OP-48(c) EXIT X1: singular product states on the III₁ Haag-dual pair exist (theorem) and constitute a framework exit, not a closable residue

Status: ATTEMPT executed — didItMove = residue-characterized on OP-48(c) exit X1. Both mission questions answered: (1) EXISTENCE is upgraded from a referee-sketched route to a theorem with named, live-verified inputs — the product state exists by Roos 1970 (on plain C*-independence) and is non-normal by Takesaki 1958 (the normal-product-state obstruction, Rédei–Summers Prop. 2); Florig–Summers 1997 is the separate C*-side faithful-product-state characterization (Rédei–Summers Prop. 4), not the obstruction — assembled via Rédei–Summers and Halvorson read at primary level this session; (2) DISQUALIFICATION resolves sharper than posed — no single reconstruction axiom fails on a singular state, so X1 does not collapse horn C to "theorem-conditional-only-on-X2"; X1 is confirmed a genuine framework exit whose price is itemized as a theorem. Medium-high confidence. The headline verdict of ../CONCLUSION.md is unaffected and tightened. Last updated: 2026-06-10 Iteration: 8 (track A4)

This note executes the first of the two named exits left by 2026-06-10-iter7-op48c-fullpair-forcing.md §1.4, where horn C of ../OPEN_PROBLEMS.md OP-48(c) was hardened to theorem-conditional-on-framework with residue relocated to {the normal-state postulate} ∪ {the composition-primitive choice → X1 singular states / X2 non-OPT embezzlement}. The iter-7 referee (R7-A4-F5, binding) voided the universal-property existence argument and prescribed the corrected route; this note makes that route rigorous and then asks the physics question the mission poses: if such states exist, can a composition rule on them count as "recovering QM"?

Scope honesty, up front. (1) The load-bearing independence theorems were read at the primary/authoritative-text level this session: Rédei–Summers arXiv:0810.5294v2 page-by-page (Defs 1–4, Props 1–4, 10–13, the hyperfinite-III example p. 10); Halvorson math-ph/0602036 §§2.5, 3.1–3.3 page-by-page (Schlieder Def 3.4, C*-independence Def 3.6, the implication box p. 28, the type-III "no pure normal states" remark p. 25). (2) The three reconstruction axiom sets are carried verbatim from iteration 7 (Hardy/CDP/MM, primary-PDF-read there); re-confirmed at statement level this session, not re-read page-by-page. (3) Buchholz 1974 normal-product-state results are cited as a contrast datum only and the tangent-region detail is [unverified this session] (Springer auth wall). (4) The Roos 1970 product-state theorem is verified via Rédei–Summers' explicit attribution, not from Roos's own PDF.

Part 1 — EXISTENCE: the corrected route, made rigorous (X1-EXIST is a theorem)

1.1 The five links, each cited and live-verified

Let $\mathcal M$ be a type III $_1$ factor in standard form on a separable Hilbert space $\mathcal H$ , $\mathcal M'$ its commutant — the Haag-dual pair. We want a product state $\omega(ab)=\varphi_A(a)\varphi_B(b)$ , $a\in\mathcal M$ , $b\in\mathcal M'$ , on the composite $AB=B(\mathcal H)$ .

Step 1 — the factor pair has the Schlieder property. [ESTABLISHED] For a von Neumann factor $\mathcal M$ , the commuting pair $(\mathcal M,\mathcal M')$ satisfies the Schlieder property ( $a\in\mathcal M,b\in\mathcal M',\,ab=0\Rightarrow a=0$ or $b=0$ ; equivalently nonzero $e\in\mathcal M$ , $f\in\mathcal M'$ ⇒ $ef\ne0$ ). The proof is the central-carrier fact (Kadison–Ringrose: every nonzero projection in a factor has central carrier $I$ ), which gives, for nonzero projections $e\in\mathcal M$ , $f\in\mathcal M'$ , that $ef\neq0$ — not Halvorson Def 3.4, which is merely the definition of the Schlieder property. So for commuting factors the Schlieder property is automatic — confirmed this session. (The AQFT route Halvorson Prop 3.16 — microcausality + property B + strong spacelike separation ⇒ Schlieder — is the net-level sufficient condition; for the abstract full Haag-dual pair the factor property alone suffices, no spacelike-separation input needed.) Type III $_1$ is a factor, so the pair has the Schlieder property unconditionally.

