Follow the source behind the result.

The registered Python source is shown as text. Displaying it does not execute the code.

"""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)

9da74d256db2e5155c970ede1bcd3ca0e1f8491aa188ffc5af57b8d06264b384

The public registry includes 116 document views, 6,693 section records and 985 physical LaTeX sources. Active publication documents, research material and archives retain their recorded roles. This is source navigation, not a new semantic review or proof run.