Iteration 19 — prove the numerics

Status: Analytic follow-through of iteration 18 (the sanctioned sharpest decidable lever from iteration 17: prove the free-field spectral fact). Two fronts: (A) a proposed lemma (LEM-K0) adversarially refereed (default REFUTE, with a primary-source novelty check), plus three refereed proof attempts on the remaining λ≠1 statement; (B) a precision-hardened embedded-eigenvalue (pinning) scan extending the iteration-18 numerics. Outcome: Verdict UNCHANGED — 18th consecutive confirmation; hedge stays R7 (the house rule requires full free-field (E_O) closure for R8; only the E=0 half closed). But the ledger genuinely moves: the E=0 / fixed-Weyl half of free-field (E_O) is retired from [unverified]/numerical to [ESTABLISHED] with a primary-source anchor and an independent second proof; the massless double-cone case is theorem-grade; the λ≠1 residual is provably narrower and provably harder (invariant-level arguments cannot decide it), with a named best-lead; and the pinning scan finds no embedded eigenvalue at any energy or mass. Last updated: 2026-07-10 Iteration: 19

1. LEM-K0 — refereed CONFIRMED_WITH_CORRECTIONS; the correction is attributional

LEM-K0 (standard-subspace factoriality lemma) [ESTABLISHED — verified re-proof; NOT novel]. Let K ⊂ H be a standard real subspace with Tomita data S = Jδ^{1/2}, K = fix(S), K′ = fix(S*), S* = Jδ^{−1/2}. (a) K ∩ K′ = (ker(δ−1))_J and ker(δ−1) = (K∩K′) + i(K∩K′); hence K factorial (K ∩ K′ = {0}) iff 1 is not an eigenvalue of δ. (b) If K is factorial, K contains no δ-eigenvector for any eigenvalue (f ∈ K, δf = λf ⟹ Jf = λ^{−1/2}f ⟹ J² = 1 forces λ = 1 ⟹ f = 0 by (a)).

Application [ESTABLISHED]: for the massive free scalar double cone, R(K_O) is a type III₁ factor (Araki; Figliolini–Guido, J. Operator Theory 31 (1994) 229–252), and Araki duality R(K)′ = R(K′) transfers factoriality to K_O. Hence 1 ∉ σ_p(δ_O): no modular-fixed one-particle vector, no fixed Weyl operator — the E=0 half of the free-field (E_O) criterion holds as a theorem.

Attribution (decisive; live-verified from the primary source). The referee downloaded the full Figliolini–Guido 1989 paper (The Tomita operator for the free scalar field, Ann. Inst. H. Poincaré 51 (1989) 419–435; numdam PDF) and verified verbatim, p. 429: "It is clear from (3.6) that 1 ∉ σ_p(δ), but it is possible to prove … that … 1 ∈ σ(δ)." FG 1989 already state and prove the E=0 exclusion, via antilocality of ω = (−Δ+m²)^{1/2} (Segal–Goodman 1965) — a different mechanism from the factoriality route above. Part (a) is standard-subspace folklore (explicit in Inclusions of Standard Subspaces, Comm. Math. Phys. 2025, DOI 10.1007/s00220-025-05458-4). LEM-K0 is therefore a verification plus an independent second proof, not new mathematics — and the iteration-17 [unverified] tag on the FG claim is retired to [ESTABLISHED] with the primary-source anchor.

