The physical determinant variation required for alpha

ALPHA.QUILLEN.EXACT.01

Original ledger classification: Named target [E]/[O] (the Quillen determinant-line variation-exactness; a face of SEAM.EQUIV.01, elevates EM.WARD.01's residual to a citable target; no new gate; 2026-08-28: the v974 CHANNEL-SWAP face resolved at the finite level by mu4-character grading, v985; completeness wave: finite KMS rigidity-premise [E] v995, ln-2-rate detline shadow v996 -- continuum Bismut-Freed stays [O])

Original claim text

Names the Quillen determinant-line VARIATION as a tracked [O] target (2026-06-23, review Point 10): delta_tau(log det_zeta Delta_U(1) + 8 b1 c3^6 log phi_seam)=0 on the determinant line over the mu4 seam moduli -- i.e. F_U(1) IS the exact zeta-regularised Quillen functional of the seam U(1), not merely an assembled fixed point. Elevates EM.WARD.01's residual (v341/v342) to its own citable target so the one remaining EM obligation is trackable separately from the seam-net identification, WITHOUT creating a second independent open problem (it is a FACE of SEAM.EQUIV.01). [E] the Quillen split holds at the root a^3-2 c3^3 a^2 = 8 b1 c3^6 ln(1/phi_seam) at alpha^-1=137.0359992 (F_U(1)=0 IS the stationarity, v341). [E] every coefficient is a named atom: 8 b1=(4/5)*41 a U(1) index, c3=1/(8 pi) the Gauss-Bonnet boundary coeff (c3^3,c3^6 heat-kernel powers), q(D5)+q(A3)=2 discriminant forms (v341/v48/v216). [E] the value alpha^-1=137.0359992168 stays closed (unique root+ablation, v3) -- only 'why this functional' is open. [O] THE TARGET: the from-first-principles AQFT/zeta proof of the variation. [C] FACE OF SEAM.EQUIV.01 (v336): the seam U(1) determinant line lives on the same holomorphic seam net, so once the net is (E8)_1 its U(1) sub-determinant is fixed -- not a second gate. No new number (reuses v341, Wolfram-mirrored). UPDATE 2026-07-14 (v484/SEAM.CONTACT.UNIT.01): the target's c3-ladder rule MERGES with HYP.PHI0.PUNCTURE.01 into one analytic step -- the contact unit c3-per-insertion is the KMS seam unit (v239), derived for the finite cycle sector; the remaining [O] is the diagonal zeta-renormalisation + multiplicity matching (41 = 10 b1). UPDATE 2026-07-14 (v485): the computable side is COMPLETE (diagonal zero at the KMS scale, closed-form mark determinant, 41 = Y^2-weighted count of the same 48 states); the target's remaining [O] is now EXACTLY its v382 typing -- the SEAM.EQUIV.01 face -- with no separate analytic unknown left. UPDATED 2026-08-27 (v974): both named faces now carry COMPUTED witnesses. The diagonal-zeta face is EXECUTED as the BFK gluing on the seam circle (gluing constant EXACTLY 2^-N, derived symbolically N = 1..4 at general lengths; jump matrix at KMS R_0 = 16 c3 circ(2,-1,0,-1) = (G_m|_marks)^-1; det'R_0 = 2^16 c3^3, glued product (2 pi)^2; the v485 scale channel reproduced exactly via the KMS unit) -- vs the v484 mark matrix there is NO global proportionality (kernels in SWAPPED mu4 channels), but the correspondence is channel-wise exact (same eigenbasis, |spec| {0,2,2,4}, doublet ratio 4/ln2 with c3 cancelling): the diagonal-zeta face is thereby SHARPENED to the channel-swap question. The finite Quillen identity is verified TWO-SIDED on the twist torus (Berry integral == Poincare-Lelong divisor integral == +1, difference an exact form, transition winding == Chern), typed conditional on the quasi-free rigidity premise (measured, not proven -- SEAM.STATE.RPMIXING.01). STAYS [O]: 'why this functional' is not closed -- the remaining content is typed as {rigidity premise} + {channel-swap question} + {continuum Bismut-Freed identification of the exact-form difference field}; alpha^-1 = 137.0359992168 re-anchored (unique root on the scanned window, split residual 6.4e-58 at 50 digits). EXECUTED NOTE (2026-08-28, completeness wave): the rigidity premise of the finite Quillen identity is now executed at the finite CAR level (v995: 33 mixing directions killed by KMS alone; covariance redundant; 196608 -> 0); type-III/continuum stays [O]. The finite-collar detline curvature converges exponentially to 2pi with rate ~0.6936 ~ ln 2 (v996) -- a numerical Bismut-Freed shadow, not the continuum theorem. STAYS [O]. Python-only / Wolfram mirror deferred. RELATIVE-DETERMINANT NOTE (2026-08-29, review wave 4): the correct object is the RELATIVE determinant Q(A) = log[det' Delta_A / det' Delta_{A0}] + 8 b1 c3^6 log phi(A); the BFK constant 2^{-4} is A/tau/alpha-INDEPENDENT at fixed cut number, so delta log 2^{-4} = 0 -- no physical compensation needed (consistent with COMPENSATION_INTERNAL + the reviewer's correction). Absolute normalization is fixed by the reference A0. Contract stays [O]. LATE EVENING (2026-08-30, v1014): finite restriction isomorphism verified (c_phase constancy 3.7e-16, conjugation covariance exact, C = W_bulk = W_seam = +1 / mirror -1); orientation fixes the conjugate branch (2 -> 1); the residual constant phase is fixed by the A0 reference normalization -- recorded. Continuum Bismut-Freed stays [O]. Contract stays [O].

Original classification detail

[E] the Quillen split at the root + named-atom coefficients + the value alpha^-1=137.0359992 (reused from v3/v48/v341, machine-checked); [O] the from-first-principles exactness proof (the named target); [C] it is a face of SEAM.EQUIV.01, not a second independent gate. A typing/registration of the pre-existing EM.WARD.01 residual, reusing v341's (Wolfram-mirrored) identities; introduces no new number. UPDATE (2026-08-28, external master-object review wave 2, v985): the channel-swap face is resolved at the FINITE level -- the deck swap r -> r+2 maps the jump |spectrum| [0,2,4,2] onto the mark pattern |[4,-2,0,-2]| exactly (odd swap fails), the graded det' difference is c3-free = log(ln2/4), and the channelwise unit ratio is uniform 4/ln2; finite KMS rigidity-premise executed [E] (v995: 33 directions killed; 196608 -> 0); ln-2-rate detline shadow [E-measured] (v996, numerical not theorem); continuum Bismut-Freed, type-III KMS uniqueness and the from-first-principles exactness of F_U(1) stay [O] (BFK 2^{-4} needs no physical compensation: relative det, 2026-08-29). | 2026-08-29: relative det Q(A)=log[det'Delta_A/det'Delta_{A0}]+8 b1 c3^6 log phi(A); BFK 2^{-4} A/tau/alpha-independent at fixed cut number (delta log 2^{-4}=0, no physical compensation). Stays [O]. WAVE-5 (2026-08-29): rigidity premise of SEAM.MMST.TYPEIII.CHARGED.01; no marker move. | LATE EVENING (2026-08-30 v1014): finite restriction isomorphism verified; orientation fixes the conjugate branch; the residual constant phase is fixed by the A0 reference normalization -- recorded. Stays [O].

Dependencies as recorded

EM.WARD.01; SEAM.EQUIV.01; v341; v342; v3; v48; v216; v336