Follow the source behind the result.

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

"""v477 -- GRAV.ENTROPIC.SCALEFLOW.01: the scale-flow representation of the
entropic action, and the R^2 route (v475's surviving interpretation 3) typed
PRECISELY -- the gap of exactly (72/17)(8 pi)^9 collapses to ONE moment
condition on the scale weight, whose value is the SAME KMS moment TFPT already
fixes (v36's f0 <-> M^2 = c3^7 Mbar^2).  Zero new dials; still [C] (the weight
is TFPT input, not derived from the entropic action); nothing closes.

Background.  v473 verified the Frullani identity -ln A = int dt/t (e^{-tA} -
e^{-t}); v475 computed the exact raw coefficients 3 beta R + (17/24) beta^2
R^2 and killed the light-trace-mode reading (raw scalaron trans-Planckian by
(72/17)(8 pi)^9 in mass^2).  This module puts the two together:

  [E] 1. THE SCALE-FLOW IDENTITY.  -ln a = int_0^inf dt/t (e^{-ta} - e^{-t})
        (symbolic, exact) = 2 int_0^inf dchi/chi (e^{-a/chi^2} - e^{-1/chi^2})
        (substitution t = 1/chi^2; verified to 1e-20, mpmath): Bianconi's log
        action IS the FLAT scale-integral of relative heat-kernel actions --
        structurally the chi-integral of TFPT-type relative spectral actions
        S_rel,chi, which TFPT instead evaluates at ONE KMS scale with a
        weight f.  The bridge between the two action principles is a choice
        of SCALE MEASURE.
  [E] 2. THE MOMENT DICTIONARY.  For a general scale weight w(t) dt/t the
        maximally-symmetric expansion (v475 eigenvalues {1; 1/4 x4; 1/6 x6})
        gives S_w = 3 beta R mu1 + (17/24) beta^2 R^2 mu2 + O(R^3) with
        mu1 = int w e^{-t} dt, mu2 = int w t e^{-t} dt (exact, symbolic).
        The flat Frullani weight w = 1 has mu1 = mu2 = 1 and reproduces the
        v475 raw coefficients identically -- the raw action is the
        unit-moment point of a moment FAMILY.
  [E] 3. THE ONE MOMENT CONDITION (interpretation 3, typed).  EH matching
        fixes beta' mu1 = c3/6 (the v473 pin, now weight-dressed); the
        Starobinsky sector then requires EXACTLY mu2/mu1^2 = (72/17)(8 pi)^9
        -- the v475 kill-test number reappears as a moment ratio.  With it,
        the scalaron mass is m^2 = (4608 pi^2/17)(mu1^2/mu2) Mbar^2 and the
        EXACT identity (4608 pi^2/17) / ((72/17)(8 pi)^9) = c3^7 closes the
        chain: m^2 = c3^7 Mbar^2 IDENTICALLY.  The '13 orders of magnitude'
        are not a mismatch of the bridge -- they are the statement that the
        R^2 coefficient is a SCALE-MEASURE datum the raw flat measure fixes
        wrongly and the TFPT KMS measure fixes correctly.
  [E] 4. CONSISTENCY WITH THE TFPT CONVENTION.  v36 writes the same physics
        as f0 = 6 (4 pi)^2 / c3^7 (spectral-action moment) <=> M^2/Mbar^2 =
        c3^7; the moment-condition route of check 3 lands on the SAME mass
        -- one KMS moment, two equivalent parametrisations, zero new dials.
  [C] 5. HONEST TYPING.  This is a CONSISTENCY statement, not a derivation:
        the KMS weight (and hence mu2/mu1^2) is TFPT input; the entropic
        action supplies the Einstein + Lambda level and the scale-flow FORM,
        TFPT supplies the measure.  Interpretation 3 stays [C]; the R^2 gate
        and v359's typing stay [O].  Also recorded (negative, scratch-level):
        a first attempt to tie the AP2 state-side reading to an EXACT lattice
        Bisognano-Wichmann commutator (Peschel-Eisler form) did NOT hold in
        the simple candidate forms on the half-filled open chain -- kept out
        of the suite, noted in the research log.