2. The λ≠1 attempts — all PARTIAL (referee grade B+, no fabrications)

Lens	Yield	Status
P1 — Mourre / positive commutator from the massless comparison dynamics	Massless (3+1) double cone: ln δ_O(0) is purely absolutely continuous, spectrum ℝ, NO eigenvalues — Hislop–Longo conformal conjugation to the wedge boost + Borchers commutation + Stone–von Neumann `[ESTABLISHED]` (the 1+1 scalar testbed needs the IR-safe derivative-field variant `[INFERENCE, high]`). Massive case: conditional theorem — under (H1) C^{1,1}-Mourre regularity and (H2) form-compactness of the mass commutator (both `[OPEN]`), point spectrum in any compact I ⊂ ℝ∖{0} is at most finite, purely a.c. otherwise.	PARTIAL
P2 — locality / standard-subspace rigidity	A genuine new rigidity constraint (proved this session): any δ_O-eigenvector extends π-strip boost-analytically in every wedge containing O, with W-independent norm; also (K_O, K_W) is not a half-sided modular inclusion for m > 0 (so the Borchers–Wiesbrock instant kill is unavailable). Strong necessary conditions; no contradiction yet extracted.	PARTIAL
P3 — literature + structure theory	The λ≠1 statement is proved NOWHERE (live-verified: Longo–Morsella's massive-ball result withdrawn; Cadamuro–Fröb–Minz 2312.04629 call it "an open problem"; all massive double-cone knowledge numerical). Invariant-level arguments provably cannot decide it (Powers/Araki–Woods ITPFI factors realize factorial standard subspaces with pure-point one-particle modular spectra; III₁ needs only cl-grp(spec δ) = ℝ₊). The only known kill mechanism in the field is Borchers commutation — exactly what bounded regions lack. Scaling-limit transport fails (spectral type not preserved under strong-resolvent limits).	PARTIAL

The named best-lead (from the FG novelty check): FG's Theorem 3.6 converts λ ≠ 1 eigenvalues of δ_O into real eigenvalues μ = (λ+1)/(λ−1), |μ| > 1, of their operator B; FG's antilocality argument (Prop. 3.5) kills exactly μ = ±1. Extending the antilocality argument from μ = ±1 to all real μ is the sharpest open proof target — a concrete, single-mechanism question for the same toolset FG already used. [OPEN — the iteration-19 deliverable target]

3. Front B — the pinning scan: no embedded eigenvalue

The iteration-18 numerics are extended by a scale-aware embedded-eigenvalue test (reproducible: scripts/eo-modular-numerics.py, iteration-19 section; precision-hardened via the symmetrized solver eig(X_A P_A) = eigh(P^{1/2}X_A P^{1/2})). Statistic per sorted mode: g_k = [ε_k(768)·ln 768]/[ε_k(96)·ln 96] — continuum ≈ 1, pinned eigenvalue = 1.456.

Result: worst g anywhere = 1.049 (m = 0.5; m = 1 gives median 1.0001 — continuum scaling to three decimals; heavier masses drift below 1, the opposite of pinning). No pinned mode at any energy, any mass up to mR = 32.
The one suspicious mid-spectrum drift resolved cleanly: ε₁₉·lnL → 21π² — a late-onset high ladder level converging upward to its (2j+1)π²/lnL slot, not an eigenvalue. (Modes with ε ≳ 22 sit at ν−½ ≲ 10⁻¹⁰ and need the symmetrized solver; flagged in the script.)
Complementarity with P1: the conditional Mourre theorem allows "at most finitely many eigenvalues per compact interval"; the scan says the actual number is zero in the accessible window. [INFERENCE, high — validated lattice computation; not a continuum proof]

4. Ledger effect (the honest accounting)

E=0 half of free-field (E_O): [ESTABLISHED] (FG 1989 + LEM-K0 factoriality re-proof). The iter-17 [unverified] tag is retired.
Massless (3+1) double cone: purely a.c., no eigenvalues — [ESTABLISHED] (upgrades the iter-16 "conformal transport sketch" at the one-particle spectral level).
λ≠1 exclusion: [OPEN], sharpened — provably undecidable by invariant-level arguments; must use locality/mass-shell data; best lead = the antilocality extension (μ = ±1 → all real μ); conditional Mourre bounds it to finite-per-interval; numerics see zero.
Hedge: HYP-CKV-VACUITY stays R7 — no movement (R8 requires full free-field (E_O) closure, referee-confirmed). Verdict unchanged, 18th consecutive confirmation; carrier-route count stays FIVE; no numeric ID consumed.
Process: all seven live-checkable citations verified verbatim; the referee read the 1989 primary source in full; no fabrications (verifier grade B+).

5. What would move it from here

(i) The antilocality extension (all real μ) → full free-field (E_O) → R7 → R8 on the free subclass — now a single-mechanism, precisely-posed question; (ii) proving Mourre hypotheses (H1)–(H2) + a Fermi-golden-rule argument; (iii) the external clocks (2027 DESI w(z)). The dossier's external-solver invitation now points at the sharpest form the gate has ever had.