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A registered result in its source context

Original ledger classification: Reformulation [E]/[O] (alpha = stationarity of a named U(1) determinant line; sharpens the existing [P] EM.WARD.01; no new gate)

Recorded statement: THE ALPHA FIXED POINT AS THE STATIONARITY OF A U(1) DETERMINANT LINE (2026-06-22): reformulates the EM closure from 'a very cleverly built fixed point' into 'the stationarity of a named functional', identifying every ingredient as an index…

Statement, assumptions and sources

Paper context: Architecture, E8 and the electromagnetic fixed point

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ALPHA.QUILLEN.01

Original ledger classification: Reformulation [E]/[O] (alpha = stationarity of a named U(1) determinant line; sharpens the existing [P] EM.WARD.01; no new gate)

Original claim text

THE ALPHA FIXED POINT AS THE STATIONARITY OF A U(1) DETERMINANT LINE (2026-06-22): reformulates the EM closure from 'a very cleverly built fixed point' into 'the stationarity of a named functional', identifying every ingredient as an index / heat-kernel coefficient / discriminant form -- the 'derive the FORM' step (external review). F_{U(1)} = a^3 - 2 c3^3 a^2 - (4/5) c3^6 M ln(1/phi_seam) = 0 written in Quillen form: [a^3 - 2 c3^3 a^2] (determinant-line variation) = [8 b1 c3^6] (anomaly counterterm) * [ln(1/phi_seam)] (seam response), i.e. delta log det Delta_U(1) + counterterm + seam response = 0. [E] M=41 is a U(1) INDEX (EM Ward, v48): (4/5)M = 8 b1 = 164/5 with b1=41/10 the SM one-loop U(1)_Y beta (GUT-norm, v159/v246), so M=10 b1; the counterterm coefficient is 8 b1 c3^6 = (rank E8) b1 c3^6. [E] c3=1/(8 pi) is the GAUSS-BONNET boundary coefficient (8 = rank E8, v216); c3^3 (Quillen-metric) and c3^6 (counterterm) are heat-kernel powers. [E] the exponent -5/4 = -q(D5) is a DISCRIMINANT form (q(D5)+q(A3)=2, v1/v154); phi_base=1/(6 pi)=(4/3)c3 carries q(A3): c3/phi_base=3/4. [E] delta_top=Omega_adm c3^4 (Omega_adm=48). [E] the Quillen split holds at the alpha^-1=137.0359992 root. [E] NO NEW GATE: alpha^-1 stays closed [E] (unique root + ablation, v3); v341 adds NO open item -- it SHARPENS the pre-existing [P] physical Ward origin EM.WARD.01 ('why THIS function') toward [E] by naming every coefficient. [O] the one still-open part IS EM.WARD.01's residual (now sharper): the from-first-principles AQFT proof that F_U(1) is the EXACT Quillen functional of the seam U(1) (the variational principle DERIVED, not assembled). Not a new gate; not closed.

Original classification detail

[E] the coefficient identifications (8 b1 = (4/5)M; c3 Gauss-Bonnet; -5/4=-q(D5); c3/phi_base=q(A3)) + the Quillen-form split at the root + the 'no new gate' clarification (machine-checked); [O] the still-open part = EM.WARD.01's residual (the first-principles AQFT derivation of the functional). Sharpens v3/v48/EM.WARD.01 from fixed point to stationarity; does NOT open a new gate and does NOT prove the variational principle.

Dependencies as recorded

EM.WARD.01; v3; v48; v159; v216; v1

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Registered verification modules

  • v341_alpha_quillen.pyRegistered verification module · em
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    The alpha fixed point reformulated as the STATIONARITY of a U(1) determinant line, every ingredient identified as an index / heat-kernel coefficient / discriminant form -- NOT a free knob (the 'derive the FORM' step). F_{U(1)} = a^3 - 2 c3^3 a^2 - (4/5) c3^6 M ln(1/phi_seam) = 0 in Quillen form: [a^3 - 2 c3^3 a^2] (det-line variation) = [8 b1 c3^6] (anomaly counterterm) * [ln(1/phi_seam)] (seam response). [E] M=41 a U(1) INDEX (EM Ward, v48): (4/5)M = 8 b1, b1=41/10 the SM 1-loop U(1)_Y beta. [E] c3=1/(8pi) GAUSS-BONNET boundary coeff (8=rank E8). [E] -5/4 = -q(D5) DISCRIMINANT; phi_base=(4/3)c3, c3/phi_base=3/4=q(A3). [E] delta_top=Omega_adm c3^4. [E] Quillen split holds at the alpha^-1=137.036 root. [E] NO NEW GATE: alpha stays [E]; v341 SHARPENS the pre-existing [P] EM.WARD.01 (no new alpha problem). [O] the one still-open part IS EM.WARD.01's residual: the AQFT proof that this is the EXACT Quillen functional (variational principle derived, not assembled). Sharpens v3/v48/EM.WARD.01 from fixed point to stationarity of a named determinant line.

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