#!/usr/bin/env python3 """ verify_P124.py Computational verification of all numerical claims in: Addendum P124 — W-Boson NLO Programme: mt Scheme Resolution and Electroweak Corrections Beyond Delta-rho. Every value is re-derived from first principles using TOE constants. Claimed values are compared with independently computed values; each check is annotated with absolute error, relative error, and PASS/FAIL. Author : Leon Fernando Vlegels License : MIT """ import math import sys # =========================================================================== # TOE mathematical constants # =========================================================================== ALPHA_INV = 4 * math.pi**3 + math.pi**2 + math.pi # ≈ 137.036 ALPHA = 1 / ALPHA_INV # fine-structure constant BREATH_PERIOD = math.pi * ALPHA_INV # ≈ 432 system units PHI = (1 + math.sqrt(5)) / 2 # golden ratio ≈ 1.618034 # Weinberg angle from J3(O) G2×2I structure (P112) SIN2_THETA_W = 3 / (8 * PHI) # ≈ 0.23173 from formula # =========================================================================== # PDG 2024 physical inputs (P124 §2 ground-truth table) # =========================================================================== MW_PDG = 80.377 # GeV (PDG 2024, ±0.012 GeV) MZ = 91.1876 # GeV (PDG 2024) GF = 1.16637e-5 # GeV⁻² (PDG 2024; paper uses 1.16637×10⁻⁵ throughout) MT_PDG = 172.69 # GeV (on-shell pole mass, PDG 2024, ±0.30 GeV) MT_TOE = 176.101 # GeV (TOE spectral / GUT-scale running mass, P100) ALPHA_S = 0.108 # alpha_s(mt) (PDG / P106) # Note: task prompt lists GF = 1.1663788e-5 (higher precision PDG value). # P124 rounds this to 1.16637e-5; both agree to all digits shown in the paper. # =========================================================================== # Verification infrastructure # =========================================================================== _results: list = [] def check(label: str, computed: float, claimed: float, tol_rel: float = 5e-3) -> bool: """ Assert |computed - claimed| / |claimed| <= tol_rel. Prints a PASS/FAIL line and accumulates results. """ if claimed == 0: err_abs = computed - claimed err_rel = abs(err_abs) else: err_abs = computed - claimed err_rel = err_abs / abs(claimed) ok = abs(err_rel) <= tol_rel _results.append((label, ok, computed, claimed, err_rel)) print(f" [{'PASS' if ok else 'FAIL'}] {len(_results):>2}. {label}") print(f" computed = {computed:.9g} | claimed = {claimed:.9g}" f" | err = {err_abs:+.3e} ({err_rel*100:+.4f}%)") return ok def note(text: str) -> None: print(f" [i] {text}") # =========================================================================== # Section 0 — Constant cross-checks # =========================================================================== print() print("=" * 72) print("Section 0 — TOE constants and setup") print("=" * 72) note(f"ALPHA_INV = {ALPHA_INV:.8f} (expect ≈ 137.036)") note(f"ALPHA = {ALPHA:.8e}") note(f"BREATH_PERIOD = {BREATH_PERIOD:.6f} (expect ≈ 432)") note(f"PHI = {PHI:.12f}") note(f"SIN2_THETA_W = 3/(8φ) = {SIN2_THETA_W:.8f}") # ------ sin²θW table-value discrepancy ------ SIN2_TW_TABLE = 0.23099 # value quoted in P124 §2 ground-truth table discrepancy = SIN2_THETA_W - SIN2_TW_TABLE mw_tree_from_table = MZ * math.sqrt(1 - SIN2_TW_TABLE) print() print(" --- sin²θW discrepancy report ---") print(f" Formula 3/(8φ): {SIN2_THETA_W:.8f}") print(f" P124 table entry: {SIN2_TW_TABLE:.8f}") print(f" Difference (formula−table): {discrepancy:+.8f} " f"({discrepancy/SIN2_TW_TABLE*100:+.4f}%)") print(f" MW_tree using table value: {mw_tree_from_table:.4f} GeV ← does NOT match 79.925") MW_tree = MZ * math.sqrt(1 - SIN2_THETA_W) print(f" MW_tree using formula: {MW_tree:.4f} GeV ← matches 79.925") print(" CONCLUSION: the table entry '0.23099' is a numerical error in P124.") print(" All subsequent calculations use the formula value 3/(8φ).") # =========================================================================== # Section 1 — Tree-level MW and P122 baseline # =========================================================================== print() print("=" * 72) print("Section 1 — Tree-level MW and P122 Δρ baseline") print("=" * 72) # (1a) MW_tree = MZ · √(1 − sin²θW) eq.(1) MW_tree = MZ * math.sqrt(1 - SIN2_THETA_W) check("MW_tree = MZ·√(1 − 3/(8φ))", MW_tree, 79.925, tol_rel=2e-3) # (1b) Δρ_TOE with mt = mt_TOE = 176.101 GeV eqs.(2–3) drho_toe_num = 3 * GF * MT_TOE**2 drho_toe_den = 8 * math.pi**2 * math.sqrt(2) drho_toe = drho_toe_num / drho_toe_den # P124 eqs.(2–3) do not quote the TOE numerator explicitly; only the final ratio. # (1.04353 is the PDG-mass numerator from Theorem 2.1 — checked in Section 2.) note(f"3·GF·mt_TOE² (numerator, not quoted in paper) = {drho_toe_num:.7f}") check("denominator 8π²√2", drho_toe_den, 111.653, tol_rel=1e-4) check("Δρ_TOE", drho_toe, 0.009718, tol_rel=5e-3) # (1c) MW_P122 = MW_tree · √(1 + Δρ_TOE) eq.(4) MW_P122 = MW_tree * math.sqrt(1 + drho_toe) check("MW_P122 = MW_tree·√(1 + Δρ_TOE)", MW_P122, 80.313, tol_rel=2e-3) # (1d) P122 residual res_P122_abs = MW_P122 - MW_PDG res_P122_pct = res_P122_abs / MW_PDG * 100 check("P122 residual [GeV]", res_P122_abs, -0.064, tol_rel=5e-2) check("P122 residual [%]", res_P122_pct, -0.080, tol_rel=5e-2) # (1e) signed gap entering P124 eq.(5) delta_0 = MW_P122 - MW_PDG check("δ₀ = MW_P122 − MW_PDG [GeV]", delta_0, -0.064, tol_rel=5e-2) # =========================================================================== # Section 2 — Step 1: scheme resolution with PDG pole mass (Theorems 2.1–2.2) # =========================================================================== print() print("=" * 72) print("Section 2 — Step 1: scheme resolution with PDG pole mass") print("=" * 72) # (2a) mt_PDG² eq. block of Theorem 2.1 mt2_pdg = MT_PDG**2 check("mt_PDG² = 172.69² [GeV²]", mt2_pdg, 29_821.84, tol_rel=1e-5) # (2b) Numerator and denominator of Δρ_PDG drho_pdg_num = 3 * GF * mt2_pdg drho_pdg_den = 8 * math.pi**2 * math.sqrt(2) check("3·GF·mt_PDG² (numerator)", drho_pdg_num, 1.04353, tol_rel=5e-3) check("8π²√2 (denominator)", drho_pdg_den, 111.653, tol_rel=1e-4) # sub-checks on individual factors the paper quotes check("8 × π² = 8 × 9.86960", 8 * math.pi**2, 8 * 9.86960, tol_rel=1e-5) check("√2 = 1.41421", math.sqrt(2), 1.41421, tol_rel=1e-5) # (2c) Δρ_PDG eq.(6) drho_pdg = drho_pdg_num / drho_pdg_den check("Δρ_PDG = 3·GF·mt_PDG²/(8π²√2)", drho_pdg, 0.009347, tol_rel=5e-3) # (2d) MW_LO = MW_tree · √(1 + Δρ_PDG) Theorem 2.2 / eq.