Step 2 — Schlieder ⟺ C*-independence (commuting case). [ESTABLISHED] Halvorson's implication box (p. 28, verbatim): $\text{Split}\Rightarrow W^*\text{-indep}\Rightarrow C^*\text{-indep}\iff\text{Schlieder property}$ , the last equivalence holding for mutually commuting pairs. Hence $(\mathcal M,\mathcal M')$ is C*-independent (Halvorson Def 3.6 / Rédei–Summers Def 1: every state of $\mathcal M$ is compatible with every state of $\mathcal M'$ ).

Step 3 — plain C*-independence ⇒ a PRODUCT-state extension exists (Roos 1970). [ESTABLISHED — Roos 1970] Rédei–Summers p. 4, verbatim (the sentence after Def 1): "if $\mathcal A_1,\mathcal A_2$ are commuting C*-algebras, then the extension state $\phi$ in Definition 1 may be chosen to be a product state [24]" — [24] = H. Roos, Commun. Math. Phys. 16, 238–246 (1970). This is exactly the existence claim X1 needs: plain C*-independence already yields a single product state $\omega(ab)=\varphi_A(a)\varphi_B(b)$ on $C^*(\mathcal M\cup\mathcal M')$ , via the Roos sentence. No product-sense isomorphism is required. (Correction, binding: one must not claim $\mathcal M\vee_{C^*}\mathcal M'\cong\mathcal M\otimes_{\min}\mathcal M'$ — "C*-independence in the product sense" — holds for every III $_1$ factor. Rédei–Summers Prop. 1 explicitly denies the general implication: plain C*-independence ⇏ C*-independence in the product sense; the product-sense iso is established by Rédei–Summers only in their hyperfinite-type-III example, not in general. The Florig–Summers characterization, Rédei–Summers Prop. 4 = ref [14], is the faithful-product-state ⟺ product-sense statement — a stronger, separate result not needed here.)

The Roos sentence makes the referee's "min-norm-domination" sub-step unnecessary directly: a single product state exists without any norm estimate, on plain C*-independence alone. (The min-norm/product-sense route would also deliver one, but only under the stronger product-sense hypothesis Rédei–Summers Prop. 1 shows is not automatic.)

Step 4 — push to $AB=B(\mathcal H)$ by positive Hahn–Banach. [ESTABLISHED] $C^*(\mathcal M\cup\mathcal M')$ is a unital C*-subalgebra of $B(\mathcal H)$ . A state on a unital C*-subalgebra extends to a state on the ambient C*-algebra: Hahn–Banach gives a norm-preserving linear extension $\tilde\omega$ with $\|\tilde\omega\|=\|\omega\|=\omega(1)=1$ ; a unital functional on a C*-algebra with $\|\tilde\omega\|=\tilde\omega(1)$ is automatically positive, hence a state. (Equivalently $B(\mathcal H)$ is injective.) So $\omega$ extends to a state on the composite $AB=B(\mathcal H)$ .

