Original ledger status
ALPHA.QUILLEN.PROGRESS.03
ALPHA.QUILLEN.PROGRESS.03
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].
Recorded status: 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)
Dependencies, status detail and source
Canonical status: [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: 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
Scripts: v434_alpha_quillen_betafunction.py
Location: tfpt_1_architecture_e8
Original CSV, line 457