(7) sqrt_lo = math.sqrt(1 + drho_pdg) check("√(1 + Δρ_PDG)", sqrt_lo, 1.004663, tol_rel=1e-4) MW_LO = MW_tree * sqrt_lo check("MW_LO = MW_tree·√(1 + Δρ_PDG) [GeV]", MW_LO, 80.298, tol_rel=2e-3) # (2e) LO residual res_LO_abs = MW_LO - MW_PDG res_LO_pct = res_LO_abs / MW_PDG * 100 check("LO residual [GeV]", res_LO_abs, -0.079, tol_rel=5e-2) check("LO residual [%]", res_LO_pct, -0.099, tol_rel=5e-2) # (2f) Scheme-correction opening the gap (P122→P124 LO) scheme_delta = MW_LO - MW_P122 note(f"Scheme correction MW_LO − MW_P122 = {scheme_delta*1000:.1f} MeV " f"(paper says ~−15 MeV)") # (2g) Informational: mt running estimate (not a precision claim) print() note("-- mt running estimate (informational, ±2% accuracy) --") mt_msbar_est = MT_TOE * 0.970 note(f"mt_MS̄(mt) ≈ mt_TOE × 0.970 = {mt_msbar_est:.1f} GeV (claimed ≈ 170.8)") pole_factor_corr = 4 * ALPHA_S / (3 * math.pi) check("4·αs/(3π) (one-loop pole-mass factor)", pole_factor_corr, 0.04584, tol_rel=5e-3) check("1 + 4·αs/(3π) (paper quotes 1.0459)", 1 + pole_factor_corr, 1.0459, tol_rel=5e-3) mt_pole_est = mt_msbar_est * (1 + pole_factor_corr) check("mt_pole estimate = mt_MS̄·(1 + 4αs/(3π)) [GeV]", mt_pole_est, 178.6, tol_rel=5e-3) # =========================================================================== # Section 3 — QCD correction to Δρ at O(αs) (§3.1) # =========================================================================== print() print("=" * 72) print("Section 3 — QCD correction to Δρ at O(αs)") print("=" * 72) # (3a) QCD factor 8·αs/(3π) qcd_factor = 8 * ALPHA_S / (3 * math.pi) check("8·αs(mt)/(3π)", qcd_factor, 0.09166, tol_rel=2e-3) note(f"Numerator 8×0.108 = {8*ALPHA_S:.4f} (paper: 0.864)") note(f"Denominator 3π = {3*math.pi:.5f} (paper: 9.4248)") check("numerator 8×αs", 8 * ALPHA_S, 0.864, tol_rel=1e-9) check("denominator 3π", 3 * math.pi, 9.4248, tol_rel=1e-4) # (3b) Δρ_QCD = Δρ_PDG · (1 − 8αs/(3π)) eq.(9) drho_qcd = drho_pdg * (1 - qcd_factor) one_m_qcd = 1 - qcd_factor check("(1 − 8αs/(3π))", one_m_qcd, 0.90834, tol_rel=2e-3) check("Δρ_QCD = Δρ_PDG·(1 − 8αs/(3π))", drho_qcd, 0.008490, tol_rel=2e-3) # (3c) δΔρ_QCD eq.(10) d_drho_qcd = drho_qcd - drho_pdg check("δΔρ_QCD = Δρ_QCD − Δρ_PDG", d_drho_qcd, -0.000857, tol_rel=5e-3) # (3d) δMW_QCD ≈ (MW_tree/2) · δΔρ_QCD eq.(11) [linearised] dMW_qcd = (MW_tree / 2) * d_drho_qcd check("δMW_QCD [GeV] (linearised)", dMW_qcd, -0.034, tol_rel=5e-2) check("δMW_QCD [MeV] (linearised)", dMW_qcd * 1000, -34, tol_rel=5e-2) # (3e) MW_QCD = MW_tree · √(1 + Δρ_QCD) (full formula, eq. after (11)) sqrt_qcd = math.sqrt(1 + drho_qcd) check("√(1 + Δρ_QCD)", sqrt_qcd, 1.004237, tol_rel=1e-4) MW_QCD = MW_tree * sqrt_qcd check("MW_QCD = MW_tree·√(1 + Δρ_QCD) [GeV]", MW_QCD, 80.264, tol_rel=2e-3) # (3f) Residual after QCD correction res_QCD_abs = MW_QCD - MW_PDG res_QCD_pct = res_QCD_abs / MW_PDG * 100 check("Residual after QCD [GeV]", res_QCD_abs, -0.113, tol_rel=5e-2) check("Residual after QCD [%]", res_QCD_pct, -0.141, tol_rel=5e-2) # =========================================================================== # Section 4 — Subleading oblique at O(GF² mt⁴) (§3.2) # =========================================================================== print() print("=" * 72) print("Section 4 — Subleading oblique correction O(GF² mt⁴)") print("=" * 72) # (4a) δΔρ⁽²⁾ ≈ (Δρ_PDG)² eq.(12) d_drho_subl = drho_pdg**2 check("δΔρ⁽²⁾ = (Δρ_PDG)²", d_drho_subl, 8.74e-5, tol_rel=5e-3) # (4b) δMW⁽²⁾ ≈ (MW_tree/2) · δΔρ⁽²⁾ eq.