OP-46 / carrier obligation (i), iteration 9: the naturality promotion — constructibility becomes a theorem, localization is the irreducible residual (and is the geometry)

Status: ATTEMPT executed — obligation (i) of the iter-8 net carrier no-go (replace the net-constructibility ANSATZ by a NATURALITY THEOREM) attacked head-on. The ansatz splits: its constructibility half (Gram operator is a function of the modular generators) is discharged to a naturality theorem via geometric-modular-action functoriality; its localization half is shown irreducible — "(loc)" is natural only relative to the net's index set, which is the geometry ( $n_1$ ). Net: the principal hedge is hardened in precision (its condition sharpens from an unproven algebra-membership ansatz to a proven-irreducible base-category hypothesis) but stays conditional HIGH, not unconditional HIGH. The carrier no-go is sharpened, not closed. Verdict unchanged (8th consecutive); carrier-convergence count stays at FIVE. Last updated: 2026-06-10 Iteration: 9

Grading instrument: 2026-06-08-iter3-encode-vs-generate-criterion.md ( $g_1$ – $g_7$ , $\kappa_\Phi$ , $\Delta_{\rm geo}$ ); net items $n_1$ – $n_3$ from 2026-06-10-iter8-net-carrier-no-go.md. Boundary inherited from iter-7 (the trichotomy + single-algebra ansatz) and iter-8 (the covariance–locality engine + net-constructibility ansatz + the two named obligations). Prior closures not relitigated.

Scope honesty: §§1–4 are this session's own mathematics, derived stepwise with every assumption flagged, and verified numerically twice (fresh RNG 20260610) in the chiral rapidity model and the half-line dilation model. The constructibility-half theorem (Lemma N1) rests on the GMA reconstruction result, web-verified live; the residual (§3) is a structural identification, not an ansatz. Citations web-verified this session unless marked unverified.

1. The naturality formulation

In the Brunetti–Fredenhagen–Verch locally-covariant idiom (fields/observables = natural transformations between functors) [ESTABLISHED — arXiv:math-ph/0112041], define:

Source $\mathsf{Net}$ : nets $\mathcal N=\{\mathcal M(O)\}_{O\in\mathcal I}$ with vacuum $\Omega$ and full modular package $\mathfrak m(\mathcal N)=(\{\Delta_O^{it}\},\{J_O\},\{\Delta_{O'|O}\},\{P_{O',O}\})$ ; automorphisms $\mathrm{Aut}(\mathcal N,\Omega)$ (which contains the modular action).
Target $\mathsf{Krein}$ : measurable fields $\{V_x\}$ with Hermitian forms $\eta_x$ .
Carrier functor $\Phi:\mathsf{Net}\to\mathsf{Krein}$ .
(n,n) as a natural transformation $\sigma:\Phi\Rightarrow\underline{(n,n)}$ (the signature flag, natural in $\mathrm{Aut}(\mathcal N,\Omega)$ ).
(loc) as a natural transformation $\ell:\Phi\Rightarrow\mathrm{Loc}_{\mathcal I}$ to a base-projection functor recording each fiber's support region in $\mathcal I$ .

A natural carrier that generates signature = a $\Phi$ with $\sigma$ indefinite and $\ell$ localized, natural, smuggle-free ( $g_1$ – $g_7$ , $n_1$ – $n_3$ false). [INFERENCE — definitions]

2. The constructibility half: discharged to a theorem

Lemma N1 (constructibility from naturality). [INFERENCE, high] An $\mathrm{Aut}(\mathcal N,\Omega)$ -natural modular-covariant carrier has Gram field $\eta\in W^*(\mathfrak m(\mathcal N))$ — forced, not assumed. Proof. Naturality includes equivariance under the modular subgroup, so $[\eta,\Delta_O^{it}]=0$ (the iter-3 C2 forcing), i.e. $\eta\in\{\Delta_O\}''$ . Globally, by Buchholz–Dreyer–Florig–Summers (arXiv:math-ph/9805026, verified) and Summers–White (arXiv:hep-th/0304179, verified), the modular conjugations $\{J_O\}$ under CGMA determine the spacetime, isometry group, and covariant representation — the geometry is a functor of the modular data, so every natural form is a natural function of it. ∎ This discharges the iter-8 net-constructibility ansatz on its constructibility half.

Lemma N2 (the engine, promoted to a naturality obstruction). [INFERENCE, high — numerically demonstrated twice] A localized indefinite natural form has a natural sign locus $\Lambda\subset\mathcal I$ , carried by every automorphism, hence by the modular flow. The flow acts as a free translation $\theta\mapsto\theta-2\pi t$ ( $p\mapsto e^{-2\pi t}p$ , Borchers); a free translation has no fixed locus in the open index set, so the only flow-invariant sign-pattern is a global constant — definite. Hence localized + natural + indefinite is impossible at finite locus.

