"""v494 -- COSMO.RUNNING.01 / COSMO.MUDIST.01: the two 2026-07-20 prediction-of-record rows (spectral-index running alpha_s/beta_s and the CMB mu-distortion branch discriminator) promoted to machine-checked verification. The Starobinsky branch is already load-bearing (v7/v28/v86: n_s = 1 - 2/N_star, r = 12/N_star^2, A_s = N_star^2 c3^7/(24 pi^2)). This module adds the SAME branch's next two observables so the paper/website prediction surfaces are machine-backed: [E] LEADING-ORDER IDENTITIES (sympy exact, no floats): alpha_s = dn_s/dln k = -2/N_star^2 = -r/6 (LO) beta_s = d alpha_s/dln k = -4/N_star^3 (LO; NOT -8/N^3) using d ln k = -dN at leading order. [E] EXACT SLOW ROLL on the Einstein-frame Starobinsky potential V = V0 (1 - e^{-sqrt(2/3) phi})^2 (third-order SR, Lyth-Riotto), cross-checked by a finite difference along ln k: alpha_s(N=51.4) = -7.09e-4, beta_s(51.4) = -2.70e-5; exact-SR band edges alpha_s(50) = -7.48e-4, alpha_s(60) = -5.23e-4; LO band over N_star in [50,60]: [-8.0e-4, -5.6e-4]. [C] DATA CONFRONTATION (external data, no marker upgrade): Planck 2018 TTTEEE+lowE+lensing alpha_s = -0.0045 +/- 0.0067 -> +0.57 sigma; P-ACT-LB (ACT DR6+Planck+lensing+DESI BAO) +0.0062 +/- 0.0052 -> -1.33 sigma; beta_s (Planck run+runrun) 0.010 +/- 0.013 -> -0.77 sigma. KILL: robust alpha_s < -5e-3 or > +3e-3 at >= 5 sigma; any robust POSITIVE running kills the plateau branch (plateau potentials give alpha_s < 0). [C] MU-DISTORTION BRANCH DISCRIMINATOR (Chluba-Erickcek-Ben-Dayan k-space window, calibrated on the LCDM Planck-2018 anchor to the literature mu_LCDM ~ 2.0e-8, a +21% window correction): sharp branch N_star = 51.4 (A_s = 1.76e-9): mu ~ 1.6e-8; profiled record N_star = 56.1 (A_s = 2.10e-9): mu ~ 2.0e-8 (1.94e-8); band N_star in [50,60]: 1.5e-8 .. 2.3e-8; branch split Delta mu = 3.0e-9 (raw window; calibrated 3.7e-9) => >= 3 sigma A_s/N_star branch decision at sigma(mu) = 1e-9 (Voyage-2050 class); the PIXIE baseline limit 9e-8 does NOT reach the band (sharp = 0.18 x limit); FIRAS 9e-5 is 4 dex above. KILL: robust mu < 0.9e-8 or > 4e-8. [E] COMB BLINDNESS IN THE MU WINDOW (honest S15 typing): the frozen recovery log-comb (omega = 2 pi/ln((3/2)^6) = 2.5827, eps = e^{-pi^2/ln((3/2)^6)} = 0.0173) is STRUCTURALLY invisible here -- window attenuation |F(omega)| = 0.033, delta mu = 7.4e-12 = 0.007 sigma even at Voyage-2050 sensitivity. The theory itself predicts where it cannot be seen; no comb-search row is opened. HONEST FENCES (do not oversell): * N_star REMAINS an external reheating input (GATE.NSTAR.01); the frozen band [50, 60] stays the prediction surface of record (v84/v86); the sharp point 51.4 is the conditional [P] reheating point. * All data confrontations are [C] external-data rows (scorecard