(13) dMW_subl = (MW_tree / 2) * d_drho_subl check("δMW⁽²⁾ [GeV]", dMW_subl, 0.0035, tol_rel=5e-2) check("δMW⁽²⁾ [MeV]", dMW_subl * 1000, 4, tol_rel=2e-1) # (4c) MW after QCD + subleading (gap-closure table row 4) MW_qcd_subl = MW_LO + dMW_qcd + dMW_subl check("MW (QCD + subleading oblique) [GeV]", MW_qcd_subl, 80.268, tol_rel=2e-3) res_qs_abs = MW_qcd_subl - MW_PDG res_qs_pct = res_qs_abs / MW_PDG * 100 check("Residual QCD + subleading [GeV]", res_qs_abs, -0.109, tol_rel=5e-2) check("Residual QCD + subleading [%]", res_qs_pct, -0.136, tol_rel=5e-2) # =========================================================================== # Section 5 — EW vertex+box corrections at O(α/π) (§3.3) # =========================================================================== print() print("=" * 72) print("Section 5 — EW vertex+box corrections O(α/π) [SM literature input]") print("=" * 72) # This correction is taken directly from SM calculations (Sirlin 1980, # Denner 1991) and cannot be independently re-derived from TOE constants # alone. The verification below checks only the arithmetic use of the # stated value. dMW_ew = -0.038 # GeV (−38 MeV, from SM literature as cited in P124) note("δMW_EW = −38 MeV (SM lit. input: Sirlin 1980 / Denner 1991)") note("This value is accepted as given; only its downstream arithmetic is verified.") check("δMW_EW [MeV] (identity check)", dMW_ew * 1000, -38, tol_rel=1e-9) # =========================================================================== # Section 6 — NLO sum and MW_NLO (§4 / Theorem 4.1) # =========================================================================== print() print("=" * 72) print("Section 6 — NLO sum and final MW_NLO (Theorem 4.1)") print("=" * 72) # (6a) integer MeV arithmetic eq.(14) check("−34 + 4 + (−38) = −68 [MeV]", -34 + 4 + (-38), -68, tol_rel=1e-9) # (6b) sum of computed shifts dMW_nlo = dMW_qcd + dMW_subl + dMW_ew check("δMW_NLO = δMW_QCD + δMW⁽²⁾ + δMW_EW [GeV]", dMW_nlo, -0.068, tol_rel=5e-2) check("δMW_NLO [MeV]", dMW_nlo * 1000, -68, tol_rel=5e-2) # (6c) individual MeV components check("δMW_QCD [MeV]", dMW_qcd * 1000, -34, tol_rel=5e-2) check("δMW⁽²⁾ [MeV]", dMW_subl * 1000, +4, tol_rel=2e-1) check("δMW_EW [MeV]", dMW_ew * 1000, -38, tol_rel=1e-9) # (6d) MW_NLO = MW_LO + δMW_NLO eq.(15) MW_NLO = MW_LO + dMW_nlo check("MW_NLO = MW_LO + δMW_NLO [GeV]", MW_NLO, 80.230, tol_rel=2e-3) # (6e) NLO residual eq.(16) res_NLO_abs = MW_NLO - MW_PDG res_NLO_pct = res_NLO_abs / MW_PDG * 100 check("NLO residual [GeV]", res_NLO_abs, -0.147, tol_rel=5e-2) check("NLO residual [%]", res_NLO_pct, -0.183, tol_rel=5e-2) # (6f) Explicit arithmetic from eq.(15): 80.298 + (−0.034) + (+0.004) + (−0.038) arithmetic_check = 80.298 + (-0.034) + 0.004 + (-0.038) check("80.298 − 0.034 + 0.004 − 0.038 = 80.230", arithmetic_check, 80.230, tol_rel=1e-5) # =========================================================================== # Section 7 — Gap-closure table (§4.2) # =========================================================================== print() print("=" * 72) print("Section 7 — Gap-closure table: all five rows") print("=" * 72) rows = [ ("Row 1 — P122 (TOE mt, Δρ only)", MW_P122, 80.313, -0.064, -0.080), ("Row 2 — P124 LO (PDG mt, Δρ only)", MW_LO, 80.298, -0.079, -0.099), ("Row 3 — After QCD correction to Δρ", MW_QCD, 80.264, -0.113, -0.141), ("Row 4 — After QCD + subleading", MW_qcd_subl, 80.268, -0.109, -0.136), ("Row 5 — After all NLO (Theorem 4.1)", MW_NLO, 80.230, -0.147, -0.183), ] for label, MW_comp, mw_claim, res_claim_gev, res_claim_pct in rows: print(f"\n {label}:") check(f" MW [GeV]", MW_comp, mw_claim, tol_rel=2e-3) check(f" residual [GeV]", MW_comp - MW_PDG, res_claim_gev, tol_rel=5e-2) check(f" residual [%]", (MW_comp - MW_PDG) / MW_PDG * 100, res_claim_pct, tol_rel=5e-2) # =========================================================================== # Section 8 — sin²θW numerical value (table cross-check) # =========================================================================== print() print("=" * 72) print("Section 8 — sin²θW formula vs. table-value cross-check") print("=" * 72) note(f"sin²θW = 3/(8φ) = {SIN2_THETA_W:.8f} (formula)") note(f"sin²θW P124 table = {SIN2_TW_TABLE:.8f} (stated in §2 table)") note(f"Difference = {SIN2_THETA_W - SIN2_TW_TABLE:+.8f}") note(f"MW_tree from table val = {mw_tree_from_table:.4f} GeV (≠ 79.925 GeV)") note(f"MW_tree from formula = {MW_tree:.4f} GeV (≈ 79.925 GeV ✓)") note("The table entry 0.23099 is numerically inconsistent with:") note(" (a) the formula 3/(8φ), and") note(" (b) the subsequent computation MW_tree = 79.925 GeV.") note("All arithmetic in this verifier uses the formula value.") # Verify the formula sin²θW = 3/(8φ) = 3(√5−1)/16 (algebraic identity) sin2_algebraic = 3 * (math.sqrt(5) - 1) / 16 check("3/(8φ) = 3(√5−1)/16 (algebraic identity)", SIN2_THETA_W, sin2_algebraic, tol_rel=1e-12) # =========================================================================== # Final summary # =========================================================================== print() print("=" * 72) print("FINAL SUMMARY") print("=" * 72) passes = sum(1 for _, ok, *_ in _results if ok) fails = sum(1 for _, ok, *_ in _results if not ok) total = len(_results) print() print(f" Arithmetic checks: {passes} PASS / {fails} FAIL (of {total} total)") if fails: print() print(" FAILED checks:") for label, ok, comp, claim, rel_err in _results: if not ok: print(f" ✗ {label}") print(f" computed={comp:.9g}, claimed={claim:.9g}, " f"rel_err={rel_err*100:+.4f}%") print() print(" Key derived values:") print(f" PHI = {PHI:.10f}") print(f" sin²θW = 3/(8φ) = {SIN2_THETA_W:.8f} " f"(table claims 0.23099 — see §8)") print(f" MW_tree = {MW_tree:.4f} GeV (claimed 79.925)") print(f" Δρ_TOE [P122, mt=176.101] = {drho_toe:.7f} (claimed 0.009718)") print(f" MW_P122 = {MW_P122:.4f} GeV (claimed 80.313)") print(f" Δρ_PDG [PDG pole mass] = {drho_pdg:.7f} (claimed 0.009347)") print(f" MW_LO = {MW_LO:.4f} GeV (claimed 80.298)") print(f" δMW_QCD = {dMW_qcd*1000:.2f} MeV (claimed −34 MeV)") print(f" δMW_subl = {dMW_subl*1000:.2f} MeV (claimed +4 MeV)") print(f" δMW_EW = {dMW_ew*1000:.1f} MeV (SM lit. input)") print(f" δMW_NLO = {dMW_nlo*1000:.2f} MeV (claimed −68 MeV)") print(f" MW_NLO = {MW_NLO:.4f} GeV (claimed 80.230)") print(f" NLO residual = {res_NLO_abs:.4f} GeV (claimed −0.147)") print(f" NLO residual = {res_NLO_pct:.4f}% (claimed −0.183%)") print() print(" Caveats:") print(" • δMW_EW = −38 MeV is a SM literature input (Sirlin 1980 / Denner 1991).") print(" Its sign and magnitude are verified arithmetically but not re-derived") print(" from TOE constants, as P124 explicitly defers this to B-1/B-2/B-3.") print(" • sin²θW table entry 0.23099 is a numerical error in P124 §2;") print(" the formula 3/(8φ) = 0.23173 is consistent with all MW computations.") print() if fails == 0: print(" OVERALL: PASS — all arithmetic claims verified within tolerance.") else: print(f" OVERALL: FAIL — {fails} check(s) failed.") print(f"\n{'='*60}\nRESULT: {passes} PASS / {fails} FAIL") sys.exit(0 if fails == 0 else 1)