Fixed-point subtlety, closed (§ engine leak): the half-line dilation fixes $p=0,\infty$ — the wedge edge, the boundary-at-infinity outside the open $\mathcal I$ ; a form concentrated there is supported on a geometric boundary = $n_1$ . So no escape. Numerics (fresh RNG): local sign form $\|m_{\rm boost}-m\|_\infty=2$ for all $t\neq0$ ; constant boost-invariant to $0$ ; the only invariant indefinite object is the Fourier multiplier $\mathrm{sgn}(\text{modular frequency})$ — natural but nonlocal ( $1/\mathrm{sep}$ Hilbert tail). The iter-8 dichotomy (positive-and-local / signature-free [by separate enumeration, not the naturality — binding iter-7 inheritance] / indefinite-but-nonlocal) survives the promotion. No natural carrier is both indefinite and localized.

3. The irreducible residual: localization is the geometry

The promotion exposes that (loc) was never a natural condition standing free of the geometry. The natural transformation $\ell:\Phi\Rightarrow\mathrm{Loc}_{\mathcal I}$ targets a base functor that is part of the source datum: "vanishing cross-coupling between disjoint regions" presupposes a region order on $\mathcal I$ — which is $n_1$ , and by BGL (positivity ⟺ isotony), Neeb–Ólafsson (Euler element), and Lashkari–Leung–Moosa–Ouseph (AdS₂-from-locality) is the geometry.

So the ansatz is replaced not by an unconditional theorem but by a sharper, provably-irreducible residual — the localization-template hypothesis $n_1$ :

Residual [OPEN — the exact naturality hypothesis that cannot be removed]. The net carrier no-go is unconditional given that (loc) is defined relative to $\mathcal I$ 's order; and that hypothesis cannot be made a theorem, because removing it removes the meaning of "localized." The geometry is the localization template.

Carrier-side confirmation: Neeb–Ólafsson (arXiv:1704.01336, verified) — modular $J$ 's ↔ antiunitary reps studied as "fiber bundles over ordered symmetric spaces": geometry in the base, no signature in the fibers. Komalan double-categorical AQFT (arXiv:2601.07807, Jan 2026, verified) — the spacetime double category (regions + inclusions) is the base the algebra-valued functor lives over; localization in the base, no indefinite fiber form.

4. The larger-structure loophole (tested, closed in the wrong direction)

A carrier natural under a larger structure than the modular generators — the obligation's named escape:

Subfactor standard invariant / planar algebra. [INFERENCE — closed] The standard invariant (arXiv:math/0304340) is a rigid C*-tensor category — hom-space inner products positive by axiom; it categorifies a positive Hilbert structure and adds only positive-definite operators (fusion, conditional expectations, arXiv:2111.04488 unverified — title-level), never a signature. Enlarging the naturality symmetry makes the constraint stronger, shrinking the pool.
Operadic / double-categorical AQFT. [INFERENCE — closed] Benini–Schenkel–Woike (arXiv:1709.08657, verified) parametrize by an orthogonality (causal) relation = $n_1$ input; the double category puts localization in the base. Neither installs an indefinite fiber form.
Modular-Berry / kinematic space — obligation (ii), not this track; imports $n_1$ by construction (lives on a space of regions). [OPEN — unchanged]

No larger structure supplies a natural localized indefinite form. [OPEN — three-session scoped negative search]

5. Consequences proposed for the wiki

HYP-CKV-VACUITY-R5 → R6 (precision hardening, grade unchanged). The condition on the HIGH hedge sharpens: from "net-constructibility ansatz" (an unproven $W^*(\mathfrak m)$ -membership claim) to the localization-template hypothesis $n_1$ (a proven-irreducible base-category datum). The constructibility half is now a theorem (Lemma N1); the localization half is the residual, and it is the geometry. The hedge stays conditional HIGH, not unconditional HIGH. Obligation (i) is partially met — the ansatz is replaced, but by a residual, not a discharge.
HYP-FACTORIZATION-IS-GEOMETRY — restated. The carrier problem's localization step is now identified at its sharpest: the localization template is the geometry; no functorial enrichment of the algebraic data supplies it. This is the naturality-promotion of route (5), not a sixth route — the count stays at FIVE (binding inheritance).
HYP-ENCODING-SCREEN — inherits the sharpened condition, grade unchanged.

