""" verify_P215.py — Verification for Addendum P215 OP-G2A2-lattice: Lattice Path Derivation of T_k from the A₂ Lepton Face. Sections verified: §1 T_k formula for k=1..5 §2 Ratios T_{k+1}/T_k = −r for k=1..4 §3 A₂ adjacency matrix Trace(A^{2k}) for k=1,2,3 §4 Seed factorisation T_k = (−1)^{k+1} · r^k · σ §5 Resolvent comparison (mismatch documented) §6 Partial sums converging to R∞ from P213 §7 Sanity checks on constants and loop-weight All assertions run at mp.dps = 60. © Léon Fernando Vlegels. MIT License. """ from mpmath import mp, mpf, pi, fabs, nstr, sqrt as mpsqrt import sys mp.dps = 60 PASS = FAIL = 0 _N = 0 def check(desc, cond): global PASS, FAIL, _N _N += 1 ok = bool(cond) PASS += ok FAIL += (not ok) print(f" [{'PASS' if ok else 'FAIL'}] {_N:>2}. {desc}") return ok # ───────────────────────────────────────────────────────────────────────────── # Constants # ───────────────────────────────────────────────────────────────────────────── ALPHA_INV = 4*pi**3 + pi**2 + pi # μ = ALPHA_INV ≈ 137.036 OMEGA_0 = pi**3 / 4 # ω = OMEGA_0 E_e = pi # e = electron sector energy = π E_mu = pi**2 # E_μ = e² = muon energy insertion TARGET = mpf('206.7682830') # CODATA-2018 muon/electron mass ratio h = mpf(3) # h∨(A₂) = 3 alpha = 1 / ALPHA_INV # fine-structure constant # Seed and loop weight sigma = ALPHA_INV / E_e**5 # σ = μ/e⁵ (bare propagator seed) r = h * E_e**2 / ALPHA_INV**2 # r = h·e²/μ² (one-loop weight) # T₀ correction (candidate-8 base) T0 = -OMEGA_0 / (E_e**3 * ALPHA_INV) # ───────────────────────────────────────────────────────────────────────────── # §1: Constants sanity # ───────────────────────────────────────────────────────────────────────────── # A1: h∨(A₂) = 3 check("A1: h∨(A₂) = 3", int(h) == 3) # A2: μ ≈ 137.036 check("A2: μ ≈ 137.036", fabs(ALPHA_INV - mpf('137.036')) < mpf('0.001')) # A3: ω = π³/4 check("A3: ω = π³/4", fabs(OMEGA_0 - E_e**3/4) < mpf('1e-58')) # A4: μ = 4e³+e²+e check("A4: μ = 4e³+e²+e", fabs(ALPHA_INV - (4*E_e**3 + E_e**2 + E_e)) < mpf('1e-58')) # A5: E_μ = e² check("A5: E_μ = e²", fabs(E_mu - E_e**2) < mpf('1e-58')) # A6: r = h·E_μ/μ² = h·e²/μ² check("A6: r = h·E_μ/μ²", fabs(r - h*E_mu/ALPHA_INV**2) < mpf('1e-58')) # A7: r ≈ 0.001577 check("A7: r ≈ 0.001577", fabs(r - mpf('0.001577')) < mpf('0.00001')) # A8: σ = μ/e⁵ = 4/e²+1/e³+1/e⁴ sigma_check = 4/E_e**2 + 1/E_e**3 + 1/E_e**4 check("A8: σ = 4/e²+1/e³+1/e⁴", fabs(sigma - sigma_check) < mpf('1e-58')) # A9: r = h∨(A₂)·α²·e² check("A9: r = h·α²·e²", fabs(r - h*alpha**2*E_e**2) < mpf('1e-55')) # ───────────────────────────────────────────────────────────────────────────── # §2: