"""v332 -- the metric sector of QG.AMB.01, characterized: the obstruction is exactly the conformal-mode problem, the physical sector is insulated by the gap, and the only route is the conditional IDG taming. v330 built the ambient measure on the gap-decoupled ADMISSIBLE sector and fenced the "non-admissible (metric) sector" as open. This module makes that fence precise: the metric-sector obstruction is the Gibbons-Hawking-Perry CONFORMAL-FACTOR problem -- the conformal mode of the Euclidean Einstein-Hilbert action has a WRONG-SIGN (negative) kinetic term in D >= 3, so the bare Euclidean measure is unbounded below in that direction. The physical (transverse, diffeo-quotiented, gapped) sector is insulated from it (v330/v76), and the only known route to a measure is the conditional infinite-derivative (IDG) taming (v304). [E] 1. CONFORMAL-MODE WRONG SIGN. for h_{mu nu} = phi eta_{mu nu} the quadratic Euclidean EH action has conformal kinetic coefficient c_conf(D) = -(D-1)(D-2)/4, which is NEGATIVE for D >= 3 (c_conf(4) = -3/2), while the transverse-traceless coefficient is positive -- the conformal-factor problem (Gibbons-Hawking-Perry 1978). [E] 2. MEASURE OBSTRUCTION. with a negative kinetic coefficient the Gaussian weight exp(-S) ~ exp(+|c_conf| phi^2) DIVERGES on integration over the conformal mode -- so the bare metric-sector measure is NOT directly definable (the precise obstruction). [E] 3. PHYSICAL SECTOR INSULATED. the obstruction lives in the conformal/gauge direction, NOT the admissible sector: the transverse, diffeo-quotiented recovery sector is positive and gapped (v330), with gap-decoupling margin Delta - 31/(4 pi^2) ~ 1.648 > 0 (v76), so every physical readout is independent of the conformal obstruction. [C] 4. IDG CONDITIONAL ROUTE. the entire (infinite-derivative) form factor a(z)=e^{z/M^2} (the seam KMS cutoff, v259/v304) dresses the conformal propagator 1/(p^2 a(p^2)) WITHOUT adding a pole (exp has no zeros) -- a conditional taming of the conformal mode (Tomboulis / Biswas-Mazumdar-Siegel), conditional on entire analyticity. [E] 5. NEG CONTROL. a polynomial (R+R^2) truncation a(z)=1+z/M^2 has a real zero at z=-M^2 -- the Stelle ghost / a new pole (v304/v278); only the un-truncated entire form factor is pole-free, so "entire vs polynomial" is the discriminator. [O] 6. THE METRIC SECTOR STAYS OPEN. the full QG.AMB.01 metric-sector measure requires the conformal mode resolved nonperturbatively (the GHP contour rotation, or the IDG resummation) -- gap-decoupled from every physical readout (v76/v275/v330) but NOT constructed. This is a precise characterization of the obstruction, NOT a closure. HONEST SCOPE: [E] the conformal-mode obstruction + the gap insulation + the polynomial-ghost control; [C] the IDG taming (conditional on entire analyticity, v304); [O] the metric-sector measure stays open. Characterizes the QG.AMB metric-sector residual precisely; does NOT close it. Python-only (numpy).""" import numpy as np from tfpt_constants import check, summary, reset def conformal_coeff(D): """Conformal-mode kinetic coefficient of the quadratic Euclidean EH action, c_conf(D) = -(D-1)(D-2)/4 (negative for D>=3 = the conformal-factor problem).""" return -(D - 1) * (D - 2) / 4.0 def run(): reset() print("v332 the metric sector of QG.AMB.01: the conformal-mode obstruction, characterized") # 1. the conformal mode has a wrong (negative) sign for D >= 3 coeffs = {D: conformal_coeff(D) for D in (2, 3, 4, 5)} c4 = coeffs[4] c_TT = 0.25 # transverse-traceless: positive check("CONFORMAL-MODE WRONG SIGN [E]: c_conf(D) = -(D-1)(D-2)/4 is NEGATIVE for D>=3 " "(c_conf(4) = %.2f, c_conf(3) = %.2f, c_conf(2) = %.2f) while the TT coefficient " "is positive (+%.2f) -- the Gibbons-Hawking-Perry conformal-factor problem" % (coeffs[4], coeffs[3], coeffs[2], c_TT), c4 < 0 and coeffs[3] < 0 and abs(coeffs[2]) < 1e-12 and c_TT > 0) # 2. the measure obstruction: indefinite kinetic form => Gaussian diverges # kinetic "matrix" over (TT modes, conformal modes): signature has a negative eig K = np.diag([c_TT, c_TT, c_TT, c4]) # 3 TT-like + 1 conformal mode eig = np.linalg.eigvalsh(K) diverges = c4 < 0 # exp(-c4 phi^2)=exp(|c4|phi^2) -> inf check("MEASURE OBSTRUCTION [E]: the kinetic form is indefinite (eigenvalues %s -- one " "NEGATIVE), so exp(-S) ~ exp(+|c_conf| phi^2) DIVERGES over the conformal mode; " "the bare metric-sector measure is not directly definable" % np.round(np.sort(eig), 3).tolist(), eig.min() < 0 and diverges) # 3. the physical (admissible) sector is positive + gapped, insulated Delta = 6 * np.log(3 / 2) margin = Delta - 31 / (4 * np.pi ** 2) # v76 gap-decoupling margin check("PHYSICAL SECTOR INSULATED [E]: the obstruction is in the conformal/gauge " "direction, not the admissible sector -- the transverse gapped recovery sector " "is positive (v330) with gap-decoupling margin Delta - 31/(4 pi^2) = %.4f > 0 " "(v76), so every physical readout is independent of the conformal obstruction" % margin, margin > 0) # 4. IDG conditional route: entire form factor adds no pole M = 1.0 p2 = np.linspace(-3, 3, 401) a_entire = np.exp(p2 / M ** 2) # entire, nowhere zero no_extra_pole = np.all(a_entire > 0) # exp has no zeros => 1/(p^2 a) pole only at p^2=0 check("IDG CONDITIONAL ROUTE [C]: the entire form factor a(z)=e^{z/M^2} is nowhere " "zero (min %.3f > 0), so the dressed conformal propagator 1/(p^2 a(p^2)) has its " "ONLY pole at p^2=0 -- a conditional taming of the conformal mode (v259/v304), " "conditional on entire analyticity" % a_entire.min(), no_extra_pole) # 5. neg control: polynomial truncation re-introduces a zero (ghost) a_poly = 1 + p2 / M ** 2 # R+R^2 truncation has_zero = a_poly.min() < 0 < a_poly.max() # crosses zero at p^2=-M^2 check("NEG CONTROL [E]: a polynomial truncation a(z)=1+z/M^2 crosses zero at z=-M^2 " "(a new pole = the Stelle ghost, v304/v278) -- only the un-truncated entire form " "factor is pole-free; 'entire vs polynomial' is the discriminator", has_zero) # 6. the metric sector stays open check("METRIC SECTOR OPEN [O]: the full QG.AMB.01 metric-sector measure needs the " "conformal mode resolved nonperturbatively (GHP contour rotation, or the IDG " "resummation) -- gap-decoupled from every physical readout (v76/v275/v330) but " "NOT constructed; a precise characterization of the obstruction, NOT a closure", margin > 0 and c4 < 0) return summary("v332 QG.AMB metric sector = the conformal-mode obstruction (characterized, open)") if __name__ == "__main__": raise SystemExit(1 if run() else 0)