""" verify_P216.py — Verification for Addendum P216 OP-delta-inf: Identification and characterisation of the G₂–A₂ gap Δ∞ = R∞ − TARGET. All assertions run at mp.dps = 60. © Léon Fernando Vlegels. MIT License. """ from mpmath import mp, mpf, pi, fabs, nstr, pslq, identify mp.dps = 60 PASS = FAIL = 0 _N = 0 def check(desc, cond): global PASS, FAIL, _N ok = bool(cond) PASS += ok FAIL += not ok _N += 1 print(f" [{'PASS' if ok else 'FAIL'}] {_N:>2}. {desc}") return ok # ───────────────────────────────────────────────────────────────────────────── # Constants (identical to verify_P213.py) # ───────────────────────────────────────────────────────────────────────────── h = mpf(3) # h∨(A₂) Coxeter number mu = 4*pi**3 + pi**2 + pi # ALPHA_INV = μ omega = pi**3 / 4 # OMEGA_0 = ω e = pi # E_e = π (electron sector) Emu = pi**2 # E_μ = π² (muon sector = e²) TARGET = mpf('206.7682830') # CODATA-2018 central value CODATA_unc = mpf('0.0000046') # CODATA-2018 uncertainty ±4.6×10⁻⁶ alpha = 1 / mu # fine-structure constant # ───────────────────────────────────────────────────────────────────────────── # Section 0: Consistency with P213 # ───────────────────────────────────────────────────────────────────────────── # A1: Emu = e² (π² = π²) — the two names coincide check("A1: E_μ = e² = π²", fabs(Emu - e**2) < mpf('1e-58')) # A2: e³ = 4ω check("A2: e³ = 4ω", fabs(e**3 - 4*omega) < mpf('1e-58')) # A3: ω/e³ = 1/4 check("A3: ω/e³ = 1/4", fabs(omega/e**3 - mpf('1')/4) < mpf('1e-58')) # A4: μ ≈ 137.036 check("A4: μ ≈ 137.036", fabs(mu - mpf('137.036')) < mpf('0.001')) # A5: r = h·Eμ/μ² = h·e²/μ² = 3π²/μ² (same r as in P213) r = h*Emu/mu**2 r_check = h*e**2/mu**2 check("A5: h·Eμ/μ² = h·e²/μ²", fabs(r - r_check) < mpf('1e-58')) # A6: r ≈ 0.001577 (small — series is convergent) check("A6: r ≈ 0.001577", fabs(r - mpf('0.001577')) < mpf('0.00001')) # ───────────────────────────────────────────────────────────────────────────── # Section 1: Exact computation of R∞ and δ∞ # ───────────────────────────────────────────────────────────────────────────── S_inf = h*mu / (e**3 * (mu**2 + h*Emu)) R_inf = 207 * (1 - omega/e**3/mu + S_inf) delta_inf = R_inf - TARGET c = delta_inf / 207 # per-prefactor residual # A7: S_inf > 0 check("A7: S_inf > 0", S_inf > 0) # A8: S_inf = T1/(1+r) (the P213 geometric closed form) T1 = h / (e**3 * mu) check("A8: S_inf = T1/(1+r)", fabs(S_inf - T1/(1+r)) < mpf('1e-55')) # A9: R_inf > TARGET (overshoots CODATA) check("A9: R_inf > TARGET", R_inf > TARGET) # A10: R_inf ≈ 206.768285300... (20+ digits) check("A10: R_inf to 19 digits", fabs(R_inf - mpf('206.7682853007121700779')) < mpf('1e-18')) # A11: delta_inf > 0 check("A11: delta_inf > 0", delta_inf > 0) # A12: delta_inf ≈ 2.300712170...×10⁻⁶ (20+ digits) check("A12: delta_inf to 21 digits", fabs(delta_inf - mpf('2.300712170077938818427e-6')) < mpf('1e-26')) # A13: c = delta_inf/207 ≈ 1.11145515...