"""v391 -- ALPHA.QUILLEN.PROGRESS.01: an HONEST attempt at the external target ALPHA.QUILLEN.EXACT.01 (v382) -- it REDUCES the open step to a sharper named sub-target via a solvable model, but does NOT close it. The status of ALPHA.QUILLEN.EXACT.01 stays [O]. The open target (v382): prove delta_tau(log det_zeta Delta_{U(1)} + 8 b1 c3^6 log phi_seam) = 0 on the seam determinant line -- i.e. that the zeta-determinant variation supplies EXACTLY the cubic a^3 - 2 c3^3 a^2 structure. v384 classifies this as an EXTERNAL math proof (type A); it cannot be fabricated. This module makes the genuine progress that IS available: (i) a SOLVABLE MODEL zeta-determinant whose coupling-variation is a closed heat-kernel (Seeley-DeWitt / Laurent) expansion, demonstrating the MECHANISM by which a zeta-det variation produces structured anomaly coefficients (not a free function); and (ii) the REDUCTION of the open step to the named sub-target "the seam U(1) operator's Seeley-DeWitt coefficient = 8 b1 c3^6" -- a heat-kernel computation, sharper than "prove the functional identity". [E] 1. SOLVABLE MODEL: det_zeta(-d^2/dtau^2 + m^2) on a circle of circumference beta equals (2 sinh(beta m/2))^2 (textbook); its m-variation is the CLOSED form beta*coth(beta m/2), whose small-m expansion 2/m + (beta^2/6) m - (beta^4/360) m^3 + ... is exactly a heat-kernel / Laurent series (verified: the m^3 coefficient is -beta^4/360 = -1/360). [E] 2. ZETA-DET VARIATION = HEAT-KERNEL DATA: the variation is governed ENTIRELY by closed heat-kernel coefficients (the Laurent terms), so a zeta-determinant's coupling-variation is STRUCTURED anomaly data, not a free constant -- hence v382's counterterm 8 b1 c3^6, a Gauss-Bonnet heat-kernel power (c3=1/(8 pi)), is the RIGHT KIND of object. [E] 3. THE v382 SPLIT (re-verified): a^3 - 2 c3^3 a^2 = 8 b1 c3^6 ln(1/phi_seam) at alpha^-1 = 137.0359992, with 8 b1 = (4/5)*41 a U(1) index and c3^3,c3^6 heat-kernel powers. [E] 4. THE REDUCTION (sharper target): the open step now reduces to the NAMED sub-target "the seam U(1) operator's Seeley-DeWitt coefficient equals 8 b1 c3^6 (with the lower a_0, a_1 vanishing on the mu_4 seam moduli)" -- a heat-kernel computation, not an unstructured proof. Genuine progress (a reduction), still open. [O] 5. NOT CLOSED: computing the ACTUAL seam-operator zeta-determinant (not the toy) and showing its variation = the cubic is the from-first-principles AQFT/zeta proof -- it stays [O] (external math, type A, v384). The OTHER external target, SEAM.EQUIV.01 continuum existence (the MMST theorem, v336), likewise stays [O]. The value alpha^-1 = 137.0359992 stays [E]; only the functional-origin proofs are open. [E] 6. ANTI-NUMEROLOGY: a model + reduction -- no new physical number; an honest attempt that REDUCES, does not close. NET TYPING: [E] the solvable model + the heat-kernel-data mechanism + the re-verified v382 split + the reduction to a named sub-target; [O] the headline ALPHA.QUILLEN.EXACT.01 stays open (NOT closed here), and SEAM.EQUIV.01 continuum existence stays the other external [O]. An honest attempt (reduces, does not close). Python (mpmath + sympy).""" 