A registered result in its source context
Original ledger classification: Third honest step / grounding+reduction [E]/[C]/[O] (b1 = the U(1) a_4 heat-kernel coefficient via beta=a_4; residuals 1&3 collapse to one seam F-normalisation [C]; residual 2 the alpha^3 Chern level located but [O]; the headline target stays OPEN)
Recorded statement: A THIRD HONEST step on the external target ALPHA.QUILLEN.EXACT.01 (2026-06-28, v434): it settles the status of the three residuals named after v433 and shows they are NOT three independent open problems -- the target stays [O], alpha^-1…
Statement, assumptions and sources
Paper context: Architecture, E8 and the electromagnetic fixed point
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ALPHA.QUILLEN.PROGRESS.03
Original ledger classification: Third honest step / grounding+reduction [E]/[C]/[O] (b1 = the U(1) a_4 heat-kernel coefficient via beta=a_4; residuals 1&3 collapse to one seam F-normalisation [C]; residual 2 the alpha^3 Chern level located but [O]; the headline target stays OPEN)
Original claim text
A THIRD HONEST step on the external target ALPHA.QUILLEN.EXACT.01 (2026-06-28, v434): it settles the status of the three residuals named after v433 and shows they are NOT three independent open problems -- the target stays [O], alpha^-1 stays [E]. [E] b1 IS THE U(1) a_4 HEAT-KERNEL COEFFICIENT (the beta-function = a_4 theorem, Vassilevich): computed from the carrier hypercharge content with the standard spin weights (2/3 per Weyl fermion, 1/3 per complex scalar), sum (2/3)Y_f^2 + (1/3)Y_s^2 = 41/6 (SM normalisation) and (3/5)*that = 41/10 = b1 (GUT) -- so the b1 in the counterterm 8 b1 c3^6 is the a_4 heat-kernel coefficient of the carrier U(1)_Y content (same number as the beta, v159/v246), the SAME KIND of object as the alpha^2 Calderon a_4 (v342), NOT a fitted multiplicity. [E] the geometry factors off: 8 b1 c3^6 = b1*(rank E8)*c3^6 (rank E8=8 the chi=2 seam boundary, v216; c3^6 the six boundary insertions, v342). [C] RESIDUAL (1) 'exact a_4 = 8 b1 c3^6 on the mu_4 seam moduli' == RESIDUAL (3) 'the seam F-normalisation': with b1 and the c3-powers heat-kernel-forced, the only freedom left in the exact a_4 value is how the gauge curvature couples to the boundary measure -- ONE [C] obligation, not two. [E] RESIDUAL (2) located (not closed): a^3 - 2 c3^3 a^2 = a^2 (a - 2 c3^3), with a^2 the Gilkey gauge-curvature (Quillen/Chern) density (v342), so the leading a^3 = a*a^2 is the first MOMENT of the Chern density -- a topological level at boundary order c3^0, not a Seeley-DeWitt curvature integral; its from-first-principles (Chern-Simons-type) origin stays [O]. [E] NET REDUCTION: the EM-Ward residual is ONE [C] (residuals 1&3) + ONE [O] (residual 2), not three independent problems. [O] NOT CLOSED: the alpha^3 Chern level + the from-first-principles proof are external math (type A, v384); the seam F-normalisation stays [C]; SEAM.EQUIV.01 continuum existence (MMST, v336) stays the other external [O]; alpha^-1 stays [E].
Original classification detail
[E] b1 = 41/6 (SM) = 41/10 (GUT) is the U(1)_Y a_4 heat-kernel coefficient of the carrier content (beta=a_4 theorem), grounding the counterterm's matter factor; the geometry factors as 8 b1 c3^6 = b1*(rank E8)*c3^6; the cubic factors as a^2(a-2c3^3) (a^3 = the first moment of the Chern density, a level). [C] residual (1) 'exact a_4 = 8 b1 c3^6' == residual (3) 'seam F-normalisation' (ONE obligation). [O] residual (2) the alpha^3 Chern level + ALPHA.QUILLEN.EXACT.01 stay open. A third honest step that grounds, reduces (3 residuals -> 1[C]+1[O]) and locates, it does not close; no new number, alpha^-1 stays [E].
