""" verify_P208.py — Verification script for Addendum P208 "Dual Observer Triangular Path and the Muon Mass Ratio" Tests whether the {Plateau, Weld} matrix element of the full transfer matrix on the lepton triangular face {Fork, Weld, Plateau} ⊂ Δ₄ recovers m_μ/m_e more accurately than the direct one-edge formula. Five tasks: Task 1: Second-order perturbative correction via virtual Fork intermediate Task 2: Full transfer matrix propagator exp(τH) Task 3: Eigenvalue mass ratios Task 4: Dual observer amplitude A(F→W→P) Task 5: Systematic scan of 8 candidates Author: Léon Fernando Vlegels © Léon Fernando Vlegels. MIT License. """ import sys from mpmath import mp, mpf, pi, exp, log, sqrt, matrix, eig, fabs, nstr, acos mp.dps = 60 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}") # ── Constants ────────────────────────────────────────────────────────────── ALPHA_INV = 4*pi**3 + pi**2 + pi # α⁻¹ = μ₀ ≈ 137.036 MU_0 = ALPHA_INV MU_1 = mpf(16)*pi**3/5 + 3*pi**2/4 + 2*pi/3 # μ₁ ≈ 108.717 MU_A36 = MU_1 / MU_0 # ≈ 0.79334 (Addendum-36 mass lever) # Sector energies (Addendum 36, Proposition 2.1) E_P = pi # Plateau / electron / S¹ E_W = pi**2 # Weld / muon / S³ E_F = 4*pi**3 # Fork / tau / B⁴ dE_PW = E_W - E_P # π² − π ≈ 6.728 # Off-diagonal coupling from 4-simplex inner product magnitude g = mpf('1') / 4 # |⟨vᵢ, vⱼ⟩| = 1/4 for all i≠j # Natural "time" scale: MU such that exp(MU·dE_PW) = 207 exactly MU = log(mpf(207)) / dE_PW # ≈ 0.79261 # CODATA CODATA = mpf("206.7682830") CODATA_tau = mpf("3477.23") # m_τ/m_e (PDG 2022) CODATA_tau_e = mpf("3477.23") CODATA_tmu = mpf("16.817") # m_τ/m_μ SEP = "=" * 72 RULE = "-" * 72 def rel_gap(x, ref=None): ref = ref or CODATA return fabs(x - ref) / ref def pct(x, ref=None): return float(rel_gap(x, ref)) * 100 # ── Matrix exponential via eigendecomposition ───────────────────────────── def mat_exp_tau(H, tau): """ Compute exp(tau * H) for a 3×3 mpmath matrix H using eigenvalue decomposition H = V @ diag(λ) @ V⁻¹. Returns (P, eigenvalues, eigenvectors). """ # eig returns (eigenvalues, left-eigenvectors) for mpmath # We need right eigenvectors: H @ v = λ v # mpmath.eig returns right eigenvectors as columns of the second return vals, vecs = eig(H) # vecs is a matrix whose columns are eigenvectors # Build diagonal of exp(tau * λ) n = len(vals) D = matrix(n, n) for i in range(n): D[i, i] = exp(tau * vals[i]) # P = vecs @ D @ vecs^{-1} from mpmath import inverse V_inv = inverse(vecs) P = vecs * D * V_inv return P, vals, vecs # ───────────────────────────────────────────────────────────────────────────── print(SEP) print("verify_P208.py — Dual Observer Triangular Path and the Muon Mass Ratio") print(f"mpmath dps = {mp.dps}") print(SEP) print("\nFUNDAMENTAL CONSTANTS") print(f" E_P (Plateau, electron, S¹) = π = {nstr(E_P, 15)}") print(f" E_W (Weld, muon, S³) = π² = {nstr(E_W, 15)}") print(f" E_F (Fork, tau, B⁴) = 4π³ = {nstr(E_F, 15)}") print(f" g (4-simplex coupling) = 1/4 = {nstr(g, 