stage prediction_of_record); nothing here closes a gate or moves a marker. * The Chluba window is imported standard physics; the +21% calibration is anchored on the LCDM literature value, not fitted to TFPT. Origin: experiments/tfpt-discovery/alpha_s_running_probe.py and mu_distortion_probe.py (2026-07-20 scorecard round), promoted 2026-07-21. Python: sympy exact (LO identities) + numpy (SR solve, window quadrature); the quadrature/ODE parts are numerical, Python-only by the suite convention. """ import numpy as np import sympy as sp import mpmath as mp from tfpt_constants import check, summary, reset, c3, PI SQ23 = np.sqrt(2.0 / 3.0) K_PIVOT = 0.05 # 1/Mpc N_SHARP = 51.4 # v86 reheating point N_PROF = 56.1 # A_s-profiled record branch LAMBDA_DSI = (3.0 / 2.0) ** 6 OMEGA = 2.0 * np.pi / np.log(LAMBDA_DSI) # frozen comb kernel EPS_COMB = np.exp(-np.pi ** 2 / np.log(LAMBDA_DSI)) MU_LCDM_LIT = 2.0e-8 # literature LCDM mu (Chluba) # ---------- Starobinsky potential + third-order slow roll (M_pl = 1) -------- def _V(phi): return (1.0 - np.exp(-SQ23 * phi)) ** 2 def _dV(phi): e = np.exp(-SQ23 * phi) return 2.0 * SQ23 * e * (1.0 - e) def _d2V(phi): e = np.exp(-SQ23 * phi) return (4.0 / 3.0) * e * (2.0 * e - 1.0) def _d3V(phi): e = np.exp(-SQ23 * phi) return (4.0 / 3.0) * SQ23 * e * (1.0 - 4.0 * e) def _d4V(phi): e = np.exp(-SQ23 * phi) return (8.0 / 9.0) * e * (8.0 * e - 1.0) def _sr_params(phi): v, v1, v2, v3, v4 = _V(phi), _dV(phi), _d2V(phi), _d3V(phi), _d4V(phi) eps = 0.5 * (v1 / v) ** 2 eta = v2 / v xi2 = v1 * v3 / v ** 2 sig3 = v1 ** 2 * v4 / v ** 3 return eps, eta, xi2, sig3 def _observables(phi): """Third-order slow-roll n_s, r, alpha_s, beta_s (Lyth-Riotto).""" eps, eta, xi2, sig3 = _sr_params(phi) ns = 1.0 - 6.0 * eps + 2.0 * eta r = 16.0 * eps alpha = 16.0 * eps * eta - 24.0 * eps ** 2 - 2.0 * xi2 beta = (-192.0 * eps ** 3 + 192.0 * eps ** 2 * eta - 32.0 * eps * eta ** 2 - 24.0 * eps * xi2 + 2.0 * eta * xi2 + 2.0 * sig3) return ns, r, alpha, beta def _phi_end(): lo, hi = 0.1, 2.0 for _ in range(200): mid = 0.5 * (lo + hi) if _sr_params(mid)[0] > 1.0: lo = mid else: hi = mid return 0.5 * (lo + hi) def _efolds(phi, phi_e): xs = np.linspace(phi_e, phi, 20001) return np.trapezoid(_V(xs) / _dV(xs), xs) def _phi_of_N(n_target, phi_e): lo, hi = phi_e, 12.0 for _ in range(200): mid = 0.5 * (lo + hi) if _efolds(mid, phi_e) < n_target: lo = mid else: hi = mid return 0.5 * (lo + hi) # ---------- Chluba k-space window for mu (khat = k * Mpc) -------------------- _LNK = np.linspace(np.log(1.0), np.log(1.0e6), 200001) _KHAT = np.exp(_LNK) _W_MU = np.exp(-_KHAT / 5400.0) - np.exp(-((_KHAT / 31.6) ** 2)) def _pzeta(a_s, ns, alpha): lnkr = _LNK - np.log(K_PIVOT) return a_s * np.exp((ns - 1.0) * lnkr + 0.5 * alpha * lnkr ** 2) def _mu_window(a_s, ns, alpha): return 2.2 * np.trapezoid(_pzeta(a_s, ns, alpha) * _W_MU, _LNK) def _As_tfpt(n): return float(n ** 2 * c3 ** 7 / (24 * PI ** 2)) def run(): reset() print("v494 cosmo running + mu-distortion record " "(COSMO.RUNNING.01 / COSMO.MUDIST.01)") # ---- [E] leading-order identities, sympy exact -------------------------- N = sp.symbols('N', positive=True) ns_lo = 1 - 2 / N r_lo = 12 / N ** 2 alpha_lo = -sp.diff(ns_lo, N) # d ln k = -dN at LO beta_lo = -sp.diff(alpha_lo, N) check("[E] LO identity alpha_s = dn_s/dlnk = -2/N^2 (sympy exact)", sp.simplify(alpha_lo + 2 / N ** 2) == 0) check("[E] LO identity alpha_s = -r/6 (sympy exact)", sp.simplify(alpha_lo + r_lo / 6) == 0) check("[E] LO identity beta_s = -4/N^3 (NOT -8/N^3; sympy exact)", sp.simplify(beta_lo + 4 / N ** 3) == 0) check("[E] LO band edges: alpha_s(50) = -8.00e-4, alpha_s(60) = -5.56e-4", sp.Rational(-2, 50 ** 2) == sp.Rational(-1, 1250) and abs(float(-2 / mp.mpf(60) ** 2) + 5.556e-4) < 1e-7) # ---- [E] exact slow roll on the Starobinsky potential -------------------- phi_e = _phi_end() obs = {} for n_star in (50.0, N_SHARP, 60.0): obs[n_star] = _observables(_phi_of_N(n_star, phi_e)) a_sharp, b_sharp = obs[N_SHARP][2], obs[N_SHARP][3] check("[E] exact-SR alpha_s(N=51.4) = -7.088e-4", a_sharp, mp.mpf('-7.0884e-4'), tol=mp.mpf('1e-3')) check("[E] exact-SR beta_s(N=51.4) = -2.702e-5", b_sharp, mp.mpf('-2.7020e-5'), tol=mp.mpf('1e-3')) check("[E] exact-SR band edges alpha_s(50) = -7.482e-4, " "alpha_s(60) = -5.234e-4", abs(obs[50.0][2] / -7.4824e-4 - 1) < 1e-3 and abs(obs[60.0][2] / -5.2337e-4 - 1) < 1e-3) # finite-difference cross-check along ln k = -N + (1/2) ln(V/3) pts = [] for n_star in (N_SHARP - 0.5, N_SHARP + 0.5): phi = _phi_of_N(n_star, phi_e) pts.append((-n_star + 0.5 * np.log(_V(phi) / 3.0), _observables(phi)[0])) alpha_fd = (pts[1][1] - pts[0][1]) / (pts[1][0] - pts[0][0]) check("[E] finite-difference dn_s/dlnk = exact-SR alpha_s to < 1%", abs(alpha_fd / a_sharp - 1) < 1e-2) # ---- [C] data confrontation (external data; no marker moves) ------------ check("[C] Planck 2018 (-0.0045 +/- 0.0067): TFPT pull = +0.57 sigma", (a_sharp - (-0.0045)) / 0.0067, mp.mpf('0.57'), tol=mp.mpf('2e-2')) check("[C] P-ACT-LB (+0.0062 +/- 0.0052): TFPT pull = -1.33 sigma " "(record leg; watch channel)", (a_sharp - 0.0062) / 0.0052, mp.mpf('-1.33'), tol=mp.mpf('2e-2')) check("[C] beta_s vs Planck run+runrun (0.010 +/- 0.013) = -0.77 sigma", (b_sharp - 0.010) / 0.013, mp.mpf('-0.77'), tol=mp.mpf('2e-2')) check("[C] kill margin: LO band edge -8.0e-4 sits 4.2e-3 INSIDE the " "negative kill edge -5e-3; any robust POSITIVE running kills the " "plateau branch", obs[50.0][2] > -5e-3 and a_sharp < 0) # ---- [C] mu-distortion branch