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Original ledger classification: Heat-kernel origin [E]/[C]/[O] (term structure [E]; exact value [C]; cubic moment + identification [O]; no new gate)

Recorded statement: HEAT-KERNEL SHARPENING OF EM.WARD.01 (2026-06-22): derives the ORIGIN and STRUCTURE of the F_U(1) determinant-line terms from textbook Seeley-DeWitt/Gilkey heat-kernel coefficients, advancing the EM-Ward origin from [P] (conjectured) to…

Statement, assumptions and sources

Paper context: Architecture, E8 and the electromagnetic fixed point

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EM.WARD.02

Original ledger classification: Heat-kernel origin [E]/[C]/[O] (term structure [E]; exact value [C]; cubic moment + identification [O]; no new gate)

Original claim text

HEAT-KERNEL SHARPENING OF EM.WARD.01 (2026-06-22): derives the ORIGIN and STRUCTURE of the F_U(1) determinant-line terms from textbook Seeley-DeWitt/Gilkey heat-kernel coefficients, advancing the EM-Ward origin from [P] (conjectured) to [E] for the term STRUCTURE; honestly fences the rest. NO new gate (sharpens EM.WARD.01). [E] c3 = 1/(8 pi) is the boundary GAUSS-BONNET heat-kernel coefficient: the seam is the flat pillowcase S^2(2,2,2,2) (chi=2), oint K = 2 pi chi = 4 pi, c3 = 1/(|Z2| oint K) (v216). [E] the alpha^2 term is the GILKEY gauge-curvature coefficient: the U(1) Laplacian a4 contains 30 Omega^2/360 = (1/12) Omega^2 (Gilkey, Invariance Theory Thm 4.8.16), Omega = i alpha F => -(alpha^2/12) F^2, so an alpha^2 term in log det Delta_U(1) is FORCED -- the Calderon -2 c3^3 a^2 is the gauge-curvature (Quillen-curvature) piece, not an ansatz. [E] the c3-ladder {0,3,6} is arithmetic (step 3 = three channels; 6 = |R^+(A3)| C6 hexagon). [E] multiplicities forced: 2 = |Z2| (sheet), 8 b1 = (4/5) M = (rank E8) b1 (b1=41/10, v48/v159). [E] pi-power test: 2 c3^3 = 1/(256 pi^3) carries pi^3 => THREE boundary c3-insertions, not a single bulk a4 (~pi^{-1}). [C] the exact coefficient 2 c3^3 needs the seam F-normalisation. [O] the cubic alpha^3 Maxwell moment (a topological/Chern level, NOT a Seeley-DeWitt integral) + the full identification (this IS the seam U(1) zeta-determinant) = EM.WARD.01's residual, tied to SEAM.EQUIV.01. NOT closed. alpha^-1 stays [E] (v3).

Original classification detail

[E] the heat-kernel STRUCTURE (c3 = Gauss-Bonnet boundary coefficient; alpha^2 = Gilkey gauge-curvature a4 term; c3-ladder; multiplicities; pi-power); [C] the exact coefficient (seam F-normalisation); [O] the cubic Maxwell moment + the full determinant identification (= EM.WARD.01, tied to SEAM.EQUIV.01). Sharpens EM.WARD.01's origin from [P] toward [E]; no new gate.

Dependencies as recorded

EM.WARD.01; v48; v216; v3; v159

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Registered verification modules

  • v342_em_ward_heatkernel.pyRegistered verification module · em
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    The heat-kernel sharpening of EM.WARD.01: DERIVES the ORIGIN and STRUCTURE of the F_{U(1)} determinant-line terms from textbook Seeley-DeWitt/Gilkey coefficients, advancing the EM-Ward origin from [P] to [E] for the STRUCTURE; NO new gate. [E] c3=1/(8pi) the boundary GAUSS-BONNET coefficient (seam = flat pillowcase chi=2, oint K = 4pi, v216). [E] the alpha^2 term is the GILKEY gauge-curvature coeff: a4 has 30 Omega^2/360 = (1/12)Omega^2, Omega=i alpha F => -(alpha^2/12)F^2, so the Calderon -2 c3^3 a^2 is the gauge-curvature piece, not an ansatz. [E] c3-ladder {0,3,6} arithmetic (3 channels; C6 hexagon at 6). [E] multiplicities 2=|Z2|, 8 b1=(rank E8)b1. [E] pi-power test: 2 c3^3 ~ pi^3 => THREE boundary insertions. [C] exact coefficient needs the seam F-normalisation. [O] the cubic alpha^3 + full identification = EM.WARD.01 residual (tied to SEAM.EQUIV.01).

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