15)}") print(f" MU = ln(207)/(π²−π) = {nstr(MU, 15)}") print(f" MU_A36 = μ₁/μ₀ = {nstr(MU_A36, 15)}") print(f" CODATA m_μ/m_e = {nstr(CODATA, 12)}") print(f" CODATA m_τ/m_e = {nstr(CODATA_tau, 9)}") print() # ───────────────────────────────────────────────────────────────────────────── # Baseline: direct edge formula (Addendum 36) baseline = exp(MU * dE_PW) print(f"BASELINE exp(MU·(E_W−E_P)) = {nstr(baseline, 12)} gap = {pct(baseline):.4f}%") print() # ───────────────────────────────────────────────────────────────────────────── # ── TASK 1: Second-order perturbative correction ─────────────────────────── print(SEP) print("TASK 1: Second-order perturbative correction via virtual Fork") print(RULE) # H_{PF} = g*(E_F - E_P), H_{FW} = g*(E_F - E_W) H_PF = g * (E_F - E_P) H_FW = g * (E_F - E_W) # Energy denominator: E_F - (E_P + E_W)/2 E_denom = E_F - (E_P + E_W) / 2 delta_H_PW = H_PF * H_FW / E_denom print(f" H_PF = g·(E_F−E_P) = {nstr(H_PF, 15)}") print(f" H_FW = g·(E_F−E_W) = {nstr(H_FW, 15)}") print(f" E_denom = E_F − (E_P+E_W)/2 = {nstr(E_denom, 15)}") print(f" δH_PW = H_PF·H_FW/E_denom = {nstr(delta_H_PW, 15)}") print() R_corrected_t1 = exp(MU * (dE_PW + delta_H_PW)) print(f" R_corrected = exp(MU·(ΔE_PW + δH_PW))") print(f" = exp(MU·{nstr(dE_PW + delta_H_PW, 12)})") print(f" = {nstr(R_corrected_t1, 12)}") print(f" gap from CODATA: {pct(R_corrected_t1):.4f}%") direction = "BETTER" if rel_gap(R_corrected_t1) < rel_gap(baseline) else "WORSE" print(f" vs baseline (0.60%): {direction}") print() # ───────────────────────────────────────────────────────────────────────────── # ── TASK 2: Full transfer matrix propagator ──────────────────────────────── print(SEP) print("TASK 2: Full transfer matrix propagator exp(τ·H)") print(RULE) # Construct H (energy-gap coupling) # H[i,j] for off-diagonals = g*(E_j - E_i) if j>i, else g*(E_i - E_j)... # Actually: H[P,W] = g*(E_W - E_P), H[P,F] = g*(E_F - E_P), H[W,F] = g*(E_F - E_W) # Symmetric; diagonals are the energies H = matrix([ [E_P, g*(E_W - E_P), g*(E_F - E_P)], [g*(E_W - E_P), E_W, g*(E_F - E_W)], [g*(E_F - E_P), g*(E_F - E_W), E_F ] ]) print(f"\n H (energy-gap coupling, rows/cols: P=0,W=1,F=2):") for i in range(3): row = [nstr(H[i,j], 8) for j in range(3)] print(f" [{', '.join(row)}]") # H2: flat coupling (adjacency model) H2 = matrix([ [E_P, g, g ], [g, E_W, g ], [g, g, E_F] ]) print(f"\n H2 (flat coupling, g on all off-diagonals):") for i in range(3): row = [nstr(H2[i,j], 8) for j in range(3)] print(f" [{', '.join(row)}]") # Try tau = MU, tau = 1, tau = pi for tau_label, tau_val in [("MU=ln(207)/ΔE", MU), ("1", mpf(1)), ("π", pi)]: print(f"\n --- τ = {tau_label} = {nstr(tau_val, 10)} ---") for Hlabel, Hmat in [("H (gap coupling)", H), ("H2 (flat coupling)", H2)]: P, vals, vecs = mat_exp_tau(Hmat, tau_val) P00 = P[0,0].real P01 = P[0,1].real P10 = P[1,0].real P11 = P[1,1].real ratio_01_00 = P01 / P00 if abs(P00) > 0 else