§ 18updated 2026-07-11
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Provenance — What Existed vs. What This Program Produced
Status: Color-coded attribution ledger, current through iteration 23. Every claim below carries its iteration of origin and lives, refereed, in the linked notes. Last updated: 2026-07-11
The color code:
- 🔵 BLUE — prior art. Existed before this program; we used it and cite it.
- 🟢 GREEN — discovered/invented here. New results produced by this program, adversarially refereed, and (per our live literature sweeps) not previously in the literature.
- 🟡 YELLOW — rediscovered here. Proved independently by this program, then found in prior art during our own verification — attribution honesty is part of the method; the prior source owns the result, our proof stands as an independent second derivation.
- 🔴 RED — impossibility results proved here. Boundaries this program established on how the remaining problem can and cannot be solved — arguably its most distinctive product.
🔵 Prior art this program stands on
| Result | Source |
|---|---|
| 🔵 Tomita–Takesaki modular theory; type III₁ structure of local algebras | Tomita, Takesaki; Fredenhagen 1985 |
| 🔵 Bisognano–Wichmann: wedge modular flow = boost; wedge vacuum ergodicity | Bisognano–Wichmann 1975/76; Borchers 1992; Driessler 1975 |
| 🔵 Massless double-cone modular flow is geometric | Hislop–Longo, CMP 84 (1982) |
| 🔵 The E=0 exclusion for the massive interval (via antilocality) | Figliolini–Guido 1989, p. 429 (see 🟡 below); Segal–Goodman 1965 |
| 🔵 Type III₁ of the second-quantization factors | Figliolini–Guido, J. Operator Theory 31 (1994) |
| 🔵 Ergodic states are generic on III₁ factors | Marrakchi–Vaes, Crelle 809 (2024) |
| 🔵 Relative bicentralizer flow ergodicity (with expectation) | Marrakchi, arXiv:2606.23636 (2026) |
| 🔵 The correlator (Peschel) method; exact massless interval modular data | Peschel; Casini–Huerta |
| 🔵 Massive interval modular operator numerics (mass/angular-momentum dependence) | Bostelmann–Cadamuro–Minz, AHP 2023; Cadamuro 2312.08525 |
| 🔵 Sine-type entire functions ⟹ exponential Riesz bases | Levin 1961; Golovin 1964; Avdonin–Ivanov 1995 |
| 🔵 Mourre theory; dilation analyticity; exponential bounds | Mourre; Aguilar–Combes 1971; Froese–Herbst 1982; Georgescu–Gérard 1999 |
| 🔵 Absence of positive eigenvalues for local short-range potentials | Kato 1959; Agmon; Simon |
| 🔵 Embedded eigenvalues occur in multi-interval finite Hilbert transforms | Bertola–Katsevich–Tovbis, arXiv:2008.10058 |
| 🔵 Definitizable operators in Krein spaces (finitely many non-real points) | Langer; Jonas; Azizov–Jonas–Trunk |
| 🔵 Entanglement-first results the wager rests on (RT; entanglement first law; Einstein equation of state; crossed-product entropy) | Ryu–Takayanagi; FGHMVR; Jacobson 1995/2015; CLPW et al. |
| 🔵 Fractional unique continuation (post-1989 toolset) | Fall–Felli 2014; Rüland 2015; Ghosh–Salo–Uhlmann 2020 |
🟢 Discovered / invented by this program
| Result | Where |
|---|---|
| 🟢 (E_O) isolated and named: vacuum ergodicity on massive double-cone algebras as the single gate of the net-naturality no-go — never previously posed for the bounded massive case (verified by dedicated literature sweep) | iter 16, iter 17 |
| 🟢 LEM-NET-NATURALITY-G: the multi-wedge classification — localized natural carriers are ±1; fully-natural non-localized carriers are gauge gradings with Δ_geo = 0; the commutant gap classified, not closed | iter 16 |
| 🟢 The η₀ vacuity chain: η₀ = sgn(ln Δ) achieves the literal "indefinite pairing from modular data" target yet is provably geometry-void, algebra-incompatible, non-localizable — and the five-route carrier-problem convergence built on it | iters 7–15; the dossier |
| 🟢 The free-field reduction of (E_O) to a one-particle spectral question, with the bosonic dΓ(A) correction refuting the naive kernel criterion | iter 17 |
