which on a net is the split construction with its collar and reference-state canonicity tax. As corrected, this requires the split property's atoms + factorization, not merely $\kappa_{\max}=2$ ; the dichotomy is real but the bridge "QM-type composition $\Leftrightarrow$ $\kappa_{\max}=2$ " is not a cited theorem and is carried as INFERENCE.

Where it breaks / what moved (didItMove = advanced)

The break is at Step 2, and iter-5 makes it a structured obstruction rather than the bare iter-4 OPEN. The exhibited state-independent, collar-free surrogate (Haagerup cone + commuting-pair operational axiomatization + Connes fusion) singles out "von Neumann / quantum-logical composition," but not quantum theory in the GPT sense, for two web-verified reasons:

Type-agnosticism. The commuting-pair structure and standard form hold for all types, so they cannot select the complex field, the tensor product, or Type I. There is no collar-free analogue of local tomography in the package.
Wrong-direction operational signature. On the physical Type III $_1$ factor the collar-free composition is a universal embezzler ( $\kappa_{\max}=0$ ): every normal state lets one extract arbitrary entanglement by local operations with arbitrarily small disturbance. Finite-dim QM composition (Type I) is the antipodal $\kappa_{\max}=2$ . So the collar-free surrogate, on III $_1$ , is operationally distinguishable from — indeed opposite to — finite-dim QM.

What moved: OPEN $\rightarrow$ a structured conditional no-go with a web-verified operational invariant ( $\kappa_{\max}$ ) doing the separating work. didItMove = advanced. This did not flip the headline verdict: no emergent Lorentzian geometry from non-symmetric data was derived, and the QM-selecting axiom is shown to coincide with the geometry-/state-importing split input, reinforcing rather than overturning encodes-not-generates (see ../README.md and ../verdicts.md).

Furthest defensible claim

INFERENCE, high — obstruction-sharpening, NOT a clean proved theorem On a single hyperfinite Type III $_1$ factor there is a strong tension — plausibly a conditional no-go modulo one named bridge — between the two desiderata of OP-48(c):

A genuinely state-independent, collar-free composition surrogate exists: the commuting pair $(\mathcal{M},\mathcal{M}')$ from the Haagerup standard form, with operations characterized as $\mathcal{M}$ -inner via the van Luijk operational axiomatization, and Connes fusion as a weight-independent associative composition law. But it is type-agnostic and, on III $_1$ , carries the universal-embezzlement signature $\kappa_{\max}=0$ — the operational antipode of finite-dim QM (Type I, $\kappa_{\max}=2$ ) — so it does not single out quantum theory.

Conversely, recovering finite-dim-QM-like composition (atoms, exact tensor factorization, local tomography, no embezzling states) on a III $_1$ region requires interposing a Type I factor via the split property, which re-imports a nonzero collar (Fewster: fails for adjacent regions) and a reference-state canonicity tax (canonical intermediate factor fixed only by $\Omega$ 's modular $J$ ; Doplicher–Longo 1984; Hollands–Longo 2020).

Hence "collar-free + state-independent" and "singles out QM" are in irreducible tension on one III $_1$ factor.

Two referee-mandated hedges baked into the statement above:

$\kappa_{\max}=2$ does not force Type I — Type II and III $_0$ share it. The Type-I interposition is supplied by the split property (atoms + factorization), not by the embezzlement value. [FLAGGED]
" $\kappa_{\max}$ signature $\Leftrightarrow$ local tomography / tensor factorization / QM-ness" is an unproven bridge — the embezzlement literature (arXiv:2401.07299/2401.07292) never connects $\kappa_{\max}$ to local tomography, the tensor product, the split property, or GPT composition. It is carried as INFERENCE, not theorem-grade. [FLAGGED]

If it fails / weaker statement

If the no-go (horn C) is wrong, the failure point is the implicit assumption that QM-selecting composition requires bounded embezzlement / a Type I factorization. The loophole: a complex-field/tensor-selecting axiom might be expressible directly at the III $_1$ commuting-pair level (e.g. a modular-compatible reversibility clause or a Connes-fusion-based duotensor condition) without demanding $\kappa_{\max}=2$ — so QM could in principle be singled out among type-agnostic von Neumann composition theories by a clause orthogonal to the embezzlement invariant. Note also that $\kappa_{\max}=0$ on III $_1$ means III $_1$ already supports a rich bipartite operational task (embezzlement) without any Type I split — evidence that III $_1$ has its own bipartite structure, which itself cuts against "QM-like composition only via splitting."

To probe the loophole one would check whether the Pati no-signalling inner-product rigidity (arXiv:2601.13012) can be re-expressed intrinsically on a III $_1$ commuting pair without presupposing the tensor structure it currently assumes. Iter-4 and iter-5 both find it presupposes $\mathcal{H}_A\otimes\mathcal{H}_B$ and local unitaries (verified verbatim), so as published it fixes the inner product on a given Hilbert space, not the composition rule, and cannot be stated on a single III $_1$ factor — it does not close the loophole. Similarly Torres Alegre (arXiv:2602.09056) fixes only the Born form via GNS, not composition.