6. Open subquestions

Obligation (ii) remains: a modular-Berry-holonomy-natural localized indefinite form not presupposing kinematic space (the residual escape; it imports $n_1$ , so the task is to show it cannot be made carrier-free). [OPEN]
Is there a precise theorem that the localization-template hypothesis is strictly necessary — i.e. a net with no causal index order provably carries no localized form of any signature (the converse to the parametrization-is-geometry corollary, a true non-existence rather than a residual)? [OPEN]
Does any non-self-adjoint / PT-symmetric relative modular structure evade Lemma N2 by being neither flow-covariant nor a base-localized form? [OPEN — inherited from iter-8]

Proposed registry items

LEM-NAT-CARRIER — (new lemma)

Statement. Every $\mathrm{Aut}(\mathcal N,\Omega)$ -natural modular-covariant carrier has Gram operator forced into $W^*(\mathfrak m(\mathcal N))$ (Lemma N1, via GMA functoriality) and a natural sign locus that the modular flow boosts away (Lemma N2); hence it is positive-and-local, signature-free, or indefinite-and-nonlocal — never both indefinite and localized. The localization predicate is irreducible: it presupposes the index-set order $n_1$ , which is the geometry. Tag [INFERENCE] (high on N1–N2; the residual is [OPEN]).

Statement. The OP-46 principal hedge's condition sharpens from the net-constructibility ansatz to the localization-template hypothesis $n_1$ ; the constructibility half is discharged to a naturality theorem, the localization half is the irreducible residual and is the geometry. Hedge stays conditional HIGH (not unconditional); convergence count stays FIVE; headline unchanged. Tag [INFERENCE].

../CONCLUSION.md §§4, 5, 8 — the hedge this note sharpens and the reopening condition (the carrier problem)
../HYPOTHESES.md — HYP-CKV-VACUITY-R5, HYP-FACTORIZATION-IS-GEOMETRY, HYP-MODULAR-TRICHOTOMY, HYP-ENCODING-SCREEN
2026-06-10-iter8-net-carrier-no-go.md — the net no-go and its two named obligations (this is obligation (i))
2026-06-10-iter7-indefinite-pairing-attempt.md — the single-algebra trichotomy + ansatz
2026-06-08-iter3-encode-vs-generate-criterion.md — grading instrument ( $g_1$ – $g_7$ , $n_1$ – $n_3$ )
../domains/mathematics.md — Tomita–Takesaki, Borchers–Wiesbrock, GMA, standard subspaces
../EPISTEMICS.md — tag discipline

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. In particular: the carrier-convergence count stays at FIVE (this is route (5)'s naturality promotion, not a sixth route); the signature-free prong rests on separate enumeration, not on the naturality; and obligation (i) is graded partially met (residual, not discharge) — not a full discharge to unconditional HIGH.

Binding note. Where the referee verdict below conflicts with the body, the referee correction governs (notably: A4 was overturned RESOLVED-NEGATIVE→SHARPENED-OPEN — the TT projector annihilates the leading divergence, de-hardening the R2 foothold; A2 "unconditional HIGH" was struck — the hedge stays MEDIUM-HIGH→HIGH conditional; the carrier-convergence count stays FIVE). The body is the pre-referee submission, retained for audit.

SHARPENED-OPEN, UPHELD WITH ONE MAJOR CORRECTION; headline unchanged (8th consecutive). The engine (Lemma N2) is sound and reproduced: by the Borchers relation the modular flow rescales momenta exponentially, so a localized indefinite form sign locus is boosted away (boost-violation max-norm 2 for all nonzero t); only positive constants are boost-invariant; sgn of modular frequency is natural but non-local; the fixed points are the wedge edge n1. Lemma N1 is a MAJOR error: commuting with the modular flow places the Gram in the commutant of Delta, not in the abelian algebra generated by Delta; for Type III-1 these differ because the modular spectrum is degenerate (numerics: 12 spectral degeneracies in a 4x4 toy), so the constructibility ansatz is only weakened, not discharged to a theorem. The GMA citations (Buchholz-Dreyer-Florig-Summers; Summers-White) reconstruct spacetime from the modular conjugations J, not a Gram from Delta. The residual is therefore two-fold: the localization-template hypothesis n1 (F3, correct and well-cited) plus the surviving commutant-to-function-of-Delta gap; the principal hedge HYP-CKV-VACUITY-R5 stays MEDIUM-HIGH-to-HIGH conditional on both, not on the localization template alone. Citations all real (BFV, Buchholz et al., Summers-White, Neeb-Olafsson 1704.01336, Komalan 2601.07807, Benini-Schenkel-Woike, Morinelli-Neeb-Olafsson 2603.26390, 2111.04488); two fixes: math 0304340 is by Bisch not Jones, and the Neeb-Olafsson fibers-carry-no-signature reading is finder inference, not a paper statement. No-go red-team clean: no conflict with dS-cardinality, the iter-7 trichotomy or iter-8 net no-go, OP-48ab, or OP-49; carrier count stays FIVE; no RESOLVED-POSITIVE. Net: a verdict-neutral sharpening whose centerpiece theorem does not hold; the hedge is unchanged in grade but its condition must be widened, not narrowed.