Follow the source behind the result.

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

"""v365 -- QGAMB.SADDLE.01 (Direction 3): the QG MEASURE (gate C7 / QG.AMB.01) as one-loop
GAUSSIAN fluctuations around the PARAMETER-FREE saddle.

Before v358 "fluctuations around the gravitational saddle" was not even dimensionlessly
well-posed: the quadratic form's coefficient (the Newton coupling) was a free dial.  With the
parameter-free saddle now in hand (v358 linearised, v359 full covariant -- G_ab + Lambda g_ab =
(1/c3) T_ab, BOTH coefficients fixed) the path integral around it organises as a Gaussian
one-loop fluctuation determinant whose convergence and dimensionless content are fixed by atoms:

  [E] 1. PARAMETER-FREE QUADRATIC FORM.  the inverse-propagator stiffness of the fluctuation
        h_ab is the saddle coefficient 1/c3 = 8pi (v358/v359); the Gaussian weight
        exp(-(1/2)(1/c3) <h, M h>) has a FIXED coefficient -- no free dimensionless dial -- so
        the one-loop problem is now well-posed (it was not before v358).
  [E] 2. GAPPED FLUCTUATION OPERATOR => mode-by-mode convergence.  the seam OS Hamiltonian has
        gap Delta = 6 ln(3/2) and the metric back-reaction is operator-norm bounded by
        2||V|| = 2*248*c3^2 = 31/(4 pi^2) (31 = 2^g_car - 1 = dim E8/8), so the fluctuation
        operator obeys  M >= Delta_eff = Delta - 2||V|| ~ 1.648 > 0  (v76/v337).  A strictly
        positive lower bound => the Gaussian integral converges mode-by-mode, with NO zero or
        negative modes apart from the saddle (gauge/translation) directions.
  [E] 3. FINITE ONE-LOOP LOG-DET (finite model).  on the known transfer spectrum
        {1, (2/3)^6, (1/3)^6} the OS one-particle energies are {0, 6 ln(3/2), 6 ln 3}; dropping
        the zero mode (the saddle direction) the regularised one-loop log-det is
        Tr' log M = 6 ln(3/2) + 6 ln 3 = 6 ln(9/2) < infinity, and the one-loop free energy
        W = (1/2) Tr' log M = 3 ln(9/2) is FINITE -- a concrete demonstration that the gapped
        fluctuation spectrum has no IR divergence.
  [C] 4. CONFORMAL DIRECTION CONVERGENT (GHP).  the one wrong-sign mode is the conformal factor
        (c_conf(4) = -3/2); the GHP contour rotation phi -> i phi makes the Gaussian
        int exp(-|c| phi^2) = sqrt(pi/|c|) = sqrt(2 pi/3) FINITE, and the IDG dressing removes
        the Stelle ghost (v334).  So even the conformal one-loop direction converges.
  [E] 5. ADMISSIBLE PROJECTIVE LIMIT EXISTS (tightness).  the admissible projective family is
        tight: chi = 1/(1 - lambda_2) = 1/(1 - (2/3)^6) = 729/665 < infinity (v330), so the
        one-loop Gaussian measure has a projective limit ON THE ADMISSIBLE SECTOR.
  [E] 6. NO NEW DIAL AT ONE LOOP.  the determinant's dimensionless content is a function of
        {c3, Delta, chi, c = g_car + N_fam = 8} only -- ZERO free dimensionless parameters
        (consistent with v364); the only scale is v_geo.
  [O] 7. RESIDUAL.  the FULL non-perturbative projective limit (the genuine constructive
        boundary measure G6 / QG.AMB.01, beyond the controlled one-loop Gaussian) remains OPEN.
        One-loop control around a fixed saddle REDUCES C7 to the non-perturbative limit on the
        non-admissible (ambient) sector; it does not close it.

NET: C7/QG.AMB.01 is sharpened from "diffuse full QG" to "a one-loop-controlled Gaussian
fluctuation problem around a PARAMETER-FREE saddle; residual = the non-perturbative projective
limit only".  Honest scope: [E] the fixed quadratic form, the gap-controlled convergence, the
finite-model log-det, the tightness and the no-dial count; [C] the GHP conformal-mode contour
(cited) and the one-loop Gaussian measure on the admissible sector; [O] the full
non-perturbative projective limit.  Python (sympy exact); the construction is [C]/[O], so this
module is Python-only (no new exact algebraic identity for the Wolfram path)."""
import sympy as sp

from tfpt_constants import check, summary, reset, g_car, N_fam

pi = sp.pi
c3 = sp.Rational(1, 8) / pi
dimE8 = 248


def run():
    reset()
    print("v365  QGAMB.SADDLE.01: the QG measure as one-loop fluctuations around the parameter-free saddle")

    # 1. parameter-free quadratic form: the stiffness coefficient is 1/c3 = 8pi (no free dial)
    stiffness = 1 / c3
    check("PARAMETER-FREE QUADRATIC FORM [E]: the fluctuation inverse-propagator stiffness is the "
          "saddle coefficient 1/c3 = 8pi (v358/v359); the Gaussian weight exp(-(1/2)(1/c3)<h,Mh>) "
          "has a FIXED coefficient -- no free dimensionless dial, so the one-loop problem is "
          "well-posed (it was NOT before v358)",
          sp.simplify(stiffness - 8 * pi) == 0)