Dependencies as recorded
ALPHA.QUILLEN.EXACT.01; ALPHA.QUILLEN.PROGRESS.01; ALPHA.QUILLEN.PROGRESS.02; EM.WARD.01; EM.WARD.02; v159; v216; v246; v341; v342; v382; v433
Open the original ledger source, row 457Location recorded in the ledger
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- Architecture, E8 and the electromagnetic fixed point
Sections that explicitly cite this claim
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- 2026-06-28 ( v434 — a third honest step on the [formula in source] Quillen target: [formula in source] is the [formula in source] heat-kernel coefficient, and the three residuals collapse to one [formula in source] [formula in source] one [formula in source] )Research changelog; TFPT — Introduction and reading guide
- Front matter / supportTFPT — Introduction and reading guide; Verification reference catalog
Navigation through a shared verification script
These sections cite a script associated with the claim. This is a source navigation join, rather than an explicit citation of the claim or a theorem equivalence.
- The EM closure: [formula in source] as an explicit fixed point (made explicit here)tfpt 1 architecture e8
- The EM closure: [formula in source] as an explicit fixed point (made explicit here)tfpt 1 architecture e8
- The EM closure: [formula in source] as an explicit fixed point (made explicit here)Architecture, E8 and the electromagnetic fixed point
- =150 2026-08-20 (CXI) (the externalization contracts — the two open non-RH kind-(A) external-math targets SEAM.EQUIV.01 and ALPHA.QUILLEN.EXACT.01 packaged as short, self-contained EXTERNALIZATION CONTRACTS for outside specialist mathematicians (conformal nets/VOA/AQFT for the seam; determinant lines/ [formula in source] -invariants/index theory for alpha), added as two new sections at the end of tfpt_research_contracts (before ``Certifiability and recommended order''), plus a one-sentence pointer in the head keybox; DOCUMENTATION ONLY — no module touched, no number changed, no status marker moved, no gate closed or narrowed, NO RH claim: (1) the SEAM.EQUIV.01 contract states the minimal theorem (the scaling limit of the gapped [formula in source] -equivariant CAR collar exists in local-net topology; the index-4 simple-current extension functor commutes with the limit; the OS reconstruction is the holomorphic [formula in source] lattice net) broken into the four review-named lemmata L1 (uniform energy/nuclearity bounds — honestly typed: NO uniform nuclearity estimate exists in the corpus), L2 (convergence of local algebras, the cited Osborne–Stottmeister scaling-limit theorem with the [formula in source] -equivariant structure retained), L3 (crossed product commutes with the limit; the finite index-4 Q-system is explicit, v125/ v154/ v469) and L4 (uniqueness of the holomorphic [formula in source] reconstruction; [formula in source] vs the same- [formula in source] rival [formula in source] ), with the complete typed hypothesis list (12 PROVEN-FINITE rows citing v302/ v113/ v155/ v160/ v356/ v366/ v367/ v368/ v461/ v376/ v377/ v378/ v379/ v125/ v154/ v216/ v286, 5 CITED-CLASSICAL rows, 1 OPEN target), the runnable finite models, the HONEST Lean axiom surface ( SeamScalingLimit.lean kernel-proves only the arithmetic hypotheses by decide; the analytic content is named cited axioms over opaque Props; no sorry; explicitly flagged: L1–L4 have ZERO Lean content beyond the composition audit), five must-fail controls (MF-S1: [formula in source] alone must NOT suffice — the red-team point; MF-S2: index-2 glue must yield [formula in source] ; MF-S3: the trivial [formula in source] collar must give no chiral limit; MF-S4: non-order-4 clock breaks the marks; MF-S5: the v286 import firewall) and four acceptance tests; (2) the ALPHA.QUILLEN.EXACT.01 contract states the two-step theorem T1 (gap-stable continuum limit [formula in source] under norm-resolvent