| 🟢 The first numerical measurements of the massive interval modular spectrum's near-zero structure: the ε_k → (2j+1)π²/ln L ladder; the pinning scan (no embedded eigenvalue); the edge-divergence probe; the resolvent-H1 probe; the Gram probe | iters 18–23; scripts/eo-modular-numerics.py |
| 🟢 The bilateral-antilocality reformulation: modular eigenvalue μ² > 1 ⟺ coupled antilocality data with coupling c = μ²/(μ²−1) > 1 (FG-1989 exactly the boundary case) | iter 20 |
| 🟢 Two rigidity theorems: every candidate eigenfunction lies in H¹₀(I) (Dirichlet rigidity, m > 0 essential) and vanishes on no open subinterval (support rigidity) | iter 20 |
| 🟢 The Corner Indicial Theorem: cos²(πs) = c; complete simple root lattice n ± iτ₀; determinant ≡ −s(c − cos²(πs)); no local kill possible | iter 21 |
| 🟢 The strip geometrization with the closed-form potential Q = m²R²(cosh ξ − cos η)⁻² and translation-invariant coupled edges | iter 21 |
| 🟢 The c-collapse: the entire free-field gate ⟺ σ_p(A) ∩ (1,∞) = ∅ for the single self-adjoint operator A = S_I^{1/2}R_I S_I^{1/2} — the operator the numerics measure | iter 21 |
| 🟢 The conditional Mourre theorem with the explicit renormalized conjugate D_g (i[A₀, D_g] = 1; A₀ = coth²(πD)); the Möbius spectral identity reading of the ladder | iter 22 |
| 🟢 The (a′) Birkhoff computation: characteristic quadratic z² − 2(2c−1)z + 1, discriminant 16c(c−1) > 0, roots e^{±2πτ₀} — machine-verified two independent ways | iters 22–23; iter 23 |
| 🟢 The named open lemmas posed for the field: LEM-A1‴-T (the transfer/Gram bound) and LEM-A1⁗ (Pontryagin-channel on-shell transversality) — the entire remaining free-field gate in two precisely-posed statements | iter 23 |
| 🟢 LEM-A1‴-T PROVED — the transverse-pencil eigenvector family is a Riesz basis for every fixed c > 1, by the elementary weighted-Fourier → unitary-folding → constant-intertwiner transfer with explicit frame bounds (assembly: PROOF_ASSEMBLES; machine-verified 1.4×10⁻¹⁵); with it, the closed-form Gram G = Toeplitz + (πC/2)J and both degeneracy laws | iter 24 |
| 🟢 P1-NF, the normal form — the entire free-field gate is EXACTLY a scalar imaginary-Aharonov–Bohm-flux zero-mode exclusion, H(iτ₀)g = 0 on the cylinder (assembler re-derived cold; 36/36 machine checks); with it the closed-form on-shell functional and the mode recast as a coupled LOCAL system | iter 25 |
| 🟢 The Bari supplement ((Q) ⟹ Riesz basis; iteration-24's exact-lattice conditionality discharged) and the complex-c extension of the transfer lemma (five elementary steps; machine-verified at five complex points) | iter 25 |
| 🟢 The order-blindness lemma (indicial corner data is invariant under both order-reversing ℤ₂ operations — neither orientation bit of n₁ is a function of it) and its honest boundary: the first mass correction is R1-odd, so the n₁ wall is conformal-scope | iter 25 |
| 🟢 The common-cause theorem (mode coordinates): the constant imaginary part of the corner lattice simultaneously powers the transfer lemma's explicitness and the carrier screen's vacuity — one structural fact behind both | iter 25 |
| 🟢 S5, the n-uniform weighted-Volterra bound (geometric-slack form) — branch (a) of the terminal-profile dichotomy forces vanishing for the INFINITE channel system; with it the {LEM-A1⁗-N} ⟺ LEM-A1⁗ equivalence and the structural correction that the un-slacked norm fails Θ(n) | iter 26 |
| 🟢 LEM-TRACE — the corner-weighted trace lemma grounding the seam identity with no extra corner term (the iteration-21 resonant-log caveat discharged); with it the graded seam chain and the Perron/positivity-improving structure of the free seam operator | iter 26 |
| 🟢 The Krein/parity dictionary Φ*J_cΦ = −((c−1)/2)·P (orientation-reversing; bilinear-only complex continuation), the P1-NF domain-closure lemma, and the explicit complex-c radius r(c₀) | iter 26 |
| 🟢 CH-26(iii) Parts 1–2 — the shell-channel limiting absorption principle (A0-LAP) and the on-band collapse ‖K^b_band q^{1/2}g‖ = O(√b) (the vanishing on-shell amplitude realized as a simple pole-cancelling zero; cold-verified 0.4999) — compressing the free-field gate to the single off-band bound (B) | iter 27 |