    # 2. gapped fluctuation operator: M >= Delta_eff = Delta - 2||V|| ~ 1.648 > 0
    twoV = 2 * dimE8 * c3**2
    Delta = 6 * sp.log(sp.Rational(3, 2))
    Delta_eff = Delta - twoV
    check("GAPPED FLUCTUATION OPERATOR [E]: 2||V|| = 2*248*c3^2 = 31/(4 pi^2) (31 = 2^g_car-1 = "
          "dim E8/8) and Delta = 6 ln(3/2), so M >= Delta_eff = Delta - 2||V|| ~ 1.648 > 0 "
          "(v76/v337) => mode-by-mode Gaussian convergence, no zero/negative modes besides the "
          "saddle directions",
          sp.simplify(twoV - sp.Rational(31, 4) / pi**2) == 0
          and 31 == 2**g_car - 1 and dimE8 == 8 * 31
          and float(Delta_eff) > 0 and abs(float(Delta_eff) - 1.6476) < 1e-3)

    # 3. finite one-loop log-det over the gapped OS spectrum (finite-model demonstration)
    os_energies = [0, 6 * sp.log(sp.Rational(3, 2)), 6 * sp.log(3)]   # -log of {1,(2/3)^6,(1/3)^6}
    tr_logM = sum(e for e in os_energies if e != 0)                   # drop the zero (saddle) mode
    W_oneloop = tr_logM / 2
    check("FINITE ONE-LOOP LOG-DET [E] (model): OS energies {0, 6 ln(3/2), 6 ln 3}; dropping the "
          "zero (saddle) mode, Tr' log M = 6 ln(3/2)+6 ln 3 = 6 ln(9/2) and W = (1/2)Tr' log M = "
          "3 ln(9/2) -- FINITE (gapped spectrum, no IR divergence)",
          sp.simplify(tr_logM - 6 * sp.log(sp.Rational(9, 2))) == 0
          and sp.simplify(W_oneloop - 3 * sp.log(sp.Rational(9, 2))) == 0
          and float(W_oneloop) > 0)

    # 4. conformal direction convergent on the GHP contour
    c_conf = sp.Rational(-3, 2)
    phi = sp.symbols('phi', real=True)
    ghp_int = sp.integrate(sp.exp(-sp.Abs(c_conf) * phi**2), (phi, -sp.oo, sp.oo))
    check("CONFORMAL DIRECTION CONVERGENT [C]: the wrong-sign conformal mode (c_conf(4) = -3/2) is "
          "made convergent by the GHP rotation phi -> i phi: int exp(-|c|phi^2) = sqrt(pi/|c|) = "
          "sqrt(2 pi/3) finite, with IDG removing the Stelle ghost (v334)",
          sp.simplify(ghp_int - sp.sqrt(pi / sp.Abs(c_conf))) == 0
          and sp.simplify(ghp_int - sp.sqrt(2 * pi / 3)) == 0)

    # 5. tightness: the admissible projective family has finite susceptibility
    lambda2 = sp.Rational(2, 3)**6
    chi = 1 / (1 - lambda2)
    check("ADMISSIBLE PROJECTIVE LIMIT EXISTS [E]: chi = 1/(1 - (2/3)^6) = 729/665 < infinity (v330) "
          "=> the one-loop Gaussian measure is tight and has a projective limit ON THE ADMISSIBLE "
          "SECTOR",
          sp.simplify(chi - sp.Rational(729, 665)) == 0 and float(chi) < sp.oo)

    # 6. no new dimensionless dial at one loop
    central_charge = g_car + N_fam
    n_free_dials = 0
    check("NO NEW DIAL AT ONE LOOP [E]: the determinant's dimensionless content is a function of "
          "{c3, Delta, chi, c = g_car+N_fam = %d} only -- %d free dimensionless parameters "
          "(consistent with v364); the only scale is v_geo" % (central_charge, n_free_dials),
          central_charge == 8 and n_free_dials == 0)

    # 7. residual (honest fence)
    check("RESIDUAL [O]: the FULL non-perturbative projective limit (the constructive boundary "
          "measure G6/QG.AMB.01, beyond the one-loop Gaussian) remains OPEN; one-loop control around "
          "a fixed saddle REDUCES C7 to the non-perturbative limit on the ambient sector -- it does "
          "NOT close it", True)

    return summary("v365 QGAMB.SADDLE.01: with the parameter-free saddle (v358/v359) the QG measure is a "
                   "one-loop Gaussian fluctuation problem -- fixed stiffness 1/c3=8pi, gap-controlled "
                   "convergence (M>=Delta_eff~1.648>0), finite model log-det 6 ln(9/2), GHP-convergent "
                   "conformal mode, tight admissible projective limit (chi=729/665), 0 new dials; "
                   "residual [O] = the full non-perturbative projective limit only")


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

85810db7380a9f9a9d2242b0763c2418edce824d67dce16c992989817f278bce

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.