A registered result in its source context

ALPHA.QUILLEN.PROGRESS.04

Original ledger classification: Fourth honest step / metric-independence test + conditional level typing [E]/[C]/[O] (the pi-power test isolates a^3 as the unique pi^0/c3^0 metric-independent topological rung; conditional on the CS/eta reading the a^3 coefficient is an integer level, =1 the unit; the EM1 CS-boundary proof and the headline target stay OPEN)

Original claim text

A FOURTH HONEST step on the external target ALPHA.QUILLEN.EXACT.01 (2026-06-29, v435): it attacks the SINGLE remaining [O] left after v434 -- residual (2), the cubic alpha^3 Chern/Maxwell moment -- and SHARPENS it with a verifiable test plus a conditional typing, it does NOT close it; the target stays [O], alpha^-1 stays [E]. [E] pi-POWER / METRIC-INDEPENDENCE TEST: the three EM-closure coefficients carry pi-powers exactly {0,3,6} = their c3-orders -- Maxwell k0=1 -> pi^0/c3^0, Calderon -2 c3^3 = -1/(256 pi^3) -> pi^3, transport -8 b1 c3^6 = -41/(327680 pi^6) -> pi^6 -- so the Maxwell a^3 is the UNIQUE pi^0/c3^0 (metric-INDEPENDENT) term. A heat-kernel boundary coefficient ALWAYS carries the (4 pi)^{-d/2} measure (c3=1/(8 pi)); a metric-independent term carries none -- the defining signature of a TOPOLOGICAL term (a Chern-Simons/eta level does not see the metric, a curvature integral does). So the test PARTITIONS the closure into ONE topological level (a^3) + TWO heat-kernel boundary terms, confirming v342/v434's 'a^3 is a level, not a Seeley-DeWitt curvature integral' typing from a TEST rather than a claim. [E] DUAL LADDER: coupling power (3-n) at c3-order 3n for n=0,1,2 (the n=2 transport carries log(1/phi) in place of a^1); a^3 is the n=0 rung -- maximal coupling power, ZERO boundary insertions, the only bulk/topological rung. [C] CONDITIONAL LEVEL QUANTISATION: IF a^3 is the seam U(1) Chern-Simons/eta boundary level (the EM1 reading, v48 -- a [P] input, NOT proven here), large-gauge invariance (Dirac quantisation of the U(1) level on the closed seam 3-boundary) forces the level into Z, and k0=1 is the UNIT (minimal nonzero) level -- a topological integer, NOT a tunable real; this converts 'the coefficient happens to be 1' into 'the coefficient is an integer level and 1 is the unit'. [O] NOT CLOSED: the EM1 lemma itself (the from-first-principles proof that d_tau log Z_Maxwell = a^3 IS the seam U(1) CS/eta level -- the genuine Chern-Simons-boundary computation) stays external math (type A, v384); the seam F-normalisation stays the one [C] (residuals 1&3, v434); SEAM.EQUIV.01 continuum existence (MMST, v336) stays the other external [O]; alpha^-1 = 137.0359992 stays [E].

Original classification detail

[E] the pi-power/metric-independence test isolates a^3 as the UNIQUE pi^0/c3^0 (metric-independent) term -- the signature of a topological level, NOT a Seeley-DeWitt curvature integral (the other two carry pi^3, pi^6); plus the dual ladder (coupling 3-n at c3-order 3n). [C] conditional on the CS/eta reading (EM1, v48) the a^3 coefficient is an integer level, =1 the unit level, not a free real. [O] the EM1 CS-boundary derivation + ALPHA.QUILLEN.EXACT.01 stay open; seam F-norm stays the one [C] (v434); alpha^-1 stays [E]. A fourth honest step that sharpens the only remaining [O]; no new number, Python-only.

Dependencies as recorded

ALPHA.QUILLEN.EXACT.01; ALPHA.QUILLEN.PROGRESS.01; ALPHA.QUILLEN.PROGRESS.02; ALPHA.QUILLEN.PROGRESS.03; EM.WARD.01; EM.WARD.02; v48; v342; v382; v384; v433; v434