Step 5 — the product state is NON-NORMAL (the load-bearing point). [ESTABLISHED — Takesaki 1958 / Florig–Summers 1997] On the III $_1$ pair, no product state can be normal. A normal product state across commuting factors $(\mathcal N_1,\mathcal N_2)$ exists iff $\eta(XY)=X\otimes Y$ extends to a $W^*$ -isomorphism $\mathcal N_1\vee\mathcal N_2\cong\mathcal N_1\,\overline\otimes\,\mathcal N_2$ (Rédei–Summers Prop 2, Takesaki 1958). For $\mathcal M$ type III $_1$ , $\mathcal M\vee\mathcal M'=B(\mathcal H)$ (Type I) while $\mathcal M\,\overline\otimes\,\mathcal M'$ is Type III (Halvorson 3.10(iii): $R\,\overline\otimes\,R'$ has the same type as $R$ ; for type II/III, $\overline{R\vee R'}$ is strictly larger than and not isomorphic to $R\,\overline\otimes\,R'$ ). The isomorphism fails ⇒ no normal product state exists. Rédei–Summers state the consequence as a worked example (p. 10, verbatim): "choosing $\mathcal N_1$ to be the hyperfinite type III factor and $\mathcal N_2=\mathcal N_1'$ , the pair $(\mathcal N_1,\mathcal N_2)$ is C*-independent in the product sense, but it is not $W^*$ -independent in the product sense." The Step-3 product state therefore exists but is non-normal — singular. (Independent cross-check: the iter-6/iter-7 T1 corollary says a normal product state would force $\mathcal M$ Type I.)

1.2 The theorem

THEOREM X1-EXIST. [INFERENCE, high — assembly mine; every link cited and live-verified] Let $\mathcal M$ be a type III $_1$ factor in standard form on a separable Hilbert space $\mathcal H$ , $\mathcal M'$ its commutant. Then:

there exists a state $\omega$ on $B(\mathcal H)$ with $\omega(ab)=\varphi_A(a)\varphi_B(b)$ for all $a\in\mathcal M,b\in\mathcal M'$ (a product state across the Haag-dual pair); and
every such product state is non-normal (singular).

Hypotheses, flagged. (i) $\mathcal M$ a factor — essential (Step 1; drop it and Schlieder/C*-independence can fail — Halvorson Ex. 3.7, Prop 3.8). (ii) Separable predual — load-bearing for the stated form of the Step-5 obstruction; the conclusion also follows from the type-independent T1 corollary, so the result is robust to this technical choice. (iii) The cited Roos/Takesaki/Florig–Summers theorems (verified this session via Rédei–Summers and Halvorson).

This promotes the iter-7 §1.4 claim "(X1) singular product states exist" from a referee-corrected route to a theorem with named inputs. The mission's verification targets are answered: yes, "product states across commuting C*-algebras exist iff the pair is C*-independent" — Roos 1970 (commuting case, the product can be chosen); yes, the Haag-dual factor pair is C*-independent (Schlieder ⟺ C*-independence, factor ⇒ Schlieder); for wedge algebras this is the standard Schlieder-property instance, and for an abstract III $_1$ factor and its commutant it holds by the factor property alone.

Part 2 — DISQUALIFICATION: does a composition on singular states "recover QM"?

The physics question. I argue both horns against the verbatim axioms iteration 7 verified.

2.1 Horn "recovers-QM = NO" (the standard folklore — correct here)

Singular states have:

no density matrix (not in the predual);
no countably-additive probability assignment (normality = $\sigma$ -additivity on orthogonal projections; a singular state is only finitely additive);
no pure-state ignorance reading — Halvorson p. 25, verbatim this session: "type III algebras have no pure normal states ... this absence of pure states is a further obstacle to an ignorance interpretation of quantum probabilities."

By the basic weak- density of normal (vector) states in the state space of $B(\mathcal H)$ * (the elementary vector-state density fact — not Fell's theorem, which is specifically the weak-* density of vector states of representations with the same kernel, i.e. a quasi-equivalence/folium result) a singular state is a weak-* limit of physical density-matrix states but is not norm-approximable by them and is not one of them. A composition rule whose preparations are singular states thus has no $\sigma$ -additive probability calculus and no tomography in the textbook sense.

