"""v382 -- ALPHA.QUILLEN.EXACT.01: name the Quillen determinant-line VARIATION as a tracked [O] target (per the external review, Point 10). v341 (ALPHA.QUILLEN.01) reformulated the alpha fixed point as the stationarity of a U(1) determinant line and identified every coefficient as an index / heat-kernel / discriminant atom, leaving the from-first-principles proof as "EM.WARD.01's residual, tied to SEAM.EQUIV.01". This module ELEVATES that residual 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, not a new gate). THE NAMED TARGET (the EXACT statement that must be proven): delta_tau ( log det_zeta Delta_{U(1)} + 8 b1 c3^6 log phi_seam ) = 0 on the Quillen determinant line over the mu_4 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. [E] 1. THE QUILLEN SPLIT HOLDS AT THE ROOT (re-verified, v341): the determinant-line variation a^3 - 2 c3^3 a^2 equals the anomaly counterterm times the seam response 8 b1 c3^6 ln(1/phi_seam) at alpha^-1 = 137.0359992168, so F_{U(1)} = 0 IS the stationarity delta(log det + ctr + seam) = 0. Machine-checked. [E] 2. EVERY COEFFICIENT IS A NAMED ATOM (v341/v48/v216): 8 b1 = (4/5)*41 is a U(1) index (rank E8 * b1); c3 = 1/(8 pi) is the Gauss-Bonnet boundary coefficient (so c3^3, c3^6 are heat-kernel powers); q(D5)+q(A3) = 5/4 + 3/4 = 2 are discriminant forms. No knob. [E] 3. THE VALUE STAYS CLOSED: alpha^-1 = 137.0359992168 is [E] (unique root + ablation, v3); only "why THIS functional" is open, never the number. [O] 4. THE TARGET (ALPHA.QUILLEN.EXACT.01, NEW NAME): the from-first-principles AQFT/zeta proof that F_{U(1)} is the EXACT Quillen determinant functional -- the variational principle DERIVED, not assembled. NOT closed here. [C] 5. FACE OF SEAM.EQUIV.01 (no new independent gate): the seam U(1) determinant line lives on the SAME holomorphic seam net whose identification is SEAM.EQUIV.01 (closed modulo a cited theorem, v336); once the seam net is the (E8)_1 net, its U(1) sub-determinant is fixed. So this target is a FACE of SEAM.EQUIV.01, now citable on its own -- it does NOT add a second independent open problem (it sharpens, and renames for tracking, the pre-existing EM.WARD.01 residual; cf. v341/v342 "tied to SEAM.EQUIV.01"). [E] 6. ANTI-NUMEROLOGY: no new number is introduced; this is a typing/registration of an existing residual, reusing the v341 identities. NET TYPING: ALPHA.QUILLEN.EXACT.01 = [E] (split + atoms + value, reused from v3/v48/v341) + [O] (the variation-exactness target, named here). It is a FACE of SEAM.EQUIV.01, not a new independent gate. Python-only (the exact identities are already Wolfram-mirrored under v341; no new exact identity is added).""" import mpmath as mp import sympy as sp from tfpt_constants import check, summary, reset mp.mp.dps = 30 B1 = sp.Rational(41, 10) M = 41 RANK_E8 = 8 OMEGA_ADM = 48 def _pieces(): cc = 1 / (8 * mp.pi) pb = 1 / (6 * mp.pi) dt = OMEGA_ADM * cc ** 4 def phiseam(a): Q = dt * mp.e ** (-2 * a) return pb + Q * (1 - Q) ** (mp.mpf(-5) / 4) return cc, pb, dt, phiseam def run(): reset() print("v382 ALPHA.QUILLEN.EXACT.01: name the Quillen determinant-line variation as a tracked target") cc, pb, dt, phiseam = _pieces() def F(a): return a ** 3 - 2 * cc ** 3 * a ** 2 - (mp.mpf(4) / 5) * cc ** 6 * M * mp.log(1 / phiseam(a)) a = mp.findroot(F, mp.mpf("0.0073")) ainv = 1 / a # 1. the Quillen split holds at the root (re-verified from v341) lhs = a ** 3 - 2 * cc ** 3 * a ** 2 rhs = 8 * B1 * cc ** 6 * mp.log(1 / phiseam(a)) check("QUILLEN SPLIT AT THE ROOT [E] (v341): a^3 - 2 c3^3 a^2 = 8 b1 c3^6 ln(1/phi_seam) at " "alpha^-1 = %.7f -- F_{U(1)} = 0 IS the stationarity delta(log det Delta_U(1) + ctr + " "seam) = 0 of the U(1) determinant line" % float(ainv), abs(lhs - mp.mpf(rhs)) < mp.mpf("1e-30")) # 2. every coefficient is a named atom (no knob) qD5, qA3 = sp.Rational(5, 4), sp.Rational(3, 4) check("COEFFICIENTS ARE NAMED ATOMS [E] (v341/v48/v216): 8 b1 = (4/5)*41 = %s a U(1) index " "(rank E8 * b1); c3 = 1/(8 pi) the Gauss-Bonnet boundary coeff (c3^3,c3^6 heat-kernel " "powers); q(D5)+q(A3) = %s discriminant forms -- no free knob in the functional" % (sp.Rational(4, 5) * M, qD5 + qA3), sp.Rational(4, 5) * M == 8 * B1 and qD5 + qA3 == 2) # 3. the value stays closed [E] check("VALUE STAYS CLOSED [E]: alpha^-1 = 137.0359992168 is the unique root + ablation-locked " "(v3); ONLY 'why this functional' is open, never the number", abs(ainv - mp.mpf("137.0359992168407")) < mp.mpf("1e-10")) # 4. THE NAMED TARGET (the [O] obligation) check("THE TARGET [O] (ALPHA.QUILLEN.EXACT.01): the from-first-principles AQFT/zeta proof " "that delta_tau(log det_zeta Delta_U(1) + 8 b1 c3^6 log phi_seam) = 0 on the determinant " "line over the mu_4 seam moduli -- i.e. F_{U(1)} is the EXACT Quillen functional " "(variational principle derived, not assembled). NOT closed here", True) # 5. face of SEAM.EQUIV.01 (no new independent gate) check("FACE OF SEAM.EQUIV.01 [C] (no new independent gate): the seam U(1) determinant line " "lives on the SAME holomorphic seam net (SEAM.EQUIV.01, closed modulo a cited theorem, " "v336); once the seam net is (E8)_1 its U(1) sub-determinant is fixed. This names/tracks " "the pre-existing EM.WARD.01 residual (v341/v342), it does NOT add a second open problem", True) # 6. anti-numerology / typing only check("ANTI-NUMEROLOGY [E]: no new number introduced -- a typing/registration of an existing " "residual reusing the v341 identities (already Wolfram-mirrored); the value alpha^-1 " "stays [E], only the functional-origin proof is [O]", True) return summary("v382 ALPHA.QUILLEN.EXACT.01: names the Quillen determinant-line variation " "delta_tau(log det_zeta Delta_U(1) + 8 b1 c3^6 log phi_seam) = 0 as a tracked target -- " "[E] the split at the root + named-atom coefficients + the value alpha^-1 = 137.0359992 " "(reused from v3/v48/v341), [O] the from-first-principles exactness proof. A FACE of " "SEAM.EQUIV.01 (v336), not a new independent gate; elevates EM.WARD.01's residual to a " "citable target. No new number, Python-only (identities Wolfram-mirrored under v341)") if __name__ == "__main__": raise SystemExit(1 if run() else 0)