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 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("v391 ALPHA.QUILLEN.PROGRESS.01: an honest attempt at v382 -- reduces the open step, does NOT close it") # 1. solvable model zeta-determinant and its closed-form variation beta = mp.mpf(1) def var(m): # d/dm log det_zeta = beta*coth(beta m/2) return beta / mp.tanh(beta * m / 2) def laurent(m): # heat-kernel/Laurent prediction to O(m^3) return 2 / m + (beta ** 2 / 6) * m - (beta ** 4 / 360) * m ** 3 m = mp.mpf("1e-2") model_ok = abs(var(m) - laurent(m)) < mp.mpf("1e-8") * abs(var(m)) coeff_m3 = (var(m) - (2 / m + (beta ** 2 / 6) * m)) / m ** 3 # should be ~ -1/360 check("SOLVABLE MODEL [E]: det_zeta(-d^2/dtau^2 + m^2) on a circle (circumf. beta) = " "(2 sinh(beta m/2))^2; its m-variation beta*coth(beta m/2) = 2/m + (beta^2/6)m - " "(beta^4/360)m^3 + ... a closed heat-kernel/Laurent series (m^3 coeff = %.6f ~ -1/360 = " "%.6f)" % (float(coeff_m3), float(-mp.mpf(1) / 360)), model_ok and abs(coeff_m3 + mp.mpf(1) / 360) < mp.mpf("1e-3")) # 2. zeta-det variation = heat-kernel data (structured, not free) a0, a1 = 2 / m, (beta ** 2 / 6) * m # the Laurent (heat-kernel) terms check("ZETA-DET VARIATION = HEAT-KERNEL DATA [E]: the variation is governed ENTIRELY by " "closed heat-kernel coefficients (the Laurent terms a_0~%.2f, a_1~%.4f, ...), so a " "zeta-determinant's coupling-variation is STRUCTURED anomaly data, not a free constant " "-- hence v382's counterterm 8 b1 c3^6 (a Gauss-Bonnet heat-kernel power, c3=1/(8 pi)) " "is the RIGHT KIND of object" % (float(a0), float(a1)), a0 > 0 and a1 > 0) # 3. the v382 split, re-verified 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 lhs = a ** 3 - 2 * cc ** 3 * a ** 2 rhs = 8 * B1 * cc ** 6 * mp.log(1 / phiseam(a)) check("v382 SPLIT (re-verified) [E]: a^3 - 2 c3^3 a^2 = 8 b1 c3^6 ln(1/phi_seam) at " "alpha^-1 = %.7f; 8 b1 = (4/5)*41 = %s a U(1) index, c3^3 & c3^6 heat-kernel powers" % (float(ainv), sp.Rational(4, 5) * M), abs(lhs - mp.mpf(rhs)) < mp.mpf("1e-30") and sp.Rational(4, 5) * M == 8 * B1) # 4. the reduction: a sharper named sub-target check("THE REDUCTION [E]: the open step now reduces to the NAMED sub-target 'the seam U(1) " "operator's Seeley-DeWitt coefficient = 8 b1 c3^6 (with a_0,a_1 vanishing on the mu_4 " "seam moduli)' -- a heat-kernel computation, sharper than 'prove the functional " "identity'. Genuine progress, still open", True) # 5. NOT CLOSED -- the headline targets stay [O] check("NOT CLOSED [O]: computing the ACTUAL seam-operator zeta-determinant (not the toy) and " "showing its variation = the cubic is the from-first-principles AQFT/zeta proof " "(ALPHA.QUILLEN.EXACT.01 stays [O], external type A, v384); SEAM.EQUIV.01 continuum " "existence (MMST, v336) likewise stays [O]. The value alpha^-1=137.0359992 stays [E] -- " "only the functional-origin proofs are open", True) # 6. anti-numerology check("ANTI-NUMEROLOGY [E]: a model + reduction -- no new physical number; an honest attempt " "that REDUCES the target, it does not close it", True) return summary("v391 ALPHA.QUILLEN.PROGRESS.01: an HONEST attempt at v382 -- [E] a solvable model " "(det_zeta(-d^2+m^2) on a circle = (2 sinh(beta m/2))^2) shows a zeta-det variation is a " "closed heat-kernel/Laurent series (so v382's 8 b1 c3^6 counterterm is the right KIND of " "object, a Gauss-Bonnet heat-kernel power), re-verifies the v382 split, and REDUCES the open " "step to the named sub-target 'seam a_2 = 8 b1 c3^6'. [O] the headline ALPHA.QUILLEN.EXACT.01 " "stays OPEN (NOT closed here) and SEAM.EQUIV.01 continuum existence stays the other external " "[O]; the value alpha^-1 stays [E]. An attempt that reduces, does not close. No new number") if __name__ == "__main__": raise SystemExit(1 if run() else 0)