2.2 Horn "recovers-QM = MAYBE" (the steelman) — and what it actually shows

The GPT frameworks nowhere demand "normal." So which stated axiom would a singular state violate? Reading the three (axioms carried verbatim from iter-7):

Reconstruction	State object (verbatim/near-verbatim)	What a singular state does
Hardy 2001	state = the finite list of $K$ fiducial-measurement probabilities; $K=\min$ #measurements determining the state; $N,K$ finite; H4: $N=N_AN_B$ , $K=K_AK_B$	$K=\infty$ ; not determined by any finite list — not a state of the theory
CDP 2011	finite-dimensional state spaces (p. 6); states = points of a finite-dim convex body; Axiom 4 local distinguishability	no finite-dim convex / density-matrix rep — not in $\mathrm{St}(AB)$
Masanes–Müller 2011	finite $d_A$ ; Req. 2 local tomography via a finite fiducial frame	not characterized by any finite frame — outside the state space

THEOREM X1-DISQ. [INFERENCE, high — from the verbatim axiom statements] In none of Hardy / CDP / Masanes–Müller does a singular state violate a stated axiom. In all three the state object is, by construction, a finite-dimensional, countably-additive (normal / density-matrix) functional, and a singular product state is not an element of the framework's state space at all — the disqualification is by non-membership in the state space, which sits upstream of every axiom. Therefore:

X1 does not strengthen horn C to "theorem-conditional-only-on-X2 with an axiom provably failing on singular states." There is no such single failing axiom.
X1 is a genuine framework exit — exactly the iter-7 referee's binding conclusion — now established as a theorem rather than asserted.

Note the asymmetry with the iter-7 LT-sep result: the separation half of local tomography holds on the III $_1$ pair (singular product states do separate normal states — von Neumann density + factor), so the exit is not at distinguishability. It is at the preparability / $\sigma$ -additivity / finiteness layer.

2.3 The price tag (itemized as a theorem)

COROLLARY X1-COST. [INFERENCE, high] To admit X1's singular product states as legitimate preparations, a reconstruction must simultaneously relinquish all three of:

(a) states = density matrices — drop the predual identification; "state" becomes an arbitrary (possibly singular) state on a C*-algebra;
(b) $\sigma$ -additivity of probabilities — Hardy's H1 / the Kolmogorov-style countable additivity over a measurement's outcomes weakens to mere finite additivity;
(c) finite tomographic dimension — $K,d,D\to\infty$ ; the local-tomography dimension count $D_{AB}=D_AD_B$ that is the operational engine of all three reconstructions loses its finite content (its separation content survives — §2.2).

These are not three independent costs one can pay piecemeal: each of the three reconstructions builds its entire derivation on the finite-dimensional, density-matrix, $\sigma$ -additive state. Paying (a)+(b)+(c) is leaving the GPT program, not weakening one of its axioms.

2.4 Horn C, updated form

Horn C stands at theorem-conditional-on-framework (iter-7). This track does not eliminate X1 — it cannot; X1's states provably exist (X1-EXIST) — but it converts X1 from an open escape into a characterized, costed exit:

existence: theorem — the singular product state exists by the Roos 1970 product-extension sentence (on plain C*-independence) and is non-normal by Takesaki 1958 (Tôhoku Math. J. 10, 116–119, via Rédei–Summers Prop. 2). (Florig–Summers 1997 is the separate C*-side faithful-product-state characterization, Rédei–Summers Prop. 4 = ref [14] — neither the existence input nor the non-normality obstruction; do not jointly attribute the obstruction to it.);
disqualification: theorem that no reconstruction axiom fails on singular states, because the normal/density-matrix state postulate is itself a load-bearing, non-derivable framework axiom (X1-DISQ);
cost: theorem that the exit forces (a)+(b)+(c) jointly (X1-COST).

So the OP-48(c) residue's two exits are now confirmed as the two independent axes along which one must leave the framework: X1 = the state-object axis (normal vs singular states — characterized and costed here), X2 = the composition-primitive axis (product preparations vs III $_1$ -embezzlement — untouched by this track, exits OPT on a different axis). Neither is a "reading"; both are exits; operator algebra and GPT-reconstruction theorems choose neither. This is the encodes-not-generates pattern recurring at the state-postulate layer, sharpened to a theorem: the normal-state postulate is an installation choice. [INFERENCE, high] — consistent with ../HYPOTHESES.md HYP-ENCODING-SCREEN.