T_k formula — direct definition # ───────────────────────────────────────────────────────────────────────────── def T_k_def(k): """T_k = (-1)^{k+1} · h^k · e^{2k-5} / μ^{2k-1}""" return ((-1)**(k+1)) * h**k * E_e**(2*k-5) / ALPHA_INV**(2*k-1) T1 = T_k_def(1) T2 = T_k_def(2) T3 = T_k_def(3) T4 = T_k_def(4) T5 = T_k_def(5) # A10: T1 = h/(e³μ) > 0 check("A10: T1 = h/(e³μ)", fabs(T1 - h/(E_e**3 * ALPHA_INV)) < mpf('1e-58')) check("A10b: T1 > 0", T1 > 0) # A11: T2 = −h²/(e·μ³) < 0 check("A11: T2 = −h²/(eμ³)", fabs(T2 - (-h**2/(E_e * ALPHA_INV**3))) < mpf('1e-58')) check("A11b: T2 < 0", T2 < 0) # A12: T3 = h³·e/μ⁵ > 0 check("A12: T3 = h³e/μ⁵", fabs(T3 - h**3*E_e/ALPHA_INV**5) < mpf('1e-58')) check("A12b: T3 > 0", T3 > 0) # A13: T4 = −h⁴·e³/μ⁷ < 0 check("A13: T4 = −h⁴e³/μ⁷", fabs(T4 - (-h**4*E_e**3/ALPHA_INV**7)) < mpf('1e-58')) check("A13b: T4 < 0", T4 < 0) # A14: T5 = h⁵·e⁵/μ⁹ > 0 check("A14: T5 = h⁵e⁵/μ⁹", fabs(T5 - h**5*E_e**5/ALPHA_INV**9) < mpf('1e-58')) check("A14b: T5 > 0", T5 > 0) # A15: magnitudes decrease monotonically check("A15: |T1|>|T2|>|T3|>|T4|>|T5|", fabs(T1) > fabs(T2) > fabs(T3) > fabs(T4) > fabs(T5)) # ───────────────────────────────────────────────────────────────────────────── # §3: Geometric ratio T_{k+1}/T_k = −r # ───────────────────────────────────────────────────────────────────────────── # A16: T2/T1 = −r check("A16: T2/T1 = −r", fabs(T2/T1 - (-r)) < mpf('1e-55')) # A17: T3/T2 = −r check("A17: T3/T2 = −r", fabs(T3/T2 - (-r)) < mpf('1e-55')) # A18: T4/T3 = −r check("A18: T4/T3 = −r", fabs(T4/T3 - (-r)) < mpf('1e-55')) # A19: T5/T4 = −r check("A19: T5/T4 = −r", fabs(T5/T4 - (-r)) < mpf('1e-55')) # A20: ratios are identical (constant common ratio) check("A20: T2/T1 = T3/T2", fabs(T2/T1 - T3/T2) < mpf('1e-60')) check("A21: T3/T2 = T4/T3", fabs(T3/T2 - T4/T3) < mpf('1e-60')) # ───────────────────────────────────────────────────────────────────────────── # §4: A₂ adjacency matrix — trace formula Tr(A^{2k}) = 4^k + 2 # ───────────────────────────────────────────────────────────────────────────── # A (K₃) has eigenvalues 2, -1, -1 # Tr(A^{2k}) = 2^{2k} + 2·(-1)^{2k} = 4^k + 2 # A22: k=1: Tr(A²) = 6 = 4¹+2 Tr_A2 = 4**1 + 2 check("A22: Tr(A²) = 6", Tr_A2 == 6) # A23: k=2: Tr(A⁴) = 18 = 4²+2 Tr_A4 = 4**2 + 2 check("A23: Tr(A⁴) = 18", Tr_A4 == 18) # A24: k=3: Tr(A⁶) = 66 = 4³+2 Tr_A6 = 4**3 + 2 check("A24: Tr(A⁶) = 66", Tr_A6 == 66) # A25: Tr(A²)/2 = 3 = h∨(A₂) ✓ (k=1 matches) check("A25: Tr(A²)/2 = h∨(A₂)", Tr_A2//2 == int(h)) # A26: Tr(A⁴)/2 = 9 = h∨(A₂)² ✓ (k=2 matches) check("A26: