Publication history: 2026-06-28

Details

• New module v434 (ALPHA.QUILLEN.PROGRESS.03) — it settles the status of the three residuals named after v433 and shows they are not three independent open problems. The target ALPHA.QUILLEN.EXACT.01 (v382) stays [O]; \ainv=137.0359992 stays [E]. itemize
• [E]b_1 is the U(1) a_4 heat-kernel coefficient (the \beta=a_4 theorem). The one-loop \beta function of a gauge coupling is the a_4 (Seeley–DeWitt a_2 in d{=}4) heat-kernel coefficient of the charged-field operator (Vassilevich). Computed from the carrier hypercharge content with the standard spin weights (\tfrac23 per Weyl fermion, \tfrac13 per complex scalar), \sum\tfrac23 Y_f^2+\tfrac13 Y_s^2 = 41/6 (SM) and (3/5)\cdot{}that =41/10=b_1 (GUT). So the b_1 in the counterterm 8b_1\cthree^6 is the a_4 coefficient of the carrier U(1)_Y content (the same number as the \beta, v159/v246), the same kind of heat-kernel object as the \alpha^2 Calderón a_4 (v342) — not a fitted multiplicity.
• [E]the geometry factors off: 8b_1\cthree^6=b_1\cdot(\operatorname{rank}E_8)\cdot\cthree^6, with \operatorname{rank}E_8=8 the \chi{=}2 seam boundary count (v216) and \cthree^6 the six boundary insertions (the \pi-power test, v342).
• [C]residual (1) \equiv residual (3). With b_1 and the \cthree-powers heat-kernel-forced, the only freedom left in the exact seam a_4 value is how the gauge curvature couples to the boundary measure — the seam F-normalisation. So “the actual seam a_4=8b_1\cthree^6” and “the seam F-normalisation” are one [C]obligation, not two.
• [E]residual (2) located (not closed): the determinant-line variation factors a^3-2\cthree^3 a^2 = a^2(a-2\cthree^3); a^2 is the Gilkey gauge-curvature (Quillen/Chern) density (v342), so the leading a^3=a\cdot a^2 is the first moment of the Chern density — a topological level at boundary order \cthree^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 — a genuine narrowing of the open surface. [O]ALPHA.QUILLEN.EXACT.01 is not closed (the \alpha^3 Chern level and the from-first-principles proof stay external, type A, v384).