discriminator ------------------------------ mu_lcdm = _mu_window(2.10e-9, 0.9649, 0.0) cal = MU_LCDM_LIT / mu_lcdm check("[C] window calibration on the LCDM anchor: mu_win(LCDM) = 1.65e-8 " "-> cal = +21% (the scorecard's '+20% to the full computation')", cal, mp.mpf('1.209'), tol=mp.mpf('2e-2')) mu_sharp_raw = _mu_window(_As_tfpt(N_SHARP), 1 - 2 / N_SHARP, -2 / N_SHARP ** 2) mu_prof_raw = _mu_window(2.10e-9, 1 - 2 / N_PROF, -2 / N_PROF ** 2) mu_50 = cal * _mu_window(_As_tfpt(50.0), 1 - 2 / 50.0, -2 / 50.0 ** 2) mu_60 = cal * _mu_window(_As_tfpt(60.0), 1 - 2 / 60.0, -2 / 60.0 ** 2) check("[C] sharp branch (N=51.4, A_s=1.76e-9): mu = 1.6e-8 (calibrated)", cal * mu_sharp_raw, mp.mpf('1.6e-8'), tol=mp.mpf('2e-2')) check("[C] profiled record (N=56.1, A_s=2.10e-9): mu = 1.94e-8 (~2.0e-8)", cal * mu_prof_raw, mp.mpf('1.94e-8'), tol=mp.mpf('2e-2')) check("[C] band N in [50,60]: mu in [1.5e-8, 2.3e-8]", abs(mu_50 / 1.5e-8 - 1) < 2e-2 and abs(mu_60 / 2.3e-8 - 1) < 2e-2) check("[C] branch split Delta mu = 3.0e-9 (raw window; calibrated 3.7e-9) " "=> >= 3 sigma at sigma(mu) = 1e-9 (Voyage-2050 class)", mu_prof_raw - mu_sharp_raw, mp.mpf('3.03e-9'), tol=mp.mpf('2e-2')) check("[C] branch ratio mu_profiled/mu_sharp = 1.23 (the 23% split)", mu_prof_raw / mu_sharp_raw, mp.mpf('1.232'), tol=mp.mpf('1e-2')) check("[C] PIXIE baseline |mu| < 9e-8 does NOT reach the band " "(sharp = 0.18 x limit); FIRAS 9e-5 is 4 dex above", cal * mu_sharp_raw / 9e-8 < 0.25 and cal * mu_sharp_raw / 9e-5 < 1e-3) # ---- [E] comb blindness in the mu window (S15 honest typing) ------------- check("[E] frozen comb kernel omega = 2 pi/ln((3/2)^6) = 2.5827, " "eps = 0.0173", abs(OMEGA - 2.5827) < 1e-4 and abs(EPS_COMB - 0.017302) < 1e-5) lnkr = _LNK - np.log(K_PIVOT) p0w = _pzeta(_As_tfpt(N_SHARP), 1 - 2 / N_SHARP, -2 / N_SHARP ** 2) * _W_MU base = np.trapezoid(p0w, _LNK) f_att = np.hypot(np.trapezoid(p0w * np.cos(OMEGA * lnkr), _LNK), np.trapezoid(p0w * np.sin(OMEGA * lnkr), _LNK)) / base dmu_comb = EPS_COMB * f_att * mu_sharp_raw check("[E] window attenuation |F(2.5827)| = 0.033 (vs 0.70 at omega=0.5)", f_att, mp.mpf('0.033'), tol=mp.mpf('3e-2')) check("[E] comb blindness: delta mu = 7.4e-12 = 0.007 sigma even at " "Voyage-2050 (structurally invisible; no comb-search row opened)", dmu_comb, mp.mpf('7.4e-12'), tol=mp.mpf('2e-2')) # ---- fences --------------------------------------------------------------- check("[O] fence: N_star stays an external reheating input " "(GATE.NSTAR.01); the frozen band [50,60] stays the surface of " "record; the sharp point is the conditional v86 [P] reading", 50.0 < N_SHARP < 60.0 and 50.0 < N_PROF < 60.0) return summary("v494 cosmo running + mu record") if __name__ == "__main__": raise SystemExit(1 if run() else 0)