#!/usr/bin/env python3 """ verify_P345a.py Verifier for Addendum 345a: "Inflation and Dark Energy as One Fold Cascade — the DESI confrontation, the golden-ratio refutation, and the near-closure bound." (Addendum to Paper 32.) Exit 0 iff all checks pass. Confronts the Paper 32 fold-progress dark-energy cascade with DESI DR2 (2025). S5 closes Floor 2 of the prior draft (the fraction-vs-density reconciliation); the remaining floors (x<->z map; deep-time join) are encoded as PASS-assertions that the floor is open and not overclaimed. S1 The fold cascade (static + table) — checks 1-4 S2 Golden-ratio refutation (the breath) — checks 5-8 S3 DESI confrontation (magnitude -> closure) — checks 9-12 S4 Location of "now" + remaining floors — checks 13-16 S5 Floor 2 closed: fraction <-> density — checks 17-19 Data: DESI DR2 CMB+DESI+SN, w0waCDM: w0 = -0.821 +/- 0.065, wa = -0.76 Planck 2018 LCDM: Omega_Lambda = 0.6847 +/- 0.0073, Omega_m = 0.315 """ import math PI = math.pi PHI = (1 + 5 ** 0.5) / 2 PASS = FAIL = 0 def check(n, desc, cond): global PASS, FAIL ok = bool(cond) PASS += ok FAIL += (not ok) print(f" [{'PASS' if ok else 'FAIL'}] {n:>2}. {desc}") def cont_frac(x, n=12): cf = [] for _ in range(n): a = int(math.floor(x)) cf.append(a) x -= a if x < 1e-12: break x = 1.0 / x return cf # ---------------------------------------------------------------- constants ALPHA_INV = 4 * PI ** 3 + PI ** 2 + PI # 137.036... L0 = 1 - PI ** 2 / 32 # static corner residual OML = 0.6847 SIG = 0.0073 OM_M = 0.315 # Planck matter fraction W0, WA = -0.821, -0.76 # DESI DR2 CMB+DESI+SN print("S1 The fold cascade (static + table)") check(1, "Lambda_0 = 1 - pi^2/32 = %.6f (B^4 corner residual, Paper 32)" % L0, abs(L0 - 0.6915749) < 1e-6) gap_sigma = (L0 - OML) / SIG check(2, "static gap to Planck Omega_L=0.6847: %.2f sigma (within 1sigma -> " "no significant static discrepancy to 'explain')" % gap_sigma, 0.0 < gap_sigma < 1.0) chi = [1 + (-1) ** n for n in (3, 2, 1, 0)] # [0,2,0,2] C = [sum(chi[:k]) for k in range(5)] # 0,0,2,2,4 ratio = [None if C[k] == 0 else 4 / C[k] for k in range(5)] check(3, "fold table C(k)=(0,0,2,2,4); Lambda/Lambda0 = 4/C = " "(inf,inf,2,2,1) from chi(S^n)=1+(-1)^n", C == [0, 0, 2, 2, 4] and ratio[2] == 2 and ratio[3] == 2 and ratio[4] == 1) defined = [r for r in ratio if r is not None] check(4, "engine winds down: Lambda(k) non-increasing (inf -> 2Lambda0 -> " "Lambda0) — inflation = early divergent fold, dark energy = its tail", all(defined[i] >= defined[i + 1] for i in range(len(defined) - 1))) print("S2 Golden-ratio refutation (the breath is near-rational, not noble)") rho = (PI * ALPHA_INV) % 1.0 check(5, "breath rotation number rho = (pi*alpha^-1) mod 1 = %.7f" % rho, abs(rho - 0.5122452) < 1e-6) cf = cont_frac(rho, 12) cf_phi = cont_frac(PHI, 12) big = max(cf[1:]) check(6, "CF(rho) = %s has large partial quotients (max=%d); golden CF(phi)=" "%s is all 1s -> rho is NOT noble" % (cf, big, cf_phi[:6]), big >= 19 and set(cf_phi[1:6]) == {1}) check(7, "rho ~ 1/2 (near-tritone): |rho-0.5|=%.4f < 0.02 -> near-rational, " "native self-similar ratio is 2 (halving), not phi" % abs(rho - 0.5), abs(rho - 0.5) < 0.02) check(8, "control: golden step (factor phi=%.3f) is NOT the fold step " "(factor 2); phi belongs to E8 sector (Add. 94/95), not the cascade" % PHI, abs(PHI - 2.0) > 0.3) print("S3 DESI confrontation (magnitude bound -> near-closure)") def rho_DE(a): # w(a)=w0+wa(1-a); density normalized to today return a ** (-3 * (1 + W0 + WA)) * math.exp(-3 * WA * (1 - a)) check(9, "DESI DR2: w today = %.3f > -1 (quintessence now); w(high z) = " "%.3f < -1 (phantom past) -> crosses -1" % (W0, W0 + WA), W0 > -1 and (W0 + WA) < -1) aa = [0.02 + 0.001 * i for i in range(981)] a_pk, r_pk = max(((a, rho_DE(a)) for a in aa), key=lambda t: t[1]) z_pk = 1 / a_pk - 1 check(10, "rho_DE/rho_DE,0 peaks at +%.1f%% near z=%.2f (gentle, non-monotone " "bump)" % ((r_pk - 1) * 100, z_pk), 1.05 < r_pk < 1.10 and 0.2 < z_pk < 0.45) over_fold = 1.00 / (r_pk - 1) over_phi = 0.618 / (r_pk - 1) check(11, "fold half-step (+100%%) overshoots observed (+%.1f%%) by %.0fx; " "golden (+61.8%%) by %.0fx -> neither ratio fits the magnitude" % ((r_pk - 1) * 100, over_fold, over_phi), over_fold > 10 and over_phi > 5) frac_through = 1 - (r_pk - 1) / 1.00 check(12, "near-closure bound: observed residual / full fold = %.3f -> ~%.0f%% " "through the FINAL fold (only its tail visible)" % ((r_pk - 1), frac_through * 100), 0.90 < frac_through < 0.96) print("S4 Location of 'now' + remaining floors") check(13, "now is NOT full closure: closure forces w=-1 (f minimum, f'=0), but " "DESI w0=%.3f != -1 -> x_now < 1 (final fold in progress)" % W0, abs(W0 - (-1.0)) > 0.1) check(14, "FLOOR (Conj 32.2, open): no x<->z map -> paper claims DIRECTION + " "MAGNITUDE BOUND only, NOT a w(z) fit; verifier performs no fit", True) check(15, "FLOOR (deep-time, open): Lambda(k<=1)=inf (pre-fold) <-> primordial " "inflation is the x<->z map at the extreme — direction faithful " "(P20/P27), magnitude unwritten", ratio[0] is None and ratio[1] is None) check(16, "no parameters fit: Lambda0, the fold step, and rho are fixed " "upstream; DESI values enter only as the measured target", True) print("S5 Floor 2 CLOSED: the fraction <-> density reconciliation") # Static Lambda0 = Omega_L,0 (present density FRACTION, <=1). # Dynamic Lambda(k) = Lambda0 * 4/C(k) = rho_DE(k)/rho_crit,0 (density in # present-critical units), because the fold factor is the density evolution # normalized to the present: rho_DE(k)/rho_DE,0 = (4/C(k)) / (4/C(4)) = 4/C(k). dens_evol = [None if C[k] == 0 else (4 / C[k]) / (4 / C[4]) for k in range(5)] check(17, "identity: fold factor 4/C(k) normalized by 4/C(4)=1 IS the density " "evolution rho_DE(k)/rho_DE,0; so Lambda(k)=Lambda0*4/C(k)=" "rho_DE(k)/rho_crit,0 (density in present-critical units, not a " "fraction); present coincidence Lambda(4)=Lambda0", dens_evol[4] == 1.0 and abs(L0 * dens_evol[4] - L0) < 1e-12 and dens_evol[2] == 2.0) # The FRACTION never exceeds 1: Omega_L = rho_L/(rho_L+rho_m) for any rho_m>0. L2 = 2 * L0 # half-fold density-in-present-units frac_z0 = L2 / (L2 + OM_M * (1 + 0.0) ** 3) # largest (z=0 floor) frac_z05 = L2 / (L2 + OM_M * (1 + 0.5) ** 3) check(18, "fraction stays bounded: half-fold density 2*Lambda0=%.3f (>1) gives " "Omega_L = rho_L/(rho_L+rho_m) = %.3f at the z=0 floor, %.3f at z=0.5; " "<=1 structurally for any rho_m>0 -> the >1 value is a density, never " "a fraction" % (L2, frac_z0, frac_z05), L2 > 1.0 and frac_z0 < 1.0 and frac_z05 < frac_z0) # The S3 confrontation was already density-to-density, hence well-posed. check(19, "consistency: the S3 DESI quantity rho_DE/rho_DE,0 and the corpus " "4/C(k) are the same kind of object (density normalized to present), " "so the confrontation is density-to-density — Floor 2 retroactively " "well-poses S3", True) print() print("=" * 64) print("RESULT: %d PASS / %d FAIL" % (PASS, FAIL)) import sys sys.exit(0 if FAIL == 0 else 1)