mpf(0) print(f" {Hlabel}:") print(f" P[0,1] (Plateau→Weld) = {nstr(P01, 10)}") print(f" P[0,0] = {nstr(P00, 10)}") print(f" P[0,1]/P[0,0] = {nstr(ratio_01_00, 10)}") # Check if |P[0,1]| is close to CODATA for label2, val2 in [("P[0,1]", P01), ("P[0,1]/P[0,0]", ratio_01_00), ("|P[0,1]|", fabs(P01))]: if val2 > 0: g2 = pct(val2) print(f" {label2:20s} = {nstr(val2,10):18s} gap={g2:.3f}%") # ───────────────────────────────────────────────────────────────────────────── # ── TASK 3: Eigenvalue mass ratios ──────────────────────────────────────── print() print(SEP) print("TASK 3: Eigenvalue mass ratios") print(RULE) for Hlabel, Hmat in [("H (gap coupling)", H), ("H2 (flat coupling)", H2)]: vals, vecs = eig(Hmat) lam = sorted([v.real for v in vals]) lam1, lam2, lam3 = lam[0], lam[1], lam[2] print(f"\n {Hlabel}:") print(f" λ₁ = {nstr(lam1, 15)}") print(f" λ₂ = {nstr(lam2, 15)}") print(f" λ₃ = {nstr(lam3, 15)}") r21 = lam2 / lam1 if lam1 != 0 else mpf(0) r31 = lam3 / lam1 if lam1 != 0 else mpf(0) r32 = lam3 / lam2 if lam2 != 0 else mpf(0) print(f" λ₂/λ₁ = {nstr(r21, 12)} (CODATA m_μ/m_e = 206.768, gap = {pct(r21):.3f}%)") print(f" λ₃/λ₁ = {nstr(r31, 12)} (CODATA m_τ/m_e = 3477.23, gap = {pct(r31, CODATA_tau):.3f}%)") print(f" λ₃/λ₂ = {nstr(r32, 12)} (CODATA m_τ/m_μ = 16.817, gap = {pct(r32, CODATA_tmu):.3f}%)") # ───────────────────────────────────────────────────────────────────────────── # ── TASK 4: Dual observer amplitude ─────────────────────────────────────── print() print(SEP) print("TASK 4: Dual observer amplitude A(F→W→P)") print(RULE) # Two-step path amplitude: g*(E_F-E_W) × g*(E_W-E_P) amp = g**2 * (E_F - E_W) * (E_W - E_P) print(f" amp = g²·(E_F−E_W)·(E_W−E_P)") print(f" = (1/4)²·{nstr(E_F-E_W,10)}·{nstr(E_W-E_P,10)}") print(f" = {nstr(amp, 15)}") amp_norm = amp / dE_PW # = g²·(E_F−E_W) print(f" amp_norm = amp/(E_W−E_P) = g²·(E_F−E_W) = {nstr(amp_norm, 15)}") R_dual_obs = exp(MU * amp_norm) print(f" exp(MU·amp_norm) = {nstr(R_dual_obs, 12)} gap = {pct(R_dual_obs):.4f}%") # Also: exp(MU * amp) directly R_dual_obs2 = exp(MU * amp) print(f" exp(MU·amp) = {nstr(R_dual_obs2, 12)} gap = {pct(R_dual_obs2):.4f}%") # Path through the propagator: P[F,W]*P[W,P] with tau=MU P_mu, _, _ = mat_exp_tau(H, MU) path_amp = (P_mu[2,1] * P_mu[1,0]).real # F->W then W->P print(f"\n Propagator path A(F→W→P) = P[F,W]·P[W,P]") print(f" P[2,1] (F→W) = {nstr(P_mu[2,1].real, 10)}") print(f" P[1,0] (W→P) = {nstr(P_mu[1,0].real, 10)}") print(f" product = {nstr(path_amp, 12)} gap = {pct(path_amp):.4f}%") # ───────────────────────────────────────────────────────────────────────────── # ── TASK 5: Systematic scan of 8 candidates ─────────────────────────────── print() print(SEP) print("TASK 5: Systematic scan of 8 candidate mass ratios") print(RULE) alpha = mpf(1) / ALPHA_INV # Precompute eigenvalues for H vals_H, _ = eig(H) lam_H = sorted([v.real for v in vals_H]) lam1_H, lam2_H = lam_H[0], lam_H[1] candidates = [] # 1. Direct edge c1 = exp(MU * dE_PW) candidates.append(("exp(MU·(E_W−E_P)) [direct edge]", c1)) # 2. Second-order corrected c2 = exp(MU * (dE_PW + delta_H_PW)) candidates.append(("exp(MU·(ΔE_PW+δH_PW)) [2nd-order]", c2)) # 3. Full propagator element P[0,1] from exp(MU*H) P3, _, _ = mat_exp_tau(H, MU) c3 = P3[0,1].real candidates.append(("P[0,1] from exp(MU·H) [propagator]", c3)) # 4. P[0,1]/P[0,0] c4 = (P3[0,1] / P3[0,0]).real candidates.append(("P[0,1]/P[0,0] from exp(MU·H)", c4)) # 5. λ₂/λ₁ c5 = lam2_H / lam1_H candidates.append(("λ₂/λ₁ [eigenvalue ratio]", c5)) # 6. exp(MU·(E_W−E_P)·(1−g)) c6 = exp(MU * dE_PW * (1 - g)) candidates.append(("exp(MU·ΔE_PW·(1−g)) [coupling suppression]", c6)) # 7. exp(MU·(E_W−E_P)·(1 − g²·(E_F−E_W)/E_denom)) factor7 = 1 - g**2 * (E_F - E_W) / E_denom c7 = exp(MU * dE_PW * factor7) candidates.append(("exp(MU·ΔE_PW·(1−g²·(E_F−E_W)/E_denom)) [perturbative]", c7)) # 8. 207·(1 − g·α) c8 = mpf(207) * (1 - g * alpha) candidates.append(("207·(1−g·α) [simplex+fine-structure]", c8)) print(f"\n {'#':<2} {'Candidate':<52} {'Value':>16} {'Gap%':>9}") print(f" {'-'*2} {'-'*52} {'-'*16} {'-'*9}") for i, (label, val) in enumerate(candidates, 1): g_pct = pct(val) marker = " ◄ BEST" if i == 1 else "" print(f" {i:<2} {label:<52} {nstr(val,10):>16} {g_pct:>8.4f}%{marker}") # Find best best_idx = min(range(len(candidates)), key=lambda i: rel_gap(candidates[i][1])) best_label, best_val = candidates[best_idx] best_gap = pct(best_val) print() print(RULE) print("RESULTS SUMMARY") print(RULE) print(f" CODATA target: m_μ/m_e = {nstr(CODATA, 12)}") print(f" Baseline (A36): = {nstr(baseline, 12)} (gap {pct(baseline):.4f}%)") print() print(f" Best candidate (#{best_idx+1}): {best_label}") print(f" Value = {nstr(best_val, 12)}") print(f" Gap = {best_gap:.4f}%") print() # Re-sort all by gap ranked = sorted(enumerate(candidates, 1), key=lambda x: rel_gap(x[1][1])) print(" All candidates ranked by gap:") for rank, (orig_idx, (label, val)) in enumerate(ranked, 1): g_pct = pct(val) flag = " ◄ BEST" if rank == 1 else "" print(f" #{rank} (cand {orig_idx}): gap={g_pct:.4f}% val={nstr(val,10):>14} {label}{flag}") print() print(SEP) print("VERDICT") print(SEP) # Threshold: ESTABLISHED < 0.1%, PARTIAL < 0.5% (better than 0.60%), OPEN otherwise if best_gap < 0.1: status = "ESTABLISHED" msg = "Approach recovers CODATA to < 0.1%. Mechanism found." elif best_gap < 0.5: status = "PARTIAL" msg = f"Best approach achieves {best_gap:.4f}%, better than baseline 0.60%." elif best_gap < 0.60: status = "PARTIAL" msg = f"Best approach achieves {best_gap:.4f}%, marginally better than baseline 0.60%." else: status = "OPEN" msg = "No improvement over the direct edge formula." check(1, f"STATUS: {status} — {msg}", best_gap < 0.1) print(f" Best candidate: #{best_idx+1} value={nstr(best_val,12)} gap={best_gap:.4f}%") print(f" Baseline gap: 0.6024%") print(f"\n{'='*60}\nRESULT: {PASS} PASS / {FAIL} FAIL") sys.exit(0) # original convention: this scan script exits 0 regardless of verdict