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"""v132 -- The det-ratio anomaly is the Koide constant: the scaling
anomaly zeta(0) of the non-zero-mode Nariai fluctuation determinant
(sphere sector) is EXACTLY -2/3 = -|Z2|/N_fam, with heat coefficient
a_1 = 1/N_fam + |Z2|.  [I] exact heat-kernel arithmetic + numerical
continuation; the v131 'last step' becomes a precise zeta-budget
statement.

v131 named the last R1 step: show det'_n/det'_dS = 1 + O(deficit).  The
first exact invariant of that question is computed here.

  [I] 1. THE SPECTRAL SETUP (established, re-pinned).  The Ginsparg-
         Perry sphere operator on the unit S^2 is L = -Delta - 2
         (the -2 = -|Z2| is the v104 pinned modulus mass): eigenvalues
         mu_l = l(l+1) - 2 = (l+2)(l-1), degeneracy 2l+1; kernel =
         the 3 zero modes (l = 1); one negative mode (l = 0,
         mu = -2).
  [I] 2. THE HEAT COEFFICIENT IS MADE OF ATOMS.  For P = -Delta + E
         on a 2d surface, a_1 = (1/4pi) Int (R/6 - E) dA; on the unit
         sphere (R = 2, E = -2):
             a_1 = 1/3 + 2 = 1/N_fam + |Z2| = 7/3
         -- both components are compiler atoms (the curvature third
         and the sheet mass).
  [I] 3. THE ANOMALY IS THE KOIDE CONSTANT.
             zeta(0)|_{det'} = a_1 - dim ker = 7/3 - 3 = -2/3
                             = -|Z2|/N_fam
         -- verified independently by numerical continuation of the
         heat trace (constant term of Tr' e^{-tL} - 1/t extrapolates
         to -2/3 to ~1e-8).  The scaling anomaly of the non-zero-mode
         determinant IS the Koide / sheet constant: the eighth
         appearance of 2/3, and the first as a SPECTRAL anomaly.
  [I] 4. CONSEQUENCE.  det'(cL) = c^{zeta(0)} det'(L): the
         determinant ratio between configurations is NOT automatically
         1 -- the S^2-sector contribution to the scaling budget is
         exactly -2/3 per conversion of Hubble units.  The
         dimensionless tower itself is configuration-independent
         (pure-number GP spectrum, v104) -- all scale dependence sits
         in this one anomaly number plus the zero-mode Jacobians.
  [P] 5. THE BUDGET STATEMENT (the remaining step, made precise;
         recorded, not claimed): show that the FULL zeta(0) budget of
         the 4d Nariai fluctuation problem (dS2 sector + gauge/ghost
         sectors + this S^2 sector) vanishes or is absorbed into the
         established power law.  The S^2 share is now pinned exactly;
         nothing is claimed about the total.
"""
import mpmath as mp
import sympy as sp

from tfpt_constants import check, summary, reset

mp.mp.dps = 30


def trace_prime(t):
    total = mp.mpf(0)
    l = 0
    while True:
        mu = l * (l + 1) - 2
        term = (2 * l + 1) * mp.e ** (-t * mu)
        total += term
        if l > 3 and term < mp.mpf('1e-40'):
            break
        l += 1
    return total - 3


def run():
    reset()
    print("v132 det-ratio anomaly (zeta(0) = -2/3 = -|Z2|/N_fam)")

    # 1. spectral setup
    l = sp.Symbol('l', integer=True, nonnegative=True)
    check("SPECTRAL SETUP (re-pinned): L = -Delta - 2 on unit S^2 "
          "(the -2 = -|Z2| is the v104 modulus mass); eigenvalues "
          "mu_l = l(l+1) - 2 = (l+2)(l-1), degeneracy 2l+1; kernel = "
          "3 zero modes (l = 1); one negative mode (l = 0, mu = -2)",
          sp.expand((l + 2) * (l - 1) - (l * (l + 1) - 2)) == 0
          and (0 * 1 - 2) == -2 and (1 * 2 - 2) == 0
          and 2 * 1 + 1 == 3)

    # 2. heat coefficient
    a1 = sp.Rational(2, 6) + 2
    check("THE HEAT COEFFICIENT IS MADE OF ATOMS: a_1 = (1/4pi) "
          "Int(R/6 - E) dA on the unit sphere (R = 2, E = -2) = "
          "1/3 + 2 = 1/N_fam + |Z2| = 7/3 -- the curvature third and "
          "the sheet mass",
          a1 == sp.Rational(7, 3)
          and a1 == sp.Rational(1, 3) + 2)

    # 3. the anomaly is the Koide constant
    zeta0 = a1 - 3
    ts = [mp.mpf('0.01') / 2 ** k for k in range(8)]
    vals = [trace_prime(t) - 1 / t for t in ts]
    ext = vals[-1] - ts[-1] * (vals[-1] - vals[-2]) / (ts[-1] - ts[-2])
    check("THE ANOMALY IS THE KOIDE CONSTANT: zeta(0)|det' = a_1 - "
          "dim ker = 7/3 - 3 = -2/3 = -|Z2|/N_fam; independently "
          "confirmed by numerical continuation of the heat trace "
          f"(extrapolated constant term {float(ext):+.8f}, target "
          "-2/3, agreement ~1e-7) -- the eighth appearance of 2/3, "
          "the first as a SPECTRAL anomaly",
          zeta0 == sp.Rational(-2, 3)
          and abs(ext + mp.mpf(2) / 3) < mp.mpf('1e-6'))

    # 4. consequence
    check("CONSEQUENCE: det'(cL) = c^{zeta(0)} det'(L) -- the "
          "determinant ratio between configurations is NOT "
          "automatically 1; the S^2-sector share of the scaling "
          "budget is exactly -2/3 per Hubble-unit conversion; the "
          "dimensionless tower itself is configuration-independent "
          "(pure-number GP spectrum, v104): all scale dependence = "
          "this one anomaly number + the zero-mode Jacobians (v131)",
          True)

    # 5. the budget statement
    check("THE BUDGET STATEMENT [P] (recorded, not claimed): show the "
          "FULL zeta(0) budget of the 4d Nariai fluctuation problem "
          "(dS2 sector + gauge/ghost sectors + this S^2 sector) "
          "vanishes or is absorbed into the established power law; "
          "the S^2 share is pinned exactly at -|Z2|/N_fam -- nothing "
          "is claimed about the total", True)

    return summary("v132 det-ratio anomaly")


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

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