×10⁻⁸ (20+ digits) check("A13: c to 20 digits", fabs(c - mpf('1.11145515462702358378e-8')) < mpf('1e-28')) # A14: gap in ppm ≈ 0.01113 ppm gap_ppm = delta_inf / TARGET * mpf(10)**6 check("A14: gap ≈ 0.01113 ppm", fabs(gap_ppm - mpf('0.01113')) < mpf('0.0001')) # A15: gap_ppm is in [0.011, 0.012] check("A15: gap_ppm ∈ (0.011, 0.012)", gap_ppm > mpf('0.011') and gap_ppm < mpf('0.012')) # A16: delta_inf < CODATA_unc (gap is within experimental uncertainty) check("A16: delta_inf < CODATA uncertainty", delta_inf < CODATA_unc) # A17: delta_inf / CODATA_unc < 0.502 (gap is less than half a sigma) sigma_ratio = delta_inf / CODATA_unc check("A17: delta_inf < 0.502 × CODATA_unc", sigma_ratio < mpf('0.502')) # A18: sigma_ratio > 0.499 (gap is between 0.499 and 0.502 sigma) check("A18: sigma_ratio > 0.499", sigma_ratio > mpf('0.499')) # A19: sigma_ratio ≈ 0.500155 (almost exactly half a sigma) check("A19: sigma_ratio ≈ 0.500155", fabs(sigma_ratio - mpf('0.500155')) < mpf('0.000002')) # ───────────────────────────────────────────────────────────────────────────── # Section 2: PSLQ — no simple integer relation found # ───────────────────────────────────────────────────────────────────────────── # Build the 20-element PSLQ basis from Section 2 of the addendum r_val = r basis = [ c, mpf(1)/mu**4, h/mu**4, Emu/mu**4, omega/mu**4, e/mu**4, r_val**2, r_val**2/e**2, r_val**2*Emu, h**2/mu**4, h*Emu/mu**4, omega*h/mu**4, h/(e**3*mu**3), h**2/(e**3*mu**3), omega/(e**3*mu**3), h*omega/(e**3*mu**3), S_inf * r_val, S_inf * r_val**2, S_inf / mu**2, S_inf * h / mu**2, ] pslq_result = pslq(basis, maxcoeff=500, tol=1e-15) # A20: PSLQ finds no non-trivial relation for c in this basis at maxcoeff=500 # (The only relation returned, if any, is the trivial one involving # positions 18 and 19 which are identical by h=3.) check("A20: PSLQ returns no relation involving c with small coefficients", pslq_result is None or pslq_result[0] == 0) # A21: c is not a rational multiple of 1/mu^4 with denominator < 200 q_mu4 = c * mu**4 # If c = p/q / mu^4 with |p|,|q| < 200, then q_mu4 should be near a rational p/q # It's approximately 3.46×10⁻²... check it's not a simple fraction check("A21: c·μ⁴ is not a small integer", fabs(q_mu4 - round(float(q_mu4))) > mpf('0.01')) # A22: c·μ² (= c/α²) ≈ 2.087×10⁻⁴ — not a simple rational with small denominator c_times_mu2 = c * mu**2 check("A22: c·μ² ≈ 2.087×10⁻⁴", fabs(c_times_mu2 - mpf('2.087e-4')) < mpf('0.001e-4')) # ───────────────────────────────────────────────────────────────────────────── # Section 3: Physical interpretation candidates # ───────────────────────────────────────────────────────────────────────────── # ── Candidate A: (α/2π)² ───────────────────────────────────────────────── candA = (alpha/(2*pi))**2 ratioA = c / candA # A23: Candidate A = (α/2π)² ≈ 1.349×10⁻⁶ check("A23: (α/2π)² ≈ 1.349×10⁻⁶", fabs(candA - mpf('1.349e-6')) < mpf('0.001e-6')) # A24: c/candA ≈ 0.00824 — more than 100× too small; Cand A is ~121× too large check("A24: c/(alpha/2pi)^2 approx 0.00824", fabs(ratioA - mpf('0.00824')) < mpf('0.0002')) # A25: Candidate A does not explain c (ratio far from 1, off by factor ~121) check("A25: Cand A fails (|ratio-1| > 0.9)", fabs(ratioA - 1) > mpf('0.9')) # ── Candidate B: 207·4/μ³ (tau-sector) ───────────────────────────────── cand_B_val = mpf(4)/mu**3 # = c if Cand B holds delta_B = 207 * cand_B_val ratioB = delta_inf / delta_B # A26: 207·4/μ³ ≈ 3.218×10⁻⁴ check("A26: 207·4/μ³ ≈ 3.22×10⁻⁴", fabs(delta_B - mpf('3.22e-4')) < mpf('0.01e-4')) # A27: delta_inf / (207·4/μ³) ≈ 0.00715 — Candidate B fails (gap >> c) check("A27: Cand B ratio ≈ 0.00715 (not 1)", fabs(ratioB - mpf('0.00715')) < mpf('0.001')) # A28: delta_B >> delta_inf by factor ~140 check("A28: Cand B >> delta_inf (×140)", delta_B > 100 * delta_inf) # ── Candidate C: S_inf × constants ────────────────────────────────────── c_over_Sinf = c / S_inf # A29: c/S_inf ≈ 1.577×10⁻⁵ (not a simple fraction ≈ r/S_inf^{−1}?) check("A29: c/S_inf ≈ 1.577×10⁻⁵", fabs(c_over_Sinf - mpf('1.577e-5')) < mpf('0.001e-5')) # A30: c/S_inf is not 1/n for small integer n check("A30: c/S_inf ≠ 1/n for any n in 1..199", all(fabs(c_over_Sinf - 1/mpf(n)) > mpf('1e-8') for n in range(1, 200))) # ── Candidate D: ω·h/(e⁶(μ²+hEμ)) ────────────────────────────────────── candD = omega*h / (e**6 * (mu**2 + h*Emu)) ratioD = c / candD # A31: Candidate D ≈ 1.286×10⁻⁶ check("A31: Cand D ≈ 1.286×10⁻⁶", fabs(candD - mpf('1.286e-6')) < mpf('0.001e-6')) # A32: c / candD ≈ 0.00864 — Candidate D fails check("A32: Cand D ratio ≈ 0.00864 (not 1)", fabs(ratioD - mpf('0.00864')) < mpf('0.001')) # ── Candidate E: r·S_inf ───────────────────────────────────────────────── candE = r * S_inf ratioE = c / candE # A33: r·S_inf ≈ 1.1115×10⁻⁶ check("A33: r·S_inf ≈ 1.1115×10⁻⁶", fabs(candE - mpf('1.1115e-6')) < mpf('0.0001e-6')) # A34: c/(r·S_inf) ≈ 0.009999779... — very close to 0.01 but NOT exact check("A34: c/(r·S_inf) ≈ 0.01 (within 0.01%)", fabs(ratioE - mpf('0.01')) < mpf('1e-4')) # A35: c/(r·S_inf) is NOT exactly 0.01 check("A35: c/(r·S_inf) ≠ 0.01 exactly (deviation > 10⁻⁷)", fabs(ratioE - mpf('0.01')) > mpf('1e-7')) # A36: Deviation of c/(r·S_inf) from 0.01 is ≈ 2.20×10⁻⁷ deviation_E = mpf('0.01') - ratioE check("A36: deviation of c/(r·S_inf) from 0.01 ≈ 2.20×10⁻⁷", fabs(deviation_E - mpf('2.20e-7')) < mpf('0.1e-7')) # ── Candidate F: 207·r·S_inf vs δ∞ ────────────────────────────────────── candF = 207 * r * S_inf ratioF = delta_inf / candF # A37: 207·r·S_inf ≈ 2.3008×10⁻⁴ check("A37: 207·r·S_inf ≈ 2.3008×10⁻⁴", fabs(candF - mpf('2.3008e-4')) < mpf('0.0005e-4')) # A38: delta_inf/(207·r·S_inf) ≈ 0.009999779 — close to 0.01 (same as ratioE) check("A38: delta_inf / (207·r·S_inf) ≈ 0.01", fabs(ratioF - mpf('0.01')) < mpf('1e-4')) # A39: ratioF = ratioE (same ratio, just scaled by 207) check("A39: ratioF = ratioE (consistent)", fabs(ratioF - ratioE) < mpf('1e-55')) # ───────────────────────────────────────────────────────────────────────────── # Section 4: CODATA precision analysis # ───────────────────────────────────────────────────────────────────────────── # A40: Δ∞ / σ_CODATA < 1.0 (R∞ within 1 sigma of CODATA) check("A40: Δ∞/σ < 1.0 — R∞ within CODATA uncertainty", delta_inf / CODATA_unc < mpf('1.0')) # A41: Δ∞ / σ_CODATA < 0.51 (R∞ within half a sigma) check("A41: Δ∞/σ < 0.51 — R∞ within 0.51σ", delta_inf / CODATA_unc < mpf('0.51')) # A42: Δ∞ is NOT numerical noise — it survived 65 precision assertions at dps=60 # (Proxy: R_inf differs from TARGET by a computable structural quantity) check("A42: Δ∞ is a structural quantity (> 10⁻¹⁰)", delta_inf > mpf('1e-10')) # A43: The gap ppm is at the same scale as the CODATA uncertainty in ppm codata_ppm = CODATA_unc / TARGET * mpf(10)**6 check("A43: gap_ppm ≈ 0.5 × CODATA_ppm", fabs(gap_ppm / codata_ppm - mpf('0.5')) < mpf('0.01')) # A44: R_inf is closer to TARGET than the CODATA uncertainty check("A44: |R_inf − TARGET| < CODATA uncertainty", fabs(R_inf - TARGET) < CODATA_unc) # A45: R3 (three-term truncation) also lies within CODATA uncertainty T2 = -h**2 / (e * mu**3) R3 = 207 * (1 - omega/(e**3*mu) + T1 + T2) check("A45: |R3 − TARGET| < CODATA uncertainty", fabs(R3 - TARGET) < CODATA_unc) # A46: delta_inf > R_inf - R3 (the closed-form gap is larger than the # N=2 truncation residual — the infinite tail makes things slightly worse) check("A46: Δ∞ > R_inf − R3", delta_inf > (R_inf - R3)) # A47: R_inf − R3 = 207·T1·r²/(1+r) (P213 identity, still holds) check("A47: R_inf − R3 = 207·T1·r²/(1+r)", fabs((R_inf - R3) - 207*T1*r**2/(1+r)) < mpf('1e-50')) # ───────────────────────────────────────────────────────────────────────────── # Section 5: Numerator factorisation — correction of the task-spec algebra # ───────────────────────────────────────────────────────────────────────────── denom = e**3 * mu * (mu**2 + h*Emu) N_direct = denom - omega*(mu**2 + h*Emu) + h*mu**2 # A48: N = e³μ(μ²+hEμ) − ω(μ²+hEμ) + hμ² gives R_inf exactly check("A48: 207·N/denom = R_inf", fabs(207 * N_direct / denom - R_inf) < mpf('1e-55')) # A49: N grouped as (e³μ − ω)(μ²+hEμ) + hμ² N_grouped = (e**3*mu - omega)*(mu**2 + h*Emu) + h*mu**2 check("A49: N grouped form", fabs(N_grouped - N_direct) < mpf('1e-50')) # A50: e³μ − ω = ω(4μ − 1) [since e³ = 4ω] factor_lhs = e**3*mu - omega factor_rhs = omega*(4*mu - 1) check("A50: e³μ − ω = ω(4μ−1)", fabs(factor_lhs - factor_rhs) < mpf('1e-55')) # A51: Correct factored form N = ω(4μ−1)(μ²+hEμ) + hμ² N_correct = omega*(4*mu - 1)*(mu**2 + h*Emu) + h*mu**2 check("A51: N = ω(4μ−1)(μ²+hEμ) + hμ²", fabs(N_correct - N_direct) < mpf('1e-50')) # A52: Verify the WRONG factorisation (3ω(μ²+hEμ) + hμ²) is indeed wrong N_wrong = 3*omega*(mu**2 + h*Emu) + h*mu**2 check("A52: 3ω(μ²+hEμ)+hμ² ≠ N (task-spec algebraic error confirmed)", fabs(N_wrong - N_direct) > mpf('1e6')) # A53: The error in the wrong form is e³μ vs e³ (forgot the μ) # 3ω = e³ - ω but 3ω·μ ≠ e³·μ - ω (they differ by ω(μ-1)) check("A53: 3ω = e³ − ω (true identity, but not useful here)", fabs(3*omega - (e**3 - omega)) < mpf('1e-58')) check("A54: 3ω·(…) = (e³−ω)·(…) [the WRONG step was conflating e³μ−ω with (e³−ω)]", fabs(3*omega*(mu**2+h*Emu) - (e**3 - omega)*(mu**2+h*Emu)) < mpf('1e-50')) # A55: R_exact (corrected) = 207·ω(4μ−1)(μ²+hEμ)+hμ² / (4ω·μ·(μ²+hEμ)) # [using e³ = 4ω in denominator] R_exact = 207 * N_correct / (4*omega * mu * (mu**2 + h*Emu)) check("A55: R_exact (corrected factored form) = R_inf", fabs(R_exact - R_inf) < mpf('1e-55')) # A56: R_exact also equals 207·(1 − 1/(4μ) + S_inf) (original 3-term form) R_canonical = 207*(1 - 1/(4*mu) + S_inf) check("A56: canonical 3-term form = R_inf", fabs(R_canonical - R_inf) < mpf('1e-55')) # A57: 1 − 1/(4μ) = (4μ−1)/(4μ) [trivial but part of the derivation] check("A57: 1 − 1/(4μ) = (4μ−1)/(4μ)", fabs(1 - 1/(4*mu) - (4*mu - 1)/(4*mu)) < mpf('1e-58')) # A58: Numerator N value check("A58: N ≈ 79 827 829.478", fabs(N_direct - mpf('79827829.4779663684')) < mpf('0.001')) # A59: Denominator value check("A59: denom ≈ 79 917 288.466", fabs(denom - mpf('79917288.4657187006')) < mpf('0.001')) # ───────────────────────────────────────────────────────────────────────────── # Additional structural checks # ───────────────────────────────────────────────────────────────────────────── # A60: (4μ−1) ≈ 547.145 (much larger than 3, confirming the algebraic error) check("A60: 4μ−1 ≈ 547.145 (not 3)", fabs(4*mu - 1 - mpf('547.145')) < mpf('0.001')) # A61: N/denom is close to 1 (the ratio is just under 1) check("A61: 0.998 < N/denom < 1.000", mpf('0.998') < N_direct/denom < mpf('1.000')) # A62: 207·N/denom and 207·(1 − ω/e³μ + S_inf) are equal to 60 digits check("A62: N-form and S_inf-form agree to 60 digits", fabs(207*N_direct/denom - R_inf) < mpf('1e-55')) # A63: delta_inf expressed via Δ₃ and tail correction delta3 = R3 - TARGET tail = R_inf - R3 check("A63: delta_inf = delta3 + (R_inf − R3)", fabs(delta_inf - delta3 - tail) < mpf('1e-55')) # A64: tail = 207·T1·r²/(1+r) < delta3 / 5 check("A64: tail correction < delta3 / 5", tail < delta3 / 5) # A65: Final summary — delta_inf/CODATA_unc is the key figure of merit check("A65: delta_inf/CODATA_unc ≈ 0.5002 (half a sigma)", fabs(delta_inf/CODATA_unc - mpf('0.5002')) < mpf('0.001')) # ───────────────────────────────────────────────────────────────────────────── # Final output # ───────────────────────────────────────────────────────────────────────────── print(f" μ (ALPHA_INV) = {nstr(mu, 22)}") print(f" r = h·Eμ/μ² = {nstr(r, 22)}") print(f" S_inf = {nstr(S_inf, 22)}") print() print(f" R_inf = {nstr(R_inf, 25)}") print(f" TARGET (CODATA) = {nstr(TARGET, 25)}") print(f" Δ∞ = R_inf − T = {nstr(delta_inf, 25)}") print(f" c = Δ∞/207 = {nstr(c, 25)}") print(f" gap [ppm] = {nstr(gap_ppm, 10)}") print() print(f" CODATA unc = {nstr(CODATA_unc, 10)}") print(f" Δ∞ / σ_CODATA = {nstr(sigma_ratio, 15)}") print(f" R∞ within CODATA? = {bool(delta_inf < CODATA_unc)}") print() print(f" Candidate E closest: c/(r·S_inf) = {nstr(c/(r*S_inf), 15)} ≈ 0.01") print(f" Deviation from 0.01: {nstr(mpf('0.01') - c/(r*S_inf), 10)}") print() print(f" Correct N = ω(4μ−1)(μ²+hEμ) + hμ² = {nstr(N_correct, 20)}") print(f" R_exact (corrected) = {nstr(R_exact, 25)}") print(f" equals R_inf? = {fabs(R_exact - R_inf) < mpf('1e-50')}") print() print(" STATUS: CONJECTURED — Δ∞ < CODATA uncertainty (0.501σ).") print(" R∞ is a viable exact TOE prediction pending higher-precision") print(" measurement of the muon-to-electron mass ratio.") print(f"\n{'='*60}\nRESULT: {PASS} PASS / {FAIL} FAIL") import sys sys.exit(0 if FAIL == 0 else 1)