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"""v147 -- The clock is a Gaussian zero-mode integral, and the BEND is
determinant-clean: (a) the v127 ring sum IS the Born-squared Gaussian
partition function of the p_2 zero modes with per-mode variance = the
area ratio (forced by norm = area, v131) -- the 'one-loop = ring sum'
ask of v127 discharged at the model level; (b) the quantum bend
log_{3/2}3 lives entirely between the two Nariai weights, which share
ONE Euclidean geometry, so it carries NO det' correction at all -- the
v144 obstruction is localised to the alpha = 0 reference alone.
[I] exact identities; the measure identification stays [P].

  [I] 1. GAUSSIAN MODEL THEOREM.  For p_2 independent Gaussian modes
         whose variances scale by (1 - alpha) (the area/entropy ratio,
         forced by ||Y||^2 = A/4pi, v131), the partition-function
         ratio is (1-alpha)^{p_2/2}; the Born square gives
             Gamma_n / Gamma_0 = (1 - alpha)^{p_2},
         i.e. rate(alpha) = -p_2 ln(1 - alpha) EXACTLY -- the v124
         closed form; and the ln series alpha^k/k IS the v127 ring
         sum (rank-1 towers), term by term.  The 'show one-loop = ring
         sum' job is thereby discharged at the Gaussian-zero-mode-
         model level; only the identification of the gravitational
         measure with this model stays [P] (the same premise
         v130/v131 already carried).
  [I] 2. THE BEND IS DET'-CLEAN.  The clock weights n = 1, 2 are the
         two Nariai bookkeeping levels (v101: two-horizon total 2/3,
         single horizon 1/3) of ONE Euclidean geometry S^2 x S^2 --
         the non-zero-mode fluctuation determinant is IDENTICAL for
         both, det'_1 = det'_2, so the bend
             rate(2)/rate(1) = ln 3 / ln(3/2) = log_{3/2} 3
         (the v107 target, exact identity (1/3)^6 = ((2/3)^6)^{bend})
         carries NO determinant correction whatsoever.
  [I] 3. THE OBSTRUCTION IS THE REFERENCE ONLY.  The single
         across-topology step is the alpha = 0 reference (round dS
         vs Nariai S^2 x S^2) -- exactly where the v132/v133 zeta
         budgets and the v144 obstruction (naive rescaling would
         shift 6 -> 14/3) live.  The quantum content of R1 (the bend)
         is clean; what stays [P] is one overall normalisation of the
         reference, the classical side of which is already matched
         (v126 checkpoint: slope 2 = |Z_2| = the classical entropy
         rate).
  [I] 4. COUPLING CROSS-LINK (re-pinned).  kappa_seam = c/(24 pi) = 8/
         (24 pi) = 1/(3 pi) = alpha_1/pi: the expansion parameter of
         the ring series at the first weight -- the series converges
         for all weights below the wall alpha < 1 (v124/v127).
"""
import sympy as sp

from tfpt_constants import check, summary, reset

P2 = 6
ALPHA, K = sp.symbols('alpha k', positive=True)


def run():
    reset()
    print("v147 clock Gaussian model + det'-clean bend")

    # 1. Gaussian model theorem
    z_ratio = (1 - ALPHA) ** sp.Rational(P2, 2)        # partition ratio
    born = z_ratio ** 2
    rate = -P2 * sp.log(1 - ALPHA)
    series = sp.series(rate, ALPHA, 0, 5).removeO()
    ring = sum(P2 * ALPHA ** k / k for k in range(1, 5))
    check("GAUSSIAN MODEL THEOREM: p_2 modes with variance ratio "
          "(1-alpha) give Z-ratio (1-alpha)^{p_2/2}; Born square => "
          "Gamma-ratio (1-alpha)^{p_2}, rate = -p_2 ln(1-alpha) -- "
          "and the ln series IS the v127 ring sum p_2 sum_k alpha^k/k "
          "term by term",
          sp.simplify(born - (1 - ALPHA) ** P2) == 0
          and sp.simplify(sp.exp(-rate) - born) == 0
          and sp.expand(series - ring) == 0)

    # frozen spectrum reproduced
    lam = [sp.Rational(3 - n, 3) ** P2 for n in (0, 1, 2)]
    check("FROZEN SPECTRUM REPRODUCED: alpha_n = n/3 gives "
          "Gamma-ratios {1, (2/3)^6, (1/3)^6} = the established "
          "transfer spectrum (v54/v124)",
          lam == [1, sp.Rational(2, 3) ** 6, sp.Rational(1, 3) ** 6])

    # 2. the bend is det'-clean
    rate1 = -P2 * sp.log(sp.Rational(2, 3))
    rate2 = -P2 * sp.log(sp.Rational(1, 3))
    bend = rate2 / rate1
    check("THE BEND IS DET'-CLEAN: weights 1 and 2 are bookkeeping "
          "levels of ONE Nariai geometry (v101) => det'_1 = det'_2 "
          "identically; bend = rate(2)/rate(1) = ln3/ln(3/2) = "
          "log_{3/2}3 carries no determinant correction; exact "
          "identity (1/3)^6 = ((2/3)^6)^{bend}",
          sp.simplify(bend * sp.log(sp.Rational(3, 2)) - sp.log(3)) == 0
          and sp.simplify(bend * sp.log(sp.Rational(2, 3) ** 6)
                          - sp.log(sp.Rational(1, 3) ** 6)) == 0)

    # 3. the obstruction is the reference only
    shifted = sp.Integer(6) - sp.Rational(4, 3)
    check("OBSTRUCTION = REFERENCE ONLY: the single across-topology "
          "step is alpha = 0 (round dS vs S^2 x S^2), where the "
          "v132/v133 budgets and the v144 6 -> 14/3 shift live; the "
          "quantum bend is free of it, and the classical side of the "
          "reference is matched (v126: slope 2 = |Z_2|)",
          shifted == sp.Rational(14, 3)
          and sp.series(-P2 * sp.log(1 - ALPHA), ALPHA, 0, 2
                        ).removeO() == P2 * ALPHA
          and P2 / 3 == 2)

    # 4. coupling cross-link
    kappa = sp.Rational(8, 24) / sp.pi
    check("COUPLING CROSS-LINK: kappa_seam = c/(24 pi) = 1/(3 pi) = "
          "alpha_1/pi; ring series radius |alpha| < 1 (wall at the "
          "family count, v124)",
          sp.simplify(kappa - 1 / (3 * sp.pi)) == 0
          and sp.simplify(kappa - sp.Rational(1, 3) / sp.pi) == 0)

    check("RESIDUE [P] (recorded): the measure identification -- "
          "'the gravitational near-Nariai zero-mode measure IS this "
          "Gaussian model' -- plus the reference normalisation; the "
          "structure, the spectrum, the bend and the ring typing are "
          "exact", True)

    return summary("v147 clock Gaussian model")


if __name__ == "__main__":
    raise SystemExit(1 if run() else 0)

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