NET TYPING: [E] the four exact identities; [C] the mechanism reading; [O]
unchanged.  Extends GRAV.ENTROPIC.ACTION.01 (v473) / .SCALARON.01 (v475).
Exact (sympy + one high-precision quadrature, Wolfram-mirrored).
"""
import mpmath as mp
import sympy as sp

from tfpt_constants import check, summary, reset

pi = sp.pi
c3 = sp.Rational(1, 8) / pi
Mbar = sp.Symbol('Mbar', positive=True)


def run():
    reset()
    print("v477  GRAV.ENTROPIC.SCALEFLOW.01: the scale-flow representation + the one moment condition")

    # 1. the scale-flow identity
    a, t = sp.symbols('a t', positive=True)
    frul = sp.simplify(sp.integrate((sp.exp(-a * t) - sp.exp(-t)) / t, (t, 0, sp.oo)))
    mp.mp.dps = 30
    chi_err = mp.mpf(0)
    for av in (mp.mpf('0.4'), mp.mpf('2.5')):
        val = mp.quad(lambda x: 2 * (mp.e ** (-av / x ** 2) - mp.e ** (-1 / x ** 2)) / x,
                      [0, 1, mp.inf])
        chi_err = max(chi_err, abs(val + mp.log(av)))
    check("SCALE-FLOW IDENTITY [E]: -ln a = int dt/t (e^{-ta} - e^{-t}) = %s "
          "(symbolic) = 2 int dchi/chi (e^{-a/chi^2} - e^{-1/chi^2}) (t = 1/chi^2; "
          "residual %s at 30 digits) -- Bianconi's log action IS the FLAT "
          "scale-integral of relative heat-kernel actions; TFPT's S_rel,chi "
          "evaluates ONE KMS scale with a weight f: the bridge is a choice of "
          "SCALE MEASURE" % (frul, mp.nstr(chi_err, 3)),
          frul == -sp.log(a) and chi_err < mp.mpf('1e-20'))

    # 2. the moment dictionary (general weight w(t) dt/t)
    b, R = sp.symbols('beta R', positive=True)
    eig = [sp.Integer(1)] + [sp.Rational(1, 4)] * 4 + [sp.Rational(1, 6)] * 6
    trace_rel = sum(sp.exp(-t * (1 - b * lam * R)) - sp.exp(-t) for lam in eig)
    series_t = sp.expand(sp.series(trace_rel, R, 0, 3).removeO())
    coeff_R = sp.simplify(series_t.coeff(R, 1))     # 3 b t e^{-t}
    coeff_R2 = sp.simplify(series_t.coeff(R, 2))    # (17/24) b^2 t^2 e^{-t}
    mu1_flat = sp.integrate(coeff_R / t, (t, 0, sp.oo))       # -> 3 b
    mu2_flat = sp.integrate(coeff_R2 / t, (t, 0, sp.oo))      # -> (17/24) b^2
    check("MOMENT DICTIONARY [E]: the maximally-symmetric relative trace expands "
          "as (%s) R + (%s) R^2 + O(R^3); a scale weight w(t) dt/t turns these "
          "into 3 beta R mu1 + (17/24) beta^2 R^2 mu2 with mu1 = int w e^{-t} dt, "
          "mu2 = int w t e^{-t} dt; the FLAT weight (w = 1) has mu1 = mu2 = 1 and "
          "reproduces the v475 raw coefficients identically -- the raw action is "
          "the unit-moment point of a moment family"
          % (coeff_R, coeff_R2),
          coeff_R == 3 * b * t * sp.exp(-t) and
          coeff_R2 == sp.Rational(17, 24) * b ** 2 * t ** 2 * sp.exp(-t) and
          mu1_flat == 3 * b and mu2_flat == sp.Rational(17, 24) * b ** 2)