Honest overall verdict

didItMove = residue-characterized (medium-high confidence). The mission's two questions are answered in order: (1) EXISTENCE is a theorem — the corrected Schlieder/Roos/Hahn–Banach route is rigorous, and the "min-norm-domination" sub-step is subsumed by C*-independence-in-the-product-sense. (2) DISQUALIFICATION is decided, sharper than posed: no single axiom fails on singular states, so X1 does not close to "theorem-conditional-only-on-X2"; X1 is a genuine framework exit, and the physics it costs is now itemized (density-matrix states + $\sigma$ -additivity + finite tomographic dimension, jointly). The headline verdict is untouched and tightened: horn C remains theorem-conditional-on-framework, with the normal-state postulate now proven to be a load-bearing, non-derivable axiom rather than a tacit convenience. No experimental channel emerged; the wager's NOT-YET-PHYSICS status is unaffected.

Open subquestions (updated)

[OPEN — carried, now the only mathematical hedge under horn C besides X2] Primary-source spot-check of the Summers–Werner 1988 hypotheses (the uniform Bell gap of the approximate reading); independent of X1, which lives on the exact preparability layer.
[OPEN — X2, the other exit, untouched] Is the III $_1$ /universal-embezzlement composition endpoint a genuine rival axiomatization (a preparability-free reconstruction that still singles out a unique theory), or only a gesture? (Carried from iter-7 §"Open subquestions" #2.)
[OPEN — new] Does any weakening of Hardy/CDP/MM that admits infinite-dimensional, non- $\sigma$ -additive states still single out a unique theory — i.e. is there a "singular-state GPT reconstruction" at all, or does dropping (a)+(b)+(c) destroy uniqueness outright? A negative answer would upgrade X1-COST from "leaves the program" to "leaves the program and loses the reconstruction theorem," closing X1 as a useless exit.

References

Read at primary/authoritative level this session unless marked otherwise.

M. Rédei, S. J. Summers, When are quantum systems operationally independent?, arXiv:0810.5294 v2 (2009). [PDF read page-by-page: Defs 1–4 pp. 2–3; Prop 1–4 pp. 3–4 incl. the Roos product-state sentence p. 4; Props 10–12 + the hyperfinite-III example p. 10] — the assembly's primary carrier.
H. Halvorson (with M. Müger), Algebraic Quantum Field Theory, arXiv:math-ph/0602036. [PDF read: §2.5 pp. 20–25 (local algebras properly infinite / type III $_1$ ; "no pure normal states" p. 25); §3.1 pp. 26–28 (Schlieder Def 3.4; C*-independence Def 3.6; the Split⇒W*⇒C*⟺Schlieder box p. 28); §3.2–3.3 pp. 28–30 (Schlieder Props 3.13/3.16; Bell Props 3.19/3.21)].
H. Roos, Independence of local algebras in quantum field theory, Commun. Math. Phys. 16, 238–246 (1970). [theorem verified via Rédei–Summers attribution this session; Roos PDF not independently read].
M. Florig, S. J. Summers, On the statistical independence of algebras of observables, J. Math. Phys. 38, 1318–1328 (1997). [the faithful-product-state ⟺ C*-independence-in-product-sense characterization (Rédei–Summers Prop 4); verified via Rédei–Summers + abstract this session].
M. Takesaki, On the direct product of $W^*$ -factors, Tôhoku Math. J. 10, 116–119 (1958). [the normal-product-state ⟺ $W^*$ -isomorphism theorem (Rédei–Summers Prop 2); verified via Rédei–Summers].
L. Hardy, Quantum Theory From Five Reasonable Axioms, arXiv:quant-ph/0101012. [axioms carried verbatim from iter-7; H1/H4 + finite $K,N$ re-confirmed at statement level this session].
G. Chiribella, G. M. D'Ariano, P. Perinotti, Informational derivation of Quantum Theory, arXiv:1011.6451; Phys. Rev. A 84, 012311 (2011). [axioms + finite-dim state-space restriction carried verbatim from iter-7; re-confirmed at statement level].
L. Masanes, M. P. Müller, A derivation of quantum theory from physical requirements, arXiv:1004.1483; New J. Phys. 13, 063001 (2011). [carried verbatim from iter-7; re-confirmed].
Weak-* density of normal (vector) states in the state space of $B(\mathcal H)$ — the elementary vector-state density fact (standard; not Fell's theorem, which is the weak-* density of vector states of representations with the same kernel — quasi-equivalence/folium). [statement verified this session].
R. V. Kadison, J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, Vols. I–II (1983/1986) — the central-carrier fact (every nonzero projection in a factor has central carrier $I$ ) justifying Step 1 (factor ⟹ Schlieder). [standard; verified this session].
D. Buchholz, Product states for local algebras, Commun. Math. Phys. 36, 287–304 (1974). [contrast datum (normal product states for non-tangent separated regions); tangent-region normal-product-state failure [unverified this session] — Springer auth wall].
Carried, not re-verified this session: S. Schlieder (1969), the strict-spacelike-separation Schlieder property (Halvorson Prop 3.13 attribution); S. J. Summers, On the independence of local algebras in QFT, Rev. Math. Phys. 2 (1990) 201–247 [the canonical review; cited via Rédei–Summers/Halvorson].

