#!/usr/bin/env python3 """ verify_P123.py ============== Standalone verification script for Addendum P123: "Fermi Coupling Constant from the J3(O) Higgs Sector — Structural Closure via the Electroweak VEV" Claims verified --------------- 1. VEV path (correct route, P89 + P123): GF = 1/(sqrt(2)*v^2) with v = 246.22 GeV (loop-exact, P89) Expected: 1.16638e-5 GeV^-2, residual +0.00046% vs PDG 2. Structural formula (Theorem 1, P123): GF = sqrt(2)*N_gen / (2*h_vee(E6)*m_H0^2) Algebraically identical to the VEV path. 3. Tree-level EW formula (documented failure, Section 1 of P123): GF_tree = pi*alpha / (2*MW^2*sin2_theta_W) Using the TOE effective angle sin2_theta_W=3/(8*phi) in place of the kinematic on-shell angle (1 - MW^2/MZ^2), and omitting the radiative- correction factor 1/(1-Delta_r), produces a ~-34% residual — confirming P123's conclusion that the tree-level route is the wrong path. 4. Kinematic on-shell sin^2(theta_W) vs. P112 effective angle: Delta(sin2_theta_W) = 3/(8*phi) - (1 - MW^2/MZ^2) ≈ +0.0075 (scheme gap) Copyright: Leon Fernando Vlegels. License: MIT. """ import math import sys # ────────────────────────────────────────────────────────────────────────────── # TOE constants (kernel/math/quat_s3.py — frozen, never approximate) # ────────────────────────────────────────────────────────────────────────────── ALPHA_INV = 4 * math.pi**3 + math.pi**2 + math.pi # ≈ 137.036 (exact TOE value) ALPHA = 1 / ALPHA_INV # fine-structure constant PHI = (1 + math.sqrt(5)) / 2 # golden ratio φ ≈ 1.61803 SIN2_THETA_W = 3 / (8 * PHI) # P112: Z-pole effective angle ≈ 0.23176 MZ = 91.1876 # GeV (PDG input, used in P122) # ────────────────────────────────────────────────────────────────────────────── # J3(O) Higgs-sector parameters (Addenda P85, P89) # ────────────────────────────────────────────────────────────────────────────── N_GEN = 3 # SM generations = rank of J3(O) over O H_VEE_E6 = 12 # dual Coxeter number of E6 LAMBDA_H_LO = N_GEN / (2 * H_VEE_E6) # = 1/8 (P85, Theorem 3.2) M_H0 = 123.11 # GeV leading-order Higgs mass from P85 spectral geometry # Electroweak VEV (P89, loop-exact): v = m_H0 / sqrt(2*lambda_H) = 2*m_H0 # Factor sqrt(h_vee/N_gen) = sqrt(12/3) = 2 is a pure Lie-algebraic integer. VEV = M_H0 / math.sqrt(2 * LAMBDA_H_LO) # = 246.22 GeV # ────────────────────────────────────────────────────────────────────────────── # W-boson mass # ────────────────────────────────────────────────────────────────────────────── # Tree-level (from MZ and sin^2 theta_W, no Veltman correction): MW_TREE = MZ * math.sqrt(1 - SIN2_THETA_W) # P122 physical value (includes Veltman Delta_rho from top-bottom Peirce imbalance): MW_P122 = 80.313 # GeV # ────────────────────────────────────────────────────────────────────────────── # PDG reference values # ────────────────────────────────────────────────────────────────────────────── GF_PDG = 1.1663788e-5 # GeV^-2 (PDG 2024) VEV_PDG = 246.22 # GeV (PDG: (sqrt(2)*GF)^{-1/2}) M_H_PDG = 125.20 # GeV (PDG Higgs mass) TAU_MU_PDG = 2.196981e-6 # s (PDG muon lifetime) M_MU_PDG = 0.105658 # GeV (PDG muon mass) # ────────────────────────────────────────────────────────────────────────────── # Computations # ────────────────────────────────────────────────────────────────────────────── # 1. VEV path (P89 + P123 Theorem 1) GF_VEV = 1.0 / (math.sqrt(2) * VEV**2) # 2. Structural formula (equivalent derivation, purely algebraic) # GF = sqrt(2)*lambda_H / m_H0^2 = sqrt(2)*N_gen / (2*h_vee*m_H0^2) GF_STRUCT = math.sqrt(2) * LAMBDA_H_LO / M_H0**2 # 3. Tree-level EW formula (documented failure — Section 1 of P123) # Standard Fermi theory: GF/sqrt(2) = pi*alpha / (2*MW^2*sin^2_theta_W) # Paper numerical check: GF_tree = pi*alpha / (2*MW^2*sin^2_theta_W) # (reproduces the 7.668e-6 / -34.3% result in P123 eq.1.5, using the P112 # effective angle in place of the on-shell angle and omitting Delta_r) GF_TREE = math.pi * ALPHA / (2 * MW_P122**2 * SIN2_THETA_W) # Kinematic on-shell sin^2(theta_W) SIN2_OS = 1 - (MW_P122**2 / MZ**2) # Bonus: leading-order muon lifetime from GF (P123 OI-2, informational) # Formula gives τ in natural units (GeV^-1); convert to seconds via ℏ = 6.582119569e-25 GeV·s HBAR_GEV_S = 6.582119569e-25 # GeV·s TAU_MU_LO_NAT = 192 * math.pi**3 / (GF_VEV**2 * M_MU_PDG**5) # GeV^-1 TAU_MU_LO = TAU_MU_LO_NAT * HBAR_GEV_S # seconds # ────────────────────────────────────────────────────────────────────────────── # Residual helper # ────────────────────────────────────────────────────────────────────────────── def pct(val, ref): return (val - ref) / ref * 100.0 res_vev = pct(GF_VEV, GF_PDG) res_struct = pct(GF_STRUCT, GF_PDG) res_tree = pct(GF_TREE, GF_PDG) res_vev_v = pct(VEV, VEV_PDG) res_tau = pct(TAU_MU_LO, TAU_MU_PDG) # ────────────────────────────────────────────────────────────────────────────── # Report # ────────────────────────────────────────────────────────────────────────────── W = 66 SEP = "─" * W print(SEP) print(" Addendum P123 — Fermi Coupling Constant Verification") print(SEP) print("\n── TOE constants ──────────────────────────────────────────────") print(f" ALPHA_INV = 4π³+π²+π = {ALPHA_INV:.10f}") print(f" α = 1/ALPHA_INV = {ALPHA:.10e}") print(f" φ = (1+√5)/2 = {PHI:.10f}") print(f" sin²θW (P112, Z-pole eff.) = 3/(8φ) = {SIN2_THETA_W:.8f}") print(f" sin²θW (on-shell, kin.) = 1-MW²/MZ² = {SIN2_OS:.8f}") print(f" Δ(sin²θW) scheme gap = {SIN2_THETA_W - SIN2_OS:+.8f} (P123 § 2.1)") print("\n── Higgs sector — P85 / P89 ───────────────────────────────────") print(f" N_gen = {N_GEN} (SM generations = rank of J3(O)/O)") print(f" h∨(E6) = {H_VEE_E6} (dual Coxeter number of E6)") print(f" λH(LO) = N_gen/(2h∨) = {LAMBDA_H_LO} = 1/8") print(f" mH(LO) = {M_H0} GeV (P85 spectral geometry)") print(f" mH(PDG) = {M_H_PDG} GeV residual {pct(M_H0,M_H_PDG):+.2f}%") print(f" v (P89) = mH0/√(2λH) = {VEV:.4f} GeV residual {res_vev_v:+.4f}% vs PDG") print(f" v = 2·mH0? = {VEV/M_H0:.15f} (expect exactly 2.0)") print("\n── W-boson mass ────────────────────────────────────────────────") print(f" MW (tree-level) = MZ·√(1−sin²θW) = {MW_TREE:.4f} GeV") print(f" MW (P122, +Δρ) = {MW_P122:.3f} GeV") print("\n── GF computation ──────────────────────────────────────────────") print(f" GF (PDG 2024) = {GF_PDG:.7e} GeV⁻²") print() print(" [1] VEV path — GF = 1/(√2·v²) (P89 → P123, correct route)") print(f" v² = {VEV**2:.4f} GeV²") print(f" √2·v² = {math.sqrt(2)*VEV**2:.2f} GeV²") print(f" GF = {GF_VEV:.7e} GeV⁻²") print(f" residual = {res_vev:+.5f}% (paper claims +0.00046%)") print() print(" [2] Structural — GF = √2·λH / mH0² = √2·N_gen/(2h∨·mH0²)") print(f" GF = {GF_STRUCT:.7e} GeV⁻²") print(f" residual = {res_struct:+.5f}%") print(f" Δ vs VEV = {abs(GF_STRUCT-GF_VEV):.2e} GeV⁻² (algebraically identical)") print() print(" [3] Tree-level — GF = πα/(2·MW²·sin²θW) (documented failure)") print(f" Uses P112 effective angle {SIN2_THETA_W:.5f} instead of") print(f" kinematic on-shell angle {SIN2_OS:.5f}") print(f" and omits radiative-correction factor 1/(1−Δr), Δr≈0.036") print(f" GF = {GF_TREE:.4e} GeV⁻²") print(f" residual = {res_tree:+.2f}% (paper claims ≈ −34.3%)") print() print(" [Bonus — P123 OI-2, informational]") print(f" Muon lifetime τμ(LO) = 192π³/(GF²·mμ⁵)") print(f" τμ(LO) = {TAU_MU_LO:.4e} s") print(f" τμ(PDG) = {TAU_MU_PDG:.4e} s") print(f" residual = {res_tau:+.2f}% (LO; QED radiative correction ≈ +0.42% expected)") # ────────────────────────────────────────────────────────────────────────────── # PASS / FAIL checks # ────────────────────────────────────────────────────────────────────────────── print(f"\n── PASS / FAIL ─────────────────────────────────────────────────") checks = [ ( "lambda_H(LO) = 1/8 (P85 Theorem 3.2)", abs(LAMBDA_H_LO - 0.125) < 1e-15, f"λH = {LAMBDA_H_LO} (exact rational)" ), ( "VEV = 2·m_H0 exactly (P89 loop-exact, √(h∨/N_gen) = 2)", abs(VEV / M_H0 - 2.0) < 1e-12, f"v/mH0 = {VEV/M_H0:.15f}" ), ( "VEV path residual < 0.01% vs PDG (paper: +0.00046%)", abs(res_vev) < 0.01, f"residual = {res_vev:+.5f}%" ), ( "Structural formula algebraically = VEV path (< 1e-14 relative)", abs(GF_STRUCT - GF_VEV) / GF_PDG < 1e-14, f"|ΔGF|/GF_PDG = {abs(GF_STRUCT-GF_VEV)/GF_PDG:.2e}" ), ( "Tree-level formula gives large negative residual (< −30%)", res_tree < -30.0, f"residual = {res_tree:+.2f}% (scheme mismatch + missing Δr)" ), ( "sin²θW scheme gap > 0.005 (P112 eff. ≠ on-shell kinematic)", (SIN2_THETA_W - SIN2_OS) > 0.005, f"Δ(sin²θW) = {SIN2_THETA_W - SIN2_OS:+.6f}" ), ( "GF_tree ≈ 7.668e-6 (paper eq. 1.5, ±1%)", abs(GF_TREE / 7.668e-6 - 1.0) < 0.01, f"GF_tree = {GF_TREE:.4e} (paper: 7.668e-6)" ), ] 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}") for i, (label, ok, detail) in enumerate(checks, 1): check(i, label, ok) print(f" {detail}") print(f"\n{'='*60}\nRESULT: {PASS} PASS / {FAIL} FAIL") sys.exit(0 if FAIL == 0 else 1)