Weakest defensible fallback (if even existence, horn A, is contested): the Haagerup standard form alone is uncontestably state-independent and collar-free, so at minimum the bare commutant pairing $(\mathcal{M},\mathcal{M}')$ is a canonical, reference-free "composition arena." The no-go would then bite only on the "singles out QM" clause — which is the published, web-verified part.

Proposed registry items (with referee verdicts)

`OP-GPT-TYPE3` — gap-refinement [bundle: MIXED]

OP-48 point (c) resolves as a conditional no-go (upgrading the iter-4 OPEN), entirely within operator-algebra / QI structure and with no geometric (Lorentzian) content. (A) ESTABLISHED A state-independent, collar-free composition surrogate exists on a III $_1$ factor: the commuting pair $(\mathcal{M},\mathcal{M}')$ from the standard form, with operations characterized as $\mathcal{M}$ -inner via the van Luijk operational axiomatization (arXiv:2510.07563), and Connes fusion / the Sauvageot–Connes relative tensor product as a weight-independent composition law. (B) ESTABLISHED It does NOT single out QM: type-agnostic; no local-tomography clause (the commuting-pair datum is strictly weaker than $\dim V_{AB}=\dim V_A\dim V_B$ , and on factors corresponds only to subfactor irreducibility); on III $_1$ carries $\kappa_{\max}=0$ (arXiv:2401.07299), opposite to finite-dim/Type I QM ( $\kappa_{\max}=2$ ). (C) INFERENCE, high Recovering QM-type composition (tensor factorization / local tomography) requires interposing a Type I factor via the split construction (Doplicher–Longo 1984), re-importing product structure and a canonical reference state; hence "collar-free + state-independent" and "singles out QM" cannot both hold on one III $_1$ factor.

Referee verdict (op48c:NI0 — keep, severity minor, isSound = true). Accepted with cosmetic flags: the "A1–A5"/"Haagerup standard form"/"canonicity tax" labels are the wiki's faithful paraphrase, not verbatim source enumerations; the Thom attribution for Connes fusion is loose (rigorously Sauvageot 1983 / Connes). Substance verified: arXiv:2510.07563 and arXiv:2401.07299 are real; the $\kappa_{\max}$ values are correct; the split property (Doplicher–Longo 1984) is correctly characterized; "Haag duality strictly weaker than local tomography" is correct, with recent work (arXiv:2509.12911, PRL) confirming Haag duality $\Rightarrow$ local tomography only for factors. No fabricated citations, theorems, or numbers. Makes no Lorentzian claim and does not purport to flip the headline. Bundle tag MIXED stands.

`HYP-EMBEZZLE-SELECT` — new-hypothesis [tag downgraded to SPECULATIVE]

The state-independent algebraic witness $\kappa_{\max}(\mathcal{M})$ vanishes ( $=0$ ) exactly on Type III factors (universal embezzlement) and is maximal ( $=2$ ) on all semifinite factors. Restricted to distinguishing Type I from Type III $_1$ (NOT Type II, which also gives $2$ ), it offers a state-independent re-encoding of the iter-4 obstruction: recovering textbook-QM (Type I) composition / local tomography on a III $_1$ net forces leaving the III $_1$ class, implemented on a net by the split (intermediate Type I factor) and therefore collar- and reference-state-dependent. This sharpens, and does not flip, encodes-not-generates, by supplying a state-independent invariant where iter-4 used a state-dependent split argument.

Referee verdict (op48c:NI1 — keep, severity MAJOR, isSound = false on the original strong form). The original phrasing — " $\kappa_{\max}$ precisely distinguishes Type I from III $_1$ ," "operationally equivalent to demanding a Type I intermediate factor," "two faces of the same requirement," tagged INFERENCE, high — was rejected for: (1) false type-discrimination ( $\kappa_{\max}=2$ for Type I and II; only III is $0$ ); the Type II $_1$ counterexample breaks " $\kappa_{\max}=2\Rightarrow$ factorization"; (2) an unsupported equivalence chain (the source never connects embezzlement to local tomography, tensor factorization, the split property, or GPT composition); (3) a single-factor vs inclusion category error ( $\kappa_{\max}$ is a single-factor invariant; the split concerns a nested pair); (4) internal tension — $\kappa_{\max}=0$ shows III $_1$ has its own bipartite operational structure without any split; (5) a semantic gloss ( $\kappa=2$ is maximal embezzlement failure, not "QM-good high value"). Recommended tag: SPECULATIVE. The refined, defensible residue is the INFERENCE, low–medium statement above: $\kappa_{\max}$ is a state-independent witness separating semifinite from Type III, and restricted to I-vs-III $_1$ it re-derives the iter-4 obstruction with a state-independent invariant in place of a state-dependent split — a modest strengthening, not a new equivalence. Recorded accordingly.