Binding note. Where a correction in the referee verdict below conflicts with the body above, the referee correction governs; the body is the pre-referee submission, retained for the audit trail per house convention.

Binding note. Where the referee verdict below conflicts with the body, the referee correction governs (notably: the carrier-convergence count stays at FIVE — the net result extends route 5, it is not a sixth; OP-49 (c′) is reframed-and-narrowed, NOT closed). The body is the pre-referee submission, retained for the audit trail.

Referee verdict — Track A4-op48c-X1-singular-states, iteration 8

Outcome: RESOLVED-POSITIVE upheld. All four findings kept; no fatals, no majors. Two minor citation-attribution corrections (binding) and two over-generalized glosses flagged. Default stance was REFUTE; the core mathematics survived a step-by-step re-derivation and a verbatim primary-source citation audit.

Citation audit (live, this session)

Every load-bearing primary source was re-fetched and the load-bearing quotes verified verbatim:

Rédei–Summers, arXiv:0810.5294v2 (PDF + ar5iv): the Roos sentence after Def 1 — "if A1,A2 are commuting C-algebras, then the extension state φ in Definition 1 may be chosen to be a product state [24], i.e. φ(XY)=φ(X)φ(Y)=φ1(X)φ2(Y)", [24]=Roos 1970 (CMP 16, 238–246) — exact. Proposition 2 (Takesaki [30]: W-iso N1∨N2≅N1⊗̄N2 ⟺ ∃ normal product state) — exact. The hyperfinite-type-III example following Prop 12 ("C-independent in the product sense, but it is not W*-independent in the product sense") — exact. Def 2 confirmed to use the minimal C-norm. Proposition 1 confirmed: plain C*-independence does not imply product-sense C*-independence (converse false). Bibliography refs [11] D'Antoni–Longo, [14] Florig–Summers (JMP 38, 1318–1328, 1997), [24] Roos, [27] Summers–Werner, [30] Takesaki (Tôhoku 10, 116–119, 1958) all confirmed.
Halvorson, arXiv:math-ph/0602036 (PDF, pdftotext): Def 3.4 Schlieder property; the implication box "Split property ⇒ W-independence ⇒ C*-independence ⟺ Schlieder property"* under mutual commutativity — exact; Remark 3.10(iii) (R∨R'=B(H), R same type, R⊗̄R' same type; type II/III ⇒ R∨R' not iso to R⊗̄R') — exact; the wedge statement *"since R(W) and R(W') are type III1 factors, there can be no -isomorphism between R(W)⊗̄R(W') and R(W)∨R(W')=B(H)" — exact; the p.24 quote "type III algebras have no pure normal states ... this absence of pure states is a further obstacle to an ignorance interpretation of quantum probabilities" — exact (track said p.25; cross-version drift). No fabrications.

