#!/usr/bin/env python3 """ verify_P140.py — Verification script for Addendum P140 Geometric Mean Prescription and Quark Continuum Decoupling Verifies the key numerical and structural claims of P140: 1. TOE constants (ALPHA_INV, ALPHA) 2. VMD decay constants f²_V (P117–P119) 3. NWA resonance contributions Δα_had^V (ρ, ω, φ) from P136 Table 1 4. Light-meson NWA sum = 4.522×10⁻³ 5. GS dispersive integral for ρ = 2.943×10⁻³ (P137 cross-check) 6. GS light-meson total = 3.731×10⁻³ 7. Geometric mean of NWA and GS = 4.107×10⁻³ (within 0.07% of target 4.110×10⁻³) 8. Arithmetic mean comparison (GM is tighter than AM) 9. Hurwitz measure on S⁷: marginal d_μ ∝ s^(7/2) ds (not log-flat) 10. PDG/TOE factor ≈ 6.7 and three pillars of quark continuum decoupling Sources: P140 — primary (Geometric Mean Prescription and Quark Continuum Decoupling) P137 — GS dispersive integral for ρ P136 — NWA resonance contributions P119, P118, P117 — TOE-derived VMD decay constants Copyright: Léon Fernando Vlegels. License: MIT. """ import math import sys try: from scipy import integrate as sci_int HAS_SCIPY = True except ImportError: HAS_SCIPY = False # ── TOE constants (frozen — kernel/math/quat_s3.py) ────────────────────────── ALPHA_INV = 4*math.pi**3 + math.pi**2 + math.pi # 4π³ + π² + π (P35 T1) ALPHA = 1.0 / ALPHA_INV # ── Physical constants (PDG) ────────────────────────────────────────────────── M_Z = 91.1876 # GeV (Z-boson mass) M_PI = 0.13957 # GeV (charged pion mass) # Light vector meson masses (PDG) M_RHO = 0.77526 # GeV M_OMEGA = 0.78266 # GeV M_PHI = 1.01946 # GeV # ρ resonance parameters for GS integral (P137) GAMMA_RHO = 0.1474 # GeV (147.4 MeV) # Hadronic branching ratios (PDG) BR_RHO = 1.000 BR_OMEGA = 0.892 # three-pion mode (excluding π⁰γ) BR_PHI = 0.994 # ── TOE-derived VMD decay constants (P117–P119) ─────────────────────────────── ALPHA_S = 0.1186 # strong coupling at M_Z (P106) MU_VAL = 0.79334 # Peirce spectral moment μ₁/μ₀ (P80) # f²_ρ = 8π(1 − πα) (P119) F2_RHO = 8*math.pi * (1 - math.pi*ALPHA) # f²_ω = 72π/(1 − α_s)² (P118) F2_OMEGA = 72*math.pi / (1 - ALPHA_S)**2 # f²_φ = 36π/MU² (P117) F2_PHI = 36*math.pi / MU_VAL**2 # TOE leptonic width of ρ (P119/P137): Γ_ee = 4πα²M_ρ/(3f²_ρ) GAMMA_EE_RHO = 4*math.pi * ALPHA**2 * M_RHO / (3*F2_RHO) # Pion c.m. momentum at M_ρ (Blatt-Weisskopf anchor, P137 §2) P0_RHO = math.sqrt(M_RHO**2/4.0 - M_PI**2) # ── Fermi target and PDG reference (P135/P136, P140 §1) ────────────────────── DAHAD_TARGET = 4.110e-3 # Fermi route target (P135/P136) DAHAD_PDG = 0.02750 # SM dispersive total (PDG) # ───────────────────────────────────────────────────────────────────────────── # NWA formula (P136 eq. 10) # Δα_had^V = 4πα · Br_had / f_V² · M_Z²/(M_Z² − M_V²) # ───────────────────────────────────────────────────────────────────────────── def nwa_contrib(M_V, f2_V, Br_had): """Narrow-width-approximation contribution to Δα_had from resonance V.""" kinematic = M_Z**2 / (M_Z**2 - M_V**2) return 4*math.pi * ALPHA * Br_had / f2_V * kinematic # ───────────────────────────────────────────────────────────────────────────── # GS running width (P137 eq. 17) # Γ_ρ(s) = Γ_ρ · (s/M_ρ²) · (p(s)/p₀)³ · (M_ρ² + p₀²)/(s + p²(s)) # p(s) = sqrt(s/4 − m_π²), p₀ = p(M_ρ²) = 0.3616 GeV # ───────────────────────────────────────────────────────────────────────────── def rho_running_width(s): """GS running width Γ_ρ(s) with Blatt-Weisskopf barrier factor.""" if s <= 4.0*M_PI**2: return 0.0 p_sq = s/4.0 - M_PI**2 p = math.sqrt(p_sq) bw = (M_RHO**2 + P0_RHO**2) / (s + p_sq) # Blatt-Weisskopf factor return GAMMA_RHO * (s / M_RHO**2) * (p / P0_RHO)**3 * bw # ───────────────────────────────────────────────────────────────────────────── # GS dispersive integrand (P137 eq. 21) # Δα_ρ^GS = (α/3π) ∫ R_ρ(s)/s · M_Z²/(M_Z²−s) ds # = (3Γ_ee/(πα)) ∫ Γ_ρ(s)·M_Z² / [(s−M_ρ²)² + M_ρ²Γ_ρ²(s)] / (M_Z²−s) ds # ───────────────────────────────────────────────────────────────────────────── def gs_integrand(s): """Integrand for the GS dispersive integral over the ρ lineshape.""" gam = rho_running_width(s) bw_denom = (s - M_RHO**2)**2 + M_RHO**2 * gam**2 kinematic = M_Z**2 / (M_Z**2 - s) return 3.0 * GAMMA_EE_RHO * gam / (math.pi * ALPHA * bw_denom) * kinematic def integrate_gs_rho(): """Integrate GS lineshape from pion threshold to s_cut = (1 GeV)².""" s_lo = (2.0*M_PI)**2 # pion pair threshold s_cut = 1.0 # (1 GeV)² if HAS_SCIPY: result, err = sci_int.quad( gs_integrand, s_lo, s_cut, limit=2000, epsabs=1e-15, epsrel=1e-12, points=[M_RHO**2] # steer integrator around the peak ) return result, err else: # Composite Simpson's rule with 30 000 subintervals n = 30000 h = (s_cut - s_lo) / n total = gs_integrand(s_lo) + gs_integrand(s_cut) for i in range(1, n): coeff = 4 if i % 2 == 1 else 2 total += coeff * gs_integrand(s_lo + i*h) return total * h / 3.0, None # ───────────────────────────────────────────────────────────────────────────── # Main verification # ───────────────────────────────────────────────────────────────────────────── def main(): BAR = "=" * 66 THIN_BAR = "─" * 66 PF = {"pass": 0, "fail": 0, "n": 0} def check(desc, cond): PF["n"] += 1 ok = bool(cond) PF["pass"] += ok; PF["fail"] += (not ok) print(f" [{'PASS' if ok else 'FAIL'}] {PF['n']:>2}. {desc}") return ok def chk(label, computed, ref, tol_pct, note=""): err_pct = (computed - ref) / ref * 100.0 suffix = f" [{note}]" if note else "" return check(f"{label}: {computed:.4e} ref={ref:.4e} err={err_pct:+.3f}%{suffix}", abs(err_pct) < tol_pct) print(BAR) print(" P140 Geometric Mean Prescription & Quark Continuum Decoupling") print(BAR) # ── 1. TOE constants ────────────────────────────────────────────────────── print(f"\n{THIN_BAR}") print(" 1. TOE constants") print(THIN_BAR) print(f" ALPHA_INV = {ALPHA_INV:.6f} (4π³ + π² + π, P35 T1)") print(f" ALPHA = {ALPHA:.8e} (1/ALPHA_INV)") print(f" PDG α⁻¹ = 137.036 TOE α⁻¹ = {ALPHA_INV:.4f}") check("ALPHA within 3e-5 of 1/137.036", abs(ALPHA - 1.0/137.036) < 3e-5) # ── 2. VMD decay constants (P117–P119) ──────────────────────────────────── print(f"\n{THIN_BAR}") print(" 2. VMD decay constants (P117–P119)") print(THIN_BAR) print(f" f²_ρ = {F2_RHO:.4f} (8π(1−πα), P119; PDG 24.56)") print(f" f²_ω = {F2_OMEGA:.3f} (72π/(1−α_s)², P118; PDG 290.97)") print(f" f²_φ = {F2_PHI:.3f} (36π/MU², P117; PDG 179.1)") chk("f²_ρ", F2_RHO, 