"""v334 -- the conformal-mode resolution: the Gibbons-Hawking-Perry contour rotation (IR sign) + the infinite-derivative (IDG) dressing (UV pole) give a CONDITIONALLY convergent metric-sector measure -- the residual is the nonperturbative validity of the contour. v332 pinned the QG.AMB metric-sector obstruction to ONE thing: the conformal mode has a negative kinetic coefficient c_conf(D) = -(D-1)(D-2)/4, so the bare Euclidean Gaussian diverges over it. This module gives the conditional resolution -- the two standard ingredients -- and states precisely what stays open. [E] 1. THE OBSTRUCTION. c_conf(4) = -3/2 < 0, so the unrotated conformal Gaussian integral(exp(+|c| phi^2)) DIVERGES (grows without bound with the cutoff). [E] 2. GHP CONTOUR ROTATION (IR sign). rotating phi -> i phi flips (i phi)^2 = -phi^2, so +|c| phi^2 -> -|c| phi^2 and integral(exp(-|c| phi^2)) = sqrt(pi/|c|) is FINITE -- the Gibbons-Hawking-Perry 1978 prescription makes the conformal measure convergent. [E] 3. IDG UV DRESSING (no pole). the entire form factor a(p^2) = exp(p^2/M^2) is nowhere zero (a > 0), so the dressed propagator 1/(p^2 a) has its only pole at p^2 = 0 (no ghost, v304); the per-mode action (1/2)|c| k^2 a(k^2) phi^2 is positive after the GHP rotation and UV-suppressed (a grows, high modes freeze). [E] 4. NEG CONTROL. 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 entire (un-truncated) form factor is pole-free. [C] 5. CONDITIONAL CONSTRUCTION. GHP (the IR sign) + IDG (the UV pole) give a formally convergent metric-sector Gaussian mode-by-mode -- a CONDITIONAL construction of the conformal-sector measure (conditional on the rotation contour AND entire analyticity, v304). [O] 6. THE RESIDUAL. whether the GHP-rotated, IDG-dressed measure is the CORRECT nonperturbative QG.AMB measure -- the nonperturbative validity of the contour rotation -- is Euclidean QG's long-standing open issue, and stays open here. It is gap-decoupled from every physical readout (Delta - 31/(4 pi^2) ~ 1.648 > 0, v76), so it blocks no test. NOT a closure. HONEST SCOPE: [E] the obstruction + the GHP sign flip + the IDG pole-freeness + the polynomial-ghost control; [C] the conditional GHP+IDG construction; [O] the contour's nonperturbative validity. Advances v332 to a conditional construction; does NOT close QG.AMB.01. Python-only (numpy + scipy).""" import numpy as np from scipy.integrate import quad from tfpt_constants import check, summary, reset def conformal_coeff(D): return -(D - 1) * (D - 2) / 4.0 def run(): reset() mid = "the conformal-mode resolution: GHP contour rotation + IDG dressing" print("v334 " + mid) c = conformal_coeff(4) # -3/2 absc = abs(c) # 1. the obstruction: the unrotated conformal Gaussian diverges def unrotated(R): val, _ = quad(lambda x: np.exp(absc * x ** 2), -R, R) return val growth = [unrotated(R) for R in (2.0, 3.0, 4.0)] diverges = growth[2] > 100 * growth[0] # grows super-fast with the cutoff check("THE OBSTRUCTION [E]: c_conf(4) = %.1f < 0, so the unrotated conformal Gaussian " "integral(exp(+|c| phi^2)) DIVERGES (R=2,3,4 give %.1e, %.1e, %.1e -- unbounded " "growth)" % (c, growth[0], growth[1], growth[2]), c < 0 and diverges) # 2. GHP contour rotation: phi -> i phi flips the sign, integral converges rotated, _ = quad(lambda x: np.exp(-absc * x ** 2), -np.inf, np.inf) analytic = np.sqrt(np.pi / absc) check("GHP CONTOUR ROTATION [E]: phi -> i phi flips (i phi)^2 = -phi^2, so " "+|c| phi^2 -> -|c| phi^2 and integral(exp(-|c| phi^2)) = sqrt(pi/|c|) = %.5f " "is FINITE (numeric %.5f) -- the Gibbons-Hawking-Perry prescription makes the " "conformal measure convergent" % (analytic, rotated), abs(rotated - analytic) < 1e-6 and np.isfinite(rotated)) # 3. IDG UV dressing: entire form factor, positive, UV-suppressing M = 1.0 ks = np.linspace(0.0, 6.0, 601) a_entire = np.exp(ks ** 2 / M ** 2) per_mode_positive = np.all(absc * ks[1:] ** 2 * a_entire[1:] > 0) # positive after GHP uv_suppressed = a_entire[-1] > 100 * a_entire[len(a_entire) // 4] # high modes frozen check("IDG UV DRESSING [E]: the entire form factor a(p^2)=exp(p^2/M^2) is nowhere " "zero (min %.3f > 0, no ghost pole, v304); the dressed per-mode action " "(1/2)|c| k^2 a(k^2) phi^2 is positive after GHP and UV-suppressed (a grows by " "%.0fx across the band, high modes freeze)" % (a_entire.min(), a_entire[-1] / a_entire[len(a_entire) // 4]), per_mode_positive and uv_suppressed) # 4. neg control: polynomial truncation re-introduces a zero (ghost) p2 = np.linspace(-3, 3, 401) a_poly = 1 + p2 / M ** 2 has_zero = a_poly.min() < 0 < a_poly.max() 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 entire form factor is " "pole-free", has_zero) # 5. conditional construction: GHP + IDG give a convergent per-mode Gaussian def dressed_mode(k): coeff = absc * k ** 2 * np.exp(k ** 2 / M ** 2) # positive (post-GHP), dressed val, _ = quad(lambda x: np.exp(-0.5 * coeff * x ** 2), -np.inf, np.inf) return val finite_modes = all(np.isfinite(dressed_mode(k)) and dressed_mode(k) > 0 for k in (0.5, 1.0, 2.0, 3.0)) check("CONDITIONAL CONSTRUCTION [C]: GHP (IR sign) + IDG (UV pole) give a formally " "convergent metric-sector Gaussian mode-by-mode (the dressed per-mode integrals " "are finite and positive) -- a CONDITIONAL construction of the conformal-sector " "measure (conditional on the contour AND entire analyticity, v304)", finite_modes) # 6. the residual: nonperturbative validity of the contour margin = 6 * np.log(3 / 2) - 31 / (4 * np.pi ** 2) check("THE RESIDUAL [O]: whether the GHP-rotated, IDG-dressed measure is the CORRECT " "nonperturbative QG.AMB measure (the contour rotation's nonperturbative " "validity) is Euclidean QG's long-standing open issue; gap-decoupled from every " "physical readout (margin %.4f > 0, v76), so it blocks no test. NOT a closure" % margin, margin > 0) return summary("v334 conformal-mode resolution: GHP + IDG conditional construction (open contour)") if __name__ == "__main__": raise SystemExit(1 if run() else 0)