| 🟢 The branch-(b) infinite-system terminal-rate dichotomy (ρ_n ≥ | n |
| 🟢 The encoding-screen corollary (no distinguishing experiment even in principle, as a corollary of encode-not-generate) and the reverse-Weinberg–Witten ⟺ carrier identification | iters 6, 9 |
🟡 Rediscovered here (independent proof; prior art found during our own verification)
| Result | The honest story |
|---|---|
| 🟡 1 is not an eigenvalue of the massive interval modular operator (the E=0 half of the free-field gate) | Proved here via factoriality (LEM-K0, iteration 19) — then our referee downloaded the Figliolini–Guido 1989 paper and found the result on p. 429, proved via antilocality. FG own it; LEM-K0 stands as an independent second proof by a different mechanism, and the iteration-17 [unverified] flag was retired against the primary source. |
| 🟡 τ₀ = ε/2π (the corner Mellin frequency equals the modular energy over 2π) | The relation is Bisognano–Wichmann/local-Rindler physics — long known. What is new (🟢) is its derivation route: it falls out of a pure boundary-value corner computation with no physics input, and was then verified on the lattice at the percent level. |
🔴 Impossibility results proved here (how this problem can NOT be solved)
| Boundary | Where |
|---|---|
| 🔴 No decay/analyticity-class theorem can close the gate: the five-line Gaussian rank-one counterexample — a self-adjoint nonlocal kernel inside every exponential class, dilation-entire, hosting an embedded eigenvalue; Kato–Agmon–Simon is irreducibly local; complex scaling only θ-persists | iter 22 |
| 🔴 No family-level/soft theorem: embedded eigenvalues genuinely occur in the ambient operator family (application of 🔵 BKT); only geometry-specific analysis can decide the single interval | iter 20 |
| 🔴 No pointwise-positive Krein symmetrizer: the unique pointwise symmetrizer of the strip pair is indefinite, [[c,−c],[−c,1]] — the Froese–Herbst exclusion is not licensed on the coupled system | iter 21 |
| 🔴 No global Krein definitizability for any c > 1 (infinitely many non-real essential branches exceed Langer's bound) — the corner exponents that define the gate kill definitizability | iter 23 |
| 🔴 (E_O) is undecidable at the invariant level (type, Connes spectrum, genericity — Powers/Araki–Woods counterexamples); the wedge scaling engine has no bounded-region analogue; scaling-limit transport fails | iter 17 |
| 🔴 The operator-algebra-side closures: the commutant gap carries no carrier (iter 10); the Krein/indefinite-metric route collapses (iter 12); every untried modern framework reproduces the encoding screen (iter 15); the causal order n₁ is presupposed by every known algebraic seed (iters 18/20) | iters 10–20 |
| 🔴 The fifth methods-negative (numerical grade): plain Konno–Kuroda norm contraction cannot close the gate (spectral radius 2.3–3.0 on every window, every refinement — the on-shell channel dominates); the naive Birman–Schwinger splitting fails (‖K‖ ≈ 9.7); the mR → ∞ norm-collapse corner is a lattice-mass artifact (σ(A) = [1,∞) a.c. for every mR). Computer-assisted exclusion survives only in on-shell-projected form — an assistant to the lemma, never a bypass | iter 25 |
Reading the ledger honestly: the 🟢 and 🔴 rows are real mathematics produced by this program — but none of them is a universal theory, and the program's verdict (unchanged through 22 consecutive refereed confirmations) is that the leading route encodes geometry without generating it. The 🟡 rows are the discipline working: independent proofs kept, credit corrected against primary sources. The full evidentiary trail for every row is in FINDINGS.md and CHANGELOG.md.
See also
- FINDINGS.md — the iteration-by-iteration record behind every row
- notes/2026-06-19-iter15-carrier-problem-dossier.md — the posed problems (🟢, open, yours to solve)
- EPISTEMICS.md — the tag discipline that makes this ledger trustworthy