""" verify_P245.py — α_s and Higgs vev from TOE moment hierarchy Addendum 245: Systematic search for closed forms for the strong coupling α_s(M_Z) and the Higgs vacuum expectation value correction δ = (μ₂−μ₃)−log(v/mₑ). Standard harness: mpmath dps=60. """ from mpmath import mp, mpf, pi, fabs, log, sqrt, quad, exp, nstr, identify, pslq, floor mp.dps = 60 PASS = 0; FAIL = 0 def check(name, condition, details=""): global PASS, FAIL ok = bool(condition) PASS += ok FAIL += (not ok) suffix = f" [{details}]" if (details and not ok) else "" print(f" [{'PASS' if ok else 'FAIL'}] {PASS + FAIL:>2}. {name}{suffix}") # ── Core constants ────────────────────────────────────────────────────────── def mu(n): """Raw n-th moment of ρ(x)=16π³x³+3π²x²+2πx. Closed form: μ_n = 16π³/(n+4) + 3π²/(n+3) + 2π/(n+2).""" return mpf(16)*pi**3/(n+4) + mpf(3)*pi**2/(n+3) + mpf(2)*pi/(n+2) Omega = mu(0) # = 4π³+π²+π = α⁻¹ ≈ 137.036 alpha = mpf(1)/Omega beta = mpf(3)*pi/20 # P17 spectral coefficient Omega0 = pi**3/4 # = Ω₀, kernel constant kappa = Omega/3 # ≈ 45.68 BREATH = pi*Omega # ≈ 430.5 # Experimental / PDG values alpha_s_MZ = mpf('0.1179') # strong coupling at M_Z pole (PDG 2022) me_GeV = mpf('0.51099895e-3')# electron mass in GeV MPl_GeV = mpf('1.22089e19') # Planck mass in GeV v_GeV = 1/sqrt(sqrt(mpf(2))*mpf('1.16638e-5')) # Higgs vev ~ 246.22 GeV GF = mpf('1.16638e-5') # Fermi constant GeV^-2 mtop = mpf('172.69') # top quark mass GeV mZ = mpf('91.1876') # Z boson mass GeV print("=" * 64) print(" A245 — α_s and Higgs vev Verifier (dps=60)") print("=" * 64) print() # ── Section 1: Moment values ───────────────────────────────────────────────── print("Section 1: Moment values and closed-form consistency") # C01: μ₀ = 4π³+π²+π = Ω = α⁻¹ Omega_cf = 4*pi**3 + pi**2 + pi check("C01: μ₀ = 4π³+π²+π (closed form exact)", fabs(mu(0) - Omega_cf) < mpf('1e-50'), f"diff={float(fabs(mu(0)-Omega_cf)):.2e}") # C02: μ₀·α = 1 exactly check("C02: μ₀·α = 1 (Ω·α = 1, EM calibration)", fabs(Omega*alpha - 1) < mpf('1e-50')) # C03: Closed form μ_n vs numerical quadrature for n=0..4 def rho(x): return 16*pi**3*x**3 + 3*pi**2*x**2 + 2*pi*x quad_ok = True for n in range(5): q = quad(lambda x: x**n * rho(x), [0, 1]) if fabs(mu(n) - q) > mpf('1e-38'): quad_ok = False break check("C03: closed form μ_n matches numerical quad for n=0..4", quad_ok) # C04: μ₂−μ₃ = 8π³/21 + π²/10 + π/10 (closed form) d23 = mu(2) - mu(3) d23_cf = 8*pi**3/21 + pi**2/10 + pi/10 check("C04: μ₂−μ₃ = 8π³/21+π²/10+π/10 (closed form)", fabs(d23 - d23_cf) < mpf('1e-40'), f"diff={float(fabs(d23-d23_cf)):.2e}") # C05: Moments strictly decreasing and positive for n=0..8 check("C05: μₙ strictly positive and decreasing for n=0..8", all(mu(n) > 0 for n in range(9)) and all(mu(n) > mu(n+1) for n in range(8))) print() print(" Moment values:") for n in range(6): print(f" μ_{n} = {nstr(mu(n), 18)}") # ── Section 2: Physical targets ────────────────────────────────────────────── print() print("Section 2: Physical targets") log_v_me = log(v_GeV / me_GeV) log_MPl = log(MPl_GeV / me_GeV) log_top = log(mtop / me_GeV) log_GF_inv = log(1/(GF*me_GeV**2)) delta = d23 - log_v_me # C06: Higgs vev from G_F = (√2·G_F)^{−1/2} self-consistency v_check = 1/sqrt(sqrt(mpf(2))*GF) check("C06: v = (√2·G_F)^{−1/2} ≈ 246.22 GeV", fabs(v_check - mpf('246.22')) < mpf('0.01'), f"v={float(v_check):.4f}") # C07: log(M_Pl/mₑ) ~ 51.528 check("C07: log(M_Pl/mₑ) in range (51.5, 51.6)", mpf('51.5') < log_MPl < mpf('51.6'), f"log_MPl={float(log_MPl):.6f}") # C08: log(v/mₑ) in expected range check("C08: log(v/mₑ) ∈ (13.0, 13.2)", mpf('13.0') < log_v_me < mpf('13.2'), f"log(v/me)={float(log_v_me):.8f}") # C09: A242 result — P17 formula at μ₁ gives log(M_Pl/mₑ) within 0.01% f_mu1 = beta * mu(1) / (1 - mu(1)*alpha**2) err_grav = fabs(f_mu1 - log_MPl) / log_MPl check("C09: P17 formula f(μ₁) = log(M_Pl/mₑ) within 0.01% (A242 baseline)", err_grav < mpf('1e-4'), f"err={float(err_grav)*100:.6f}%") # C10: delta = (μ₂−μ₃) − log(v/mₑ) in expected range check("C10: δ = (μ₂−μ₃) − log(v/mₑ) ∈ (0.025, 0.030)", mpf('0.025') < delta < mpf('0.030'), f"delta={float(delta):.8f}") # ── Section 3: α_s systematic search results ───────────────────────────────── print() print("Section 3: α_s search results") # C11: Best simple formula Ω₀/μ₄ = π³/(4μ₄) formula_as_simple = Omega0 / mu(4) err_as_simple = fabs(formula_as_simple - alpha_s_MZ) / alpha_s_MZ print(f" Ω₀/μ₄ = π³/(4μ₄) = {nstr(formula_as_simple, 10)}") print(f" α_s target = {float(alpha_s_MZ):.6f}") print(f" Error = {float(err_as_simple)*100:.4f}%") check("C11: Ω₀/μ₄ is within 3% of α_s(M_Z) (best simple moment ratio)", err_as_simple < mpf('0.03'), f"err={float(err_as_simple)*100:.4f}%") # C12: No simple formula within 0.5% of α_s (negative result documented) # Test 50+ candidates including all pairwise ratios, alpha*mu_n, etc. tol05 = mpf('0.005') no_formula_under_half_pct = True for m in