    # 3. the one moment condition closes the chain exactly
    mu1, mu2 = sp.symbols('mu1 mu2', positive=True)
    beta_val = (c3 / 6) / (8 * pi * Mbar ** 2) / mu1     # EH match: beta' mu1 = c3/6
    lp4 = (8 * pi * Mbar ** 2) ** 2
    eh_res = sp.simplify(lp4 * 3 * beta_val * mu1 - Mbar ** 2 / 2)
    m2s = sp.Symbol('m2s', positive=True)
    m2_w = sp.solve(sp.Eq(lp4 * sp.Rational(17, 24) * beta_val ** 2 * mu2,
                          Mbar ** 2 / (12 * m2s)), m2s)[0]
    ratio_needed = sp.solve(sp.Eq(m2_w, c3 ** 7 * Mbar ** 2), mu2)[0] / mu1 ** 2
    closure = sp.simplify((sp.Rational(4608, 17) * pi ** 2) /
                          (sp.Rational(72, 17) * (8 * pi) ** 9))
    check("THE ONE MOMENT CONDITION [E] (interpretation 3, typed): with the "
          "weight-dressed EH pin beta' mu1 = c3/6 (residual %s) the scalaron is "
          "m^2 = (4608 pi^2/17)(mu1^2/mu2) Mbar^2; demanding m^2 = c3^7 Mbar^2 "
          "forces EXACTLY mu2/mu1^2 = %s = (72/17)(8 pi)^9 -- the v475 kill-test "
          "number IS a moment ratio; and the closure identity (4608 pi^2/17) / "
          "((72/17)(8 pi)^9) = %s = c3^7 holds IDENTICALLY -- the 13 orders are "
          "not a bridge mismatch but a scale-measure datum"
          % (eh_res, sp.simplify(ratio_needed), closure),
          eh_res == 0 and
          sp.simplify(m2_w - sp.Rational(4608, 17) * pi ** 2 * (mu1 ** 2 / mu2) * Mbar ** 2) == 0 and
          sp.simplify(ratio_needed - sp.Rational(72, 17) * (8 * pi) ** 9) == 0 and
          sp.simplify(closure - c3 ** 7) == 0)

    # 4. consistency with the v36 spectral-action convention
    f0 = 6 * (4 * pi) ** 2 / c3 ** 7
    m2_v36 = 6 * (4 * pi) ** 2 / f0
    check("TFPT-CONVENTION CONSISTENCY [E]: v36 writes the same physics as f0 = "
          "6 (4 pi)^2 / c3^7 (KMS spectral-action moment) <=> M^2/Mbar^2 = "
          "6 (4 pi)^2 / f0 = %s = c3^7 -- the moment-condition route (check 3) "
          "and the spectral-action route land on the SAME scalaron mass: one KMS "
          "moment, two parametrisations, ZERO new dials" % m2_v36,
          sp.simplify(m2_v36 - c3 ** 7) == 0)

    # 5. honest typing
    check("HONEST TYPING [C/O]: a CONSISTENCY statement, not a derivation -- the "
          "KMS weight (hence mu2/mu1^2) is TFPT input; the entropic action "
          "supplies the Einstein + Lambda level and the scale-flow FORM, TFPT "
          "supplies the measure. Interpretation 3 stays [C]; the R^2 gate and "
          "v359's equation-of-state typing stay [O]. Negative side-record: the "
          "exact lattice-BW commutator attempt for AP2 (Peschel-Eisler simple "
          "forms) did NOT hold on the half-filled open chain -- kept out of the "
          "suite, logged in the research notes", True)

    return summary("v477 GRAV.ENTROPIC.SCALEFLOW.01: the scale-flow representation -- [E] "
                   "-ln A = flat scale-integral of relative heat kernels (Frullani, symbolic + "
                   "1e-20 quadrature), moment dictionary (flat weight = unit moments = the v475 "
                   "raw coefficients), the ONE moment condition mu2/mu1^2 = (72/17)(8 pi)^9 with "
                   "the EXACT closure (4608 pi^2/17)/((72/17)(8 pi)^9) = c3^7 => m^2 = c3^7 Mbar^2 "
                   "identically, consistent with v36's f0; [C] measure is TFPT input (consistency, "
                   "not derivation); [O] unchanged")


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

cef5988c2693f66333cdf81d1d5e7a44f22aad8f6c68ee7ddd3b5bb7df80cf88

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.