Re-derivation of X1-EXIST (the product)

The chain is logically valid and the conclusion is sound: (1) M factor ⟹ Schlieder for (M,M') [TRUE via central carriers]; (2) Schlieder ⟺ C*-independence [verbatim]; (3) plain C*-independence + commuting ⟹ a product state on C*(M∪M') [Roos, verbatim]; (4) Hahn–Banach to a state on B(H) [standard, correctly stated]; (5) every product state on B(H) is non-normal because M∨M'=B(H) is type I while M⊗̄M' is type III, so the Takesaki W*-iso fails [verbatim + type theory]. This faithfully executes the binding iter-7 referee route R7-A4-F5 and promotes X1 from a sketched route to a theorem. The factor hypothesis is essential (used in Step 1) and the track correctly flags it.

Corrections (binding)

Attribution of the non-normality obstruction. The headline/F1 jointly credit "Takesaki/Florig–Summers 1997". The obstruction (Prop 2: no normal product state) is Takesaki [30] alone. Florig–Summers [14] is the C-side* faithful-product-state characterization (Prop 4) — real, correctly cited elsewhere, but not the obstruction. Drop the joint attribution. [minor]
Step 1 citation. "factor ⟹ Schlieder" is cited to Halvorson Def 3.4, which is only the definition. The proof is the central-carrier fact (nonzero projections in a factor have central carrier I). Cite that, not a definition. [minor]
"Fell's theorem" (F3). The weak-* density of normal states in the state space is the basic vector-state density fact, not Fell's theorem (which is the folium/quasi-equivalence density result). Fix or drop the label. [minor]

Flags (not corrections, but the wiki text must not over-generalize)

The gloss "M∨_{C}M' ≅ M⊗_min M' (C*-independence in the product sense), so the min-norm-domination sub-step is subsumed"* over-generalizes the Rédei–Summers hyperfinite-type-III example to all III1. Rédei–Summers Prop 1 explicitly denies the general implication (plain ⇏ product-sense). This is harmless because the existence of one product state needs only plain C*-independence + the Roos sentence — the product-sense iso is not load-bearing. State it that way.
F2's "singular product states still separate normal states" is an extension of the iter-7 LT-sep lemma (proven for normal states), not a re-use of it; tag accordingly.

No-go and closure audit (track level)

The track is confined to the state-object axis of OP-48c (normal vs singular states) and engages no other closure. dS-cardinality no-go: untouched. HYP-CKV-VACUITY-R3 + extended smuggle list (indefinite-pairing/(n,n)-flag axiom; signature-valued template): not engaged — this track installs no readout, no η-form, no foliation; the state-existence theorem carries no signature content. The net's index set does not smuggle a localization (no net is used; the construction is a single Hahn–Banach extension on a fixed B(H)). Iteration-7 trichotomy lemma / modular-constructibility ansatz: not engaged (OP-46, different axis). OP-49 near-no-go: untouched (no pseudo-entropy/dS first law claim). HYP-ENCODING-SCREEN: reinforced at the state-postulate layer, consistent with iter-7; not relitigated. No circularity: horn C says no normal product state exists (PS fails); the track exhibits a singular product state — different state classes, fully consistent. Bell/CHSH, Reeh–Schlieder, Haag, PBR, Kochen–Specker, Coleman–Mandula, Weinberg–Witten, spin–statistics: not implicated.

Net

The X1-EXIST theorem is correct and citation-clean after the three minor attribution fixes. RESOLVED-POSITIVE is warranted as the execution of a named decidable step (the iter-7-prescribed route), not a headline flip — the track honestly leaves horn C at theorem-conditional-on-framework and confirms X1 as a characterized, costed framework exit. Headline verdict unchanged and tightened, 8th consecutive iteration. Object level: still NOT-YET-PHYSICS; no distinguishing experimental channel emerged.