Tr(A⁴)/2 = h∨(A₂)²", Tr_A4//2 == int(h)**2) # A27: Tr(A⁶)/2 = 33 ≠ 27 = h∨(A₂)³ ✗ (k=3 fails) check("A27: Tr(A⁶)/2 = 33", Tr_A6//2 == 33) check("A28: 33 ≠ 27 = h∨(A₂)³ (pattern breaks at k=3)", 33 != int(h)**3) # A29: General formula: Tr(A^{2k}) = 4^k + 2 (verified for k=1,2,3,4,5) for k in range(1, 6): formula_val = 4**k + 2 # Verify by expansion: eigenvalues 2,-1,-1 → Tr = 2^{2k}+2·(-1)^{2k} = 4^k+2 eigen_trace = 2**(2*k) + 2*(-1)**(2*k) check(f"A29k{k}: eigenvalue formula at k={k}", formula_val == eigen_trace) # A30: Trace-half matches h^k only for k=1,2 for k in [1, 2]: check(f"A30: Tr(A^{{2k}})/2 = h^k at k={k}", (4**k + 2)//2 == 3**k) for k in [3, 4, 5]: check(f"A31: Tr(A^{{2k}})/2 ≠ h^k at k={k} (pattern breaks)", (4**k + 2)//2 != 3**k) # ───────────────────────────────────────────────────────────────────────────── # §5: Seed factorisation T_k = (−1)^{k+1} · r^k · σ # ───────────────────────────────────────────────────────────────────────────── def T_k_seed(k): """T_k = (−1)^{k+1} · r^k · σ (Lemma 3.1)""" return ((-1)**(k+1)) * r**k * sigma # A32: Seed form matches definition for k=1,2,3,4,5 for k in range(1, 6): T_def = T_k_def(k) T_seed = T_k_seed(k) check(f"A32k{k}: seed form = def at k={k}", fabs(T_def - T_seed) < mpf('1e-55')) # A33: σ = μ/e⁵ check("A33: σ = μ/e⁵", fabs(sigma - ALPHA_INV/E_e**5) < mpf('1e-58')) # A34: r = h·e²/μ² (r already verified; reconfirm link to seed) check("A34: r·σ = T1", fabs(r**1 * sigma - T1) < mpf('1e-58')) check("A35: r²·σ = -T2", fabs(r**2 * sigma - (-T2)) < mpf('1e-58')) # A36: T_k/σ = (−r)^k × (−1) ... actually (−1)^{k+1}·r^k, verified: for k in range(1, 5): ratio = T_k_def(k) / sigma expected = ((-1)**(k+1)) * r**k check(f"A36k{k}: T_k/σ = (−1)^{{k+1}}·r^k", fabs(ratio - expected) < mpf('1e-55')) # ───────────────────────────────────────────────────────────────────────────── # §6: Resolvent comparison (mismatch documented) # ───────────────────────────────────────────────────────────────────────────── # A₂ adjacency resolvent at λ = μ/e: # G(λ) = 1/(λ-2) + 2/(λ+1) = 3(λ-1)/((λ-2)(λ+1)) lam = ALPHA_INV / E_e G_lam = 1/(lam - 2) + 2/(lam + 1) # Closed-form sum S = T1/(1+r) = hμ/(e³(μ²+he²)) S_closed = h * ALPHA_INV / (E_e**3 * (ALPHA_INV**2 + h*E_e**2)) # A37: Resolvent at λ=μ/e is computed correctly G_check = 3*(lam - 1)/((lam - 2)*(lam + 1)) check("A37: resolvent formula consistent", fabs(G_lam - G_check) < mpf('1e-55')) # A38: G(μ/e) ≠ S_closed (mismatch documented) mismatch = fabs(G_lam - S_closed) check("A38: G(μ/e) ≠ S_closed (mismatch > 0.06)", mismatch > mpf('0.06')) # A39: Mismatch factor ≈ 97.7 (not a simple integer ratio) factor = G_lam / S_closed check("A39: mismatch factor ≈ 97.7", factor > mpf('90') and factor < mpf('105')) # A40: S_closed = σ·r/(1+r) exactly (Lemma + geometric series) S_seed_form = sigma * r / (1 + r) check("A40: S_closed = σ·r/(1+r)", fabs(S_closed - S_seed_form) < mpf('1e-55')) # A41: S_closed = T1/(1+r) (from geometric series) check("A41: S_closed = T1/(1+r)", fabs(S_closed - T1/(1+r)) < mpf('1e-55')) # ───────────────────────────────────────────────────────────────────────────── # §7: Partial sums converging to R∞ # ───────────────────────────────────────────────────────────────────────────── R_closed = 207 * (1 + T0 + S_closed) gap_closed = fabs(R_closed - TARGET)/TARGET * 1e6 # A42: R∞ computed from closed form check("A42: R∞ = 207(1+T0+T1/(1+r))", fabs(R_closed - 207*(1 + T0 + T1/(1+r))) < mpf('1e-50')) # A43: R∞ gap ≈ 0.01113 ppm check("A43: R∞ gap ≈ 0.01113 ppm", fabs(gap_closed - mpf('0.01113')) < mpf('0.0001')) # A44: R∞ > TARGET check("A44: R∞ > TARGET", R_closed > TARGET) # A45: Partial sum k=1 (R_star) S_partial1 = T1 R_star = 207*(1 + T0 + S_partial1) gap_star = fabs(R_star - TARGET)/TARGET * 1e6 check("A45: R_star gap ∈ [1.1, 1.2] ppm", gap_star > mpf('1.1') and gap_star < mpf('1.2')) # A46: Partial sum k=2 (R3) S_partial2 = T1 + T2 R3 = 207*(1 + T0 + S_partial2) gap_R3 = fabs(R3 - TARGET)/TARGET * 1e6 check("A46: R3 gap ≈ 0.009373 ppm", fabs(gap_R3 - mpf('0.009373')) < mpf('0.0001')) # A47: Partial sum k=3 increases gap (R3 closest to TARGET among partial sums) S_partial3 = T1 + T2 + T3 R4 = 207*(1 + T0 + S_partial3) gap_R4 = fabs(R4 - TARGET)/TARGET * 1e6 check("A47: k=3 partial sum gap > k=2 (R3 is optimal truncation)", gap_R4 > gap_R3) # A48: Convergence: by k=4 the sum is within 10⁻¹⁰ ppm of R∞ S_partial4 = T1 + T2 + T3 + T4 R5 = 207*(1 + T0 + S_partial4) gap_to_Rinf = fabs(R5 - R_closed) check("A48: partial sum k=4 within 10⁻⁹ of R∞", gap_to_Rinf < mpf('1e-9')) # A49: The series converges — |r| < 1 check("A49: r < 1 (series converges)", r < mpf('1')) check("A50: r < 0.002 (rapid convergence)", r < mpf('0.002')) # ───────────────────────────────────────────────────────────────────────────── # §8: Loop-weight structure checks # ───────────────────────────────────────────────────────────────────────────── # A51: r = h∨(A₂) · α² · E_μ (loop-weight decomposition) check("A51: r = h∨·α²·E_μ", fabs(r - h*alpha**2*E_mu) < mpf('1e-55')) # A52: T1 = r·σ (first loop = seed × loop-weight) check("A52: T1 = r·σ", fabs(T1 - r*sigma) < mpf('1e-58')) # A53: normalised ratio T_k·e³·μ/h = (−1)^{k+1}·(−r)^{k−1} for k in range(1, 5): lhs = T_k_def(k) * E_e**3 * ALPHA_INV / h rhs = ((-1)**(k+1)) * ((-r)**(k-1)) * ((-1)**(k-1)) # = (-1)^{k+1}·r^{k-1} # Actually: T_k·e³μ/h = (-1)^{k+1}·r^{k-1}·r·σ·e³μ/h ... simplify: # T_k = (-1)^{k+1}·r^k·σ; T_k·e³μ/h = (-1)^{k+1}·r^k·(μ/e⁵)·e³μ/h # = (-1)^{k+1}·r^k·μ²/(h·e²) = (-1)^{k+1}·r^{k-1}·(r·μ²/(h·e²)) # and r·μ²/(h·e²) = h·e²/μ²·μ²/(h·e²) = 1 simple_lhs = T_k_def(k) * E_e**3 * ALPHA_INV / h simple_rhs = ((-1)**(k+1)) * r**(k-1) check(f"A53k{k}: T_k·e³μ/h = (−1)^{{k+1}}·r^{{k−1}}", fabs(simple_lhs - simple_rhs) < mpf('1e-55')) # A54: The resolvent two-solution quadratic has no natural TOE solution # (just verify the mismatch is large — detailed root computation in §3) check("A54: resolvent mismatch is large (>0.015)", mismatch > mpf('0.015')) # ───────────────────────────────────────────────────────────────────────────── # Final report # ───────────────────────────────────────────────────────────────────────────── print() print(f" ALPHA_INV (μ) = {nstr(ALPHA_INV, 20)}") print(f" h = h∨(A₂) = {h}") print(f" r = h·e²/μ² = {nstr(r, 20)}") print(f" σ = μ/e⁵ = {nstr(sigma, 20)}") print() print(f" T1 = {nstr(T1, 15)}") print(f" T2 = {nstr(T2, 15)}") print(f" T3 = {nstr(T3, 15)}") print(f" T4 = {nstr(T4, 15)}") print(f" T5 = {nstr(T5, 15)}") print() print(f" Ratio T2/T1 = {nstr(T2/T1, 15)} (should be −r = {nstr(-r, 15)})") print(f" Ratio T3/T2 = {nstr(T3/T2, 15)} (should be −r)") print(f" Ratio T4/T3 = {nstr(T4/T3, 15)} (should be −r)") print() print(f" Tr(A²) = {4**1+2} = 4¹+2; /2 = {(4**1+2)//2} = h∨ ✓") print(f" Tr(A⁴) = {4**2+2} = 4²+2; /2 = {(4**2+2)//2} = h∨² ✓") print(f" Tr(A⁶) = {4**3+2} = 4³+2; /2 = {(4**3+2)//2} ≠ {3**3} = h∨³ ✗ (pattern breaks)") print() print(f" Resolvent G(μ/e) = {nstr(G_lam, 15)}") print(f" S_closed = {nstr(S_closed, 15)}") print(f" Mismatch factor = {nstr(G_lam/S_closed, 10)}") print(f" → Resolvent interpretation: MISMATCH (status: CONJECTURED)") print() print(f" R_star gap = {nstr(fabs(R_star-TARGET)/TARGET*1e6, 8)} ppm") print(f" R3 gap = {nstr(gap_R3, 8)} ppm") print(f" R4 gap = {nstr(gap_R4, 8)} ppm (k=3 term worsens)") print(f" R∞ gap = {nstr(gap_closed, 8)} ppm (closed form)") print() print("Programme status: CONJECTURED") print(" Loop-order factorisation T_k = (−1)^{k+1}·r^k·σ proved.") print(" h∨(A₂)^k from A₂ adjacency matrix holds for k=1,2 only.") print(" Full G₂ holonomy derivation remains open (OP-G2A2-lattice).") print(f"\n{'='*60}\nRESULT: {PASS} PASS / {FAIL} FAIL") sys.exit(0 if FAIL == 0 else 1)