Verdict

SETTLED as a clean conditional no-go for OP-48(c) (medium-high confidence); the headline did NOT flip — encodes-not-generates STANDS and is tightened.

(i) Existence. A state-independent, collar-free composition surrogate genuinely exists — the commuting pair $(\mathcal{M},\mathcal{M}')$ from the Haagerup standard form, operationally axiomatized by van Luijk et al., with Connes fusion as its weight-independent composition law ESTABLISHED, web-confirmed. This refutes the iter-3/4 presumption that any canonical composition must cost a collar and a reference state.
(ii) No selection. But it does not single out QM: type-agnostic, no local-tomography clause, and on the physical Type III $_1$ factor a universal embezzler ( $\kappa_{\max}=0$ ), the operational opposite of finite-dim QM (Type I, $\kappa_{\max}=2$ ) ESTABLISHED, web-confirmed.
(iii) The tax returns. Any QM-selecting composition (atoms / tensor factorization / local tomography) on a III $_1$ region forces a Type I intermediate factor via the split property, re-importing collar- and state-dependence INFERENCE, high. So "collar-free + state-independent" and "singles out QM" are incompatible on one III $_1$ factor.

The 2026 rigidity results do not rescue selection: Pati (arXiv:2601.13012) fixes the inner product but presupposes the bipartite tensor product (verified verbatim) and is not statable on a single III $_1$ factor; Torres Alegre (arXiv:2602.09056) fixes only the Born form, not composition.

didItMove = advanced (OPEN $\to$ structured conditional no-go with a web-verified operational invariant doing the work). Not flips-headline-verdict: no emergent Lorentzian geometry from non-symmetric data was derived, and the QM-selecting axiom is shown to coincide with the geometry/state-importing input. Confidence medium-high: horns (A),(B) are ESTABLISHED and triple-web-verified; horn (C)'s "must interpose Type I" step is INFERENCE — strongly supported by the type classification but not a cited theorem in the exact phrasing, and the embezzlement $\leftrightarrow$ composition bridge is explicitly conjectural.

Open subquestions

OPEN — decides HYP-EMBEZZLE-SELECT Is horn (C) a theorem: must every composition surrogate with QM-type behavior (local tomography / genuine tensor factorization) on a III $_1$ net factor through a Type I intermediate factor (the split)? Or is there a complex-field/tensor-selecting axiom expressible directly at the III $_1$ commuting-pair level, orthogonal to the embezzlement invariant?
OPEN Can the Pati no-signalling inner-product rigidity (arXiv:2601.13012) be reformulated intrinsically on a III $_1$ commuting pair $(\mathcal{M},\mathcal{M}')$ without presupposing $\mathcal{H}_A\otimes\mathcal{H}_B$ — e.g. via the natural cone / a Connes-fusion duotensor — turning an inner-product-fixing result into a composition-fixing one? As published it cannot.
OPEN Does the van Luijk axiom set admit a strengthening (a tomographic / reversibility / complex-structure clause) that singles out Type I among all von Neumann types without a collar — or is type-agnosticism provably irreducible for any operational axiomatization on a shared Hilbert space?
OPEN Connes fusion is weight-independent up to unique isomorphism: is the natural transformation implementing that isomorphism itself free of any modular/geometric input, or does canonicity of the fusion associator covertly re-introduce a preferred weight (a higher-categorical analogue of the iter-4 canonicity tax)?
OPEN — verification Peer-review status of the 2026 preprints (Pati arXiv:2601.13012; Torres Alegre arXiv:2602.09056) remains unconfirmed; the van Luijk PRL (133, 261602, 2024) and the embezzlement-classification results are published/peer-reviewed and are the load-bearing citations for the no-go.

OP-48 point (c): a state-/collar-free composition surrogate on Type III₁?

Goal

Setup (real math)

The attempt

Where it breaks / what moved (didItMove = advanced)

Furthest defensible claim

If it fails / weaker statement

Proposed registry items (with referee verdicts)

`OP-GPT-TYPE3` — gap-refinement [bundle: MIXED]

`HYP-EMBEZZLE-SELECT` — new-hypothesis [tag downgraded to SPECULATIVE]

Verdict

Open subquestions

See also

Key results cited

OP-48 point (c): a state-/collar-free composition surrogate on Type III₁?

Goal

Setup (real math)

The attempt

Where it breaks / what moved (didItMove = advanced)

Furthest defensible claim

If it fails / weaker statement

Proposed registry items (with referee verdicts)

OP-GPT-TYPE3 — gap-refinement [bundle: MIXED]

HYP-EMBEZZLE-SELECT — new-hypothesis [tag downgraded to SPECULATIVE]

Verdict

Open subquestions

See also

Key results cited

`OP-GPT-TYPE3` — gap-refinement [bundle: MIXED]

`HYP-EMBEZZLE-SELECT` — new-hypothesis [tag downgraded to SPECULATIVE]