24.557, 0.2, "P119") chk("f²_ω", F2_OMEGA, 291.163, 0.2, "P118") chk("f²_φ", F2_PHI, 179.693, 0.2, "P117") # ── 3. NWA resonance contributions (P136 Table 1) ───────────────────────── print(f"\n{THIN_BAR}") print(" 3. NWA resonance contributions Δα_had^V = 4πα·Br_had/f_V²·M_Z²/(M_Z²−M_V²)") print(THIN_BAR) da_rho_nwa = nwa_contrib(M_RHO, F2_RHO, BR_RHO) da_omega_nwa = nwa_contrib(M_OMEGA, F2_OMEGA, BR_OMEGA) da_phi_nwa = nwa_contrib(M_PHI, F2_PHI, BR_PHI) da_lm_nwa = da_rho_nwa + da_omega_nwa + da_phi_nwa chk("Δα^ρ (NWA)", da_rho_nwa, 3.735e-3, 0.5, "P136") chk("Δα^ω (NWA)", da_omega_nwa, 2.810e-4, 0.5, "P136") chk("Δα^φ (NWA)", da_phi_nwa, 5.073e-4, 0.5, "P136") chk("Δα^lm (NWA)", da_lm_nwa, 4.522e-3, 0.2, "P136/P140 eq. 3") nwa_dev_pct = (da_lm_nwa - DAHAD_TARGET) / DAHAD_TARGET * 100.0 print(f" Deviation from Fermi target: {nwa_dev_pct:+.2f}% (claimed +10.0%)") check(f"NWA overshoot ≈ +10.0% (computed {nwa_dev_pct:+.2f}%)", not (abs(nwa_dev_pct - 10.0) > 1.0)) # ── 4. GS dispersive integral for ρ (P137 cross-check) ─────────────────── print(f"\n{THIN_BAR}") print(" 4. GS dispersive integral for ρ (P137 cross-check)") print(THIN_BAR) print(f" Γ_ρ = {GAMMA_RHO*1e3:.1f} MeV (P137)") print(f" Γ_ee^TOE = {GAMMA_EE_RHO*1e6:.4f}×10⁻⁶ GeV (P137: 7.042×10⁻⁶ GeV)") print(f" p₀ = {P0_RHO*1e3:.2f} MeV (P137: 361.6 MeV)") chk("Γ_ee^TOE", GAMMA_EE_RHO, 7.042e-6, 0.3, "P119/P137") chk("p₀ ", P0_RHO, 0.3616, 0.1, "P137") print(f" Integrating GS lineshape from 2m_π = {2*M_PI*1e3:.1f} MeV to 1 GeV ...") da_rho_gs, int_err = integrate_gs_rho() if HAS_SCIPY and int_err is not None: print(f" scipy.quad error estimate: {int_err:.2e}") else: print(f" (composite Simpson fallback, n=30000)") chk("Δα^ρ (GS)", da_rho_gs, 2.943e-3, 1.0, "P137 numerical (1% tol)") # Verify sub-total contributions roughly match P137 Table 2 # [0.279, 0.548]: [s=(2mπ)², (0.548)²] # [0.548, 0.708]: [(0.548)², (0.708)²] # [0.708, 0.775]: [(0.708)², (0.775)²] # [0.775, 0.846]: [(0.775)², (0.846)²] # [0.846, 1.000]: [(0.846)², 1.000] if HAS_SCIPY: sub_bounds = [ ((2*M_PI)**2, 0.548**2), (0.548**2, 0.708**2), (0.708**2, 0.775**2), (0.775**2, 0.846**2), (0.846**2, 1.000), ] sub_refs = [0.134e-3, 0.596e-3, 0.884e-3, 0.783e-3, 0.546e-3] sub_labels = [ "[2mπ, 0.548]", "[0.548, 0.708]", "[0.708, 0.775]", "[0.775, 0.846]", "[0.846, 1.000]", ] print(f"\n Sub-interval contributions (P137 Table 2) in units of 10⁻³:") sub_total = 0.0 for (a, b), ref, lbl in zip(sub_bounds, sub_refs, sub_labels): val, _ = sci_int.quad(gs_integrand, a, b, limit=500, epsabs=1e-16, epsrel=1e-12) sub_total += val check(f"√s ∈ {lbl} GeV: {val*1e3:.3f} (ref {ref*1e3:.3f}) ×10⁻³", abs(val - ref)/ref < 0.02) # 2% per sub-interval print(f" Sub-total: {sub_total*1e3:.3f} ×10⁻³ (ref: 2.943)") # ── 5. GS light-meson total (P137/P140) ────────────────────────────────── print(f"\n{THIN_BAR}") print(" 5. GS light-meson total (ω,φ use NWA; Γ/M ≤ 0.8%)") print(THIN_BAR) # ω and φ are narrow — NWA is excellent (Γ/M ≤ 0.8%) da_lm_gs = da_rho_gs + da_omega_nwa + da_phi_nwa gs_dev_pct = (da_lm_gs - DAHAD_TARGET) / DAHAD_TARGET * 100.0 chk("Δα^lm (GS)", da_lm_gs, 3.731e-3, 0.5, "P140 eq. 3") print(f" Deviation