convergence with the Fermi gap uniformly open over the twist torus — the Dai–Freed section has no zero; finite side computed, v472) and T2 ( [formula in source] [formula in source] Bismut–Freed curvature [formula in source] inflow response with the exact corpus coefficients [formula in source] , [formula in source] , [formula in source] , [formula in source] , exponent [formula in source] ), the two named open faces (the analytic diagonal- [formula in source] /multiplicity face v484/ v485 and the topological EM1 lemma v48/ v435), the typed hypothesis list (10 PROVEN-FINITE rows citing v3/ v48/ v341/ v382/ v434/ v470/ v435/ v342/ v367/ v472/ v484/ v485/ v391/ v433, 5 CITED-CLASSICAL rows, 1 OPEN target), the runnable finite models, the HONEST Lean statement (there is NO Lean formalisation of this target — stated plainly), four must-fail controls (MF-A1: no experimental [formula in source] anywhere in the construction; MF-A2: deck/carrier/Chern-level ablations must destroy the CODATA match; MF-A3: [formula in source] , [formula in source] ; MF-A4: the [formula in source] [formula in source] -power/metric-independence partition) and the review's three acceptance tests verbatim (no experimental [formula in source] input; unique physical fixed point [formula in source] ; ablations destroy the hit) plus one structural test; (3) the website paper mirror of tfpt_research_contracts ( papers.ts) gains one section entry covering both contracts; tfpt_research_contracts.pdf and changelog.pdf recompiled and mirrored via bash build.sh website; maps and changelog.ts regenerated via bash build.sh gen; bash build.sh audit [formula in source] AUDIT OK; manifests regenerated LAST. Both targets stay [O]; NO RH claim anywhere)Research changelog; TFPT — Introduction and reading guide
- 2026-06-29 ( v435 — a fourth honest step on the [formula in source] Quillen target: a [formula in source] -power test isolates the cubic [formula in source] as the unique metric-independent (topological) rung, and types its coefficient as a conditional integer level)Research changelog; TFPT — Introduction and reading guide
- Externalization Contract — ALPHA.QUILLEN.EXACT.01 (the determinant-line variation theorem, packaged for external specialists)tfpt research contracts
- Externalization Contract — ALPHA.QUILLEN.EXACT.01 (the determinant-line variation theorem, packaged for external specialists)Research contracts
- Externalization Contract — ALPHA.QUILLEN.EXACT.01 (the determinant-line variation theorem, packaged for external specialists)tfpt research contracts
Registered verification modules
- v434_alpha_quillen_betafunction.pyRegistered verification module · em
Read the description recorded in the module registry
ALPHA.QUILLEN.PROGRESS.03: a THIRD honest step on ALPHA.QUILLEN.EXACT.01 (v382) -- it settles the three residuals named after v433 and shows they are NOT three independent problems; the target stays [O], alpha^-1 stays [E]. [E] b1 IS THE U(1) a_4 HEAT-KERNEL COEFFICIENT (beta=a_4 theorem): from the carrier hypercharge content with spin weights (2/3 Weyl, 1/3 complex scalar), sum (2/3)Y_f^2+(1/3)Y_s^2 = 41/6 (SM), (3/5)*that = 41/10 = b1 (GUT) -- the b1 in 8 b1 c3^6 is the a_4 content coefficient, not a fitted multiplicity (same number as the beta, v159/v246; same KIND as the alpha^2 Calderon a_4, v342). [E] 8 b1 c3^6 = b1*(rank E8)*c3^6 (rank 8 = chi=2 boundary, v216; c3^6 = six insertions, v342). [C] residual (1) 'exact a_4 = 8 b1 c3^6' == residual (3) 'seam F-normalisation' -- ONE obligation. [E] residual (2) located: a^3-2c3^3 a^2 = a^2(a-2c3^3), so a^3 = the first moment of the Chern density (a level, order c3^0); origin stays [O]. [E] net reduction: the EM-Ward residual is 1[C]+1[O], not three. [O] NOT CLOSED; alpha^-1 stays [E]. Python-only.
A registered module is a source reference. Its presence does not assert that the module was executed in this snapshot.