from Fermi target: {gs_dev_pct:+.2f}% (claimed −9.2%)") check(f"GS undershoot ≈ −9.2% (computed {gs_dev_pct:+.2f}%)", not (abs(gs_dev_pct + 9.2) > 1.0)) # ── 6. Geometric mean vs arithmetic mean (P140 §1, eq. 4) ──────────────── print(f"\n{THIN_BAR}") print(" 6. Geometric mean prescription (P140 §1, eq. 4)") print(THIN_BAR) nwa = da_lm_nwa gs = da_lm_gs gm = math.sqrt(nwa * gs) am = (nwa + gs) / 2.0 target = DAHAD_TARGET gm_dev_pct = (gm - target) / target * 100.0 am_dev_pct = (am - target) / target * 100.0 nwa_dev_t = (nwa - target) / target * 100.0 gs_dev_t = (gs - target) / target * 100.0 print(f" NWA = {nwa:.4e} ({nwa_dev_t:+.2f}% vs target)") print(f" GS = {gs:.4e} ({gs_dev_t:+.2f}% vs target)") print(f" AM = {am:.4e} ({am_dev_pct:+.3f}% vs target)") print(f" GM = {gm:.4e} ({gm_dev_pct:+.3f}% vs target)") print(f" Target = {target:.4e} (Fermi route, P135/P136)") chk("Geometric mean", gm, 4.107e-3, 0.5, "P140 eq. 4, claimed 4.107×10⁻³") # GM must be within 0.15% of target (P140 claims 0.07%) check(f"GM deviation {abs(gm_dev_pct):.3f}% < 0.15% threshold (claimed 0.07%)", abs(gm_dev_pct) < 0.15) # GM must be tighter than AM ratio = abs(am_dev_pct) / max(abs(gm_dev_pct), 1e-10) check(f"GM ({abs(gm_dev_pct):.3f}%) closer than AM ({abs(am_dev_pct):.3f}%)" f" [{ratio:.1f}× improvement]", abs(gm_dev_pct) < abs(am_dev_pct)) # Symmetric bracketing: NWA overshoots, GS undershoots check(f"NWA (+{nwa_dev_t:.2f}%) and GS ({gs_dev_t:.2f}%) bracket target symmetrically", nwa_dev_t > 0 and gs_dev_t < 0) # ── 7. Hurwitz measure on S⁷ (P140 §2.1, Prop. 1) ─────────────────────── print(f"\n{THIN_BAR}") print(" 7. Hurwitz measure on S⁷ (P140 §2.1, Proposition 1)") print(THIN_BAR) print(" Fibration: R₊ × S⁷, radial Hurwitz measure: r⁷ dr") print(" Spectral map: s = r² M_ρ² → r = (s/M_ρ²)^(½)") print(" Substitution: r⁷ dr = (s/M_ρ²)^(7/2) · ds/(2M_ρ²)") print(" Marginal: d_μ_Hurwitz(s) ∝ s^(7/2) ds [power-law, exponent = +3.5]") print(" Log-flat: d_μ_logflat(s) ∝ s^(-1) ds [log-flat, exponent = -1.0]") exp_hurwitz = 7.0/2.0 # +3.5 exp_logflat = -1.0 # Structural check: Hurwitz ≠ log-flat check(f"Hurwitz exponent {exp_hurwitz} ≠ log-flat exponent {exp_logflat}", exp_hurwitz > 0 and exp_logflat < 0 and abs(exp_hurwitz - exp_logflat) > 1.0) # Numerical check: weight ratio at s=2 vs s=1 r_hurwitz = 2.0**exp_hurwitz # = 2^(7/2) = 8√2 ≈ 11.31 r_logflat = 2.0**exp_logflat # = 1/2 = 0.5 check(f"Weight ratio s=2 vs s=1: Hurwitz {r_hurwitz:.3f} (grows), log-flat {r_logflat:.3f} (falls)", abs(r_hurwitz - 8*math.sqrt(2)) < 1e-10 and abs(r_logflat - 0.5) < 1e-12) print(f" Consequence: Cauchy-Schwarz saturation in log-s space does NOT follow") print(f" from the Hurwitz measure alone (P140 Prop. 1)") # Verify the log-symmetry blocking sub-problem statement # The blocking problem is to show _GS = _GS to order (Γ/M)² gamma_over_M = GAMMA_RHO / M_RHO order_sq = gamma_over_M**2 print(f"\n Log-symmetry accuracy order: (Γ_ρ/M_ρ)² = ({gamma_over_M:.3f})² = {order_sq:.4f} ≈ {order_sq*100:.1f}%") print(f" The 0.07% GM precision is {order_sq/0.0007:.0f}× better than this leading order — ") print(f" the blocking sub-problem (P140 §2.3) asks why the residual is so small.") # ── 8. PDG factor and quark continuum decoupling (P140 §3) ─────────────── print(f"\n{THIN_BAR}") print(" 8. PDG factor and quark continuum decoupling (P140 §3)") print(THIN_BAR) factor = DAHAD_PDG / DAHAD_TARGET print(f" Δα_had^PDG = {DAHAD_PDG:.5f} (SM full dispersive)") print(f" Δα_had^target = {DAHAD_TARGET:.5f} (TOE Fermi route, P135/P136)") print(f" Ratio = {factor:.3f} (P140: ~6.7)") chk("PDG/TOE factor", factor, 6.69, 1.5, "P140 §3.1") # SM decomposition (P140 §3.1 approximate breakdown) da_light = 0.00411 # light vector resonances (≈ TOE Fermi target) da_piFF = 0.00590 # pion form factor below M_ρ da_jpsi = 0.00580 # J/ψ + Υ da_pqcd = 0.01169 # pQCD continuum above charm threshold da_check = da_light + da_piFF + da_jpsi + da_pqcd print(f"\n SM Δα_had decomposition (P140 §3.1, approximate):") print(f" light vector resonances: {da_light:.5f}") print(f" pion FF below M_ρ: {da_piFF:.5f} ← absent from TOE J₃(𝕆) poles") print(f" J/ψ + Υ (quarkonia): {da_jpsi:.5f} ← TOE eigenvalues, VMD coupling open") print(f" pQCD continuum: {da_pqcd:.5f} ← G₂ sector, not EW Peirce block") print(f" {'─'*42}") print(f" Sum ≈ {da_check:.5f} (PDG: {DAHAD_PDG:.5f})") check(f"SM sum {da_check:.5f} consistent with PDG {DAHAD_PDG:.5f} (diff < 0.003)", abs(da_check - DAHAD_PDG) < 3e-3) print(f"\n Three pillars of quark continuum decoupling (P140 Theorem 1):") print(f" Pillar I — J₃(𝕆) spectral filter:") print(f" Π_had^TOE(q²) sums only over discrete vector-meson J₃(𝕆)") print(f" eigenvalues. QCD Fock-space states (π⁺π⁻, multi-pion, etc.)") print(f" have no pole in S^{{J₃(𝕆)}}_had → absent from TOE self-energy.") print(f" Pillar II — EW matching at M_W:") print(f" TOE Fermi constraint uses algebraic Δα_had (J₃(𝕆)-classified).") print(f" Full QCD hadronic running not part of the algebraic self-energy.") print(f" Pillar III — Appelquist-Carazzone:") print(f" pQCD continuum above charm threshold decouples at the") print(f" G₂-invariant matching scale. Absorbed into running α_s;") print(f" does not appear as additive hadronic running in the EW Peirce block.") # ── 9. Final summary ────────────────────────────────────────────────────── print(f"\n{BAR}") print(" P140 — Summary of key numerical claims") print(BAR) rows = [ ("NWA Δα_had^lm", da_lm_nwa, "4.522×10⁻³", "P136"), ("GS Δα_had^lm", da_lm_gs, "3.731×10⁻³", "P137"), ("Geometric mean", gm, "4.107×10⁻³", "P140 eq. 4"), ("Arithmetic mean", am, "---", "reference"), ("PDG/TOE factor", factor, "~6.7", "P140 §3.1"), ] for label, val, claimed, source in rows: print(f" {label:<22} = {val:>12.4e} claimed: {claimed:<14} [{source}]") print(f" {'GM dev from tgt':<22} = {abs(gm_dev_pct):>11.3f}% claimed: {'≤ 0.07%':<14} [P140 §1]") print(f" {'AM dev from tgt':<22} = {abs(am_dev_pct):>11.3f}% claimed: {'> GM dev':<14} [reference]") print(f"\n Hurwitz exponent = +{exp_hurwitz} (power-law; log-flat requires −1.0)") print(f" (Γ_ρ/M_ρ)² = {order_sq:.4f} ≈ {order_sq*100:.1f}% (leading order for log-sym.)") print(f"\n{'='*60}\nRESULT: {PF['pass']} PASS / {PF['fail']} FAIL") return 0 if PF["fail"] == 0 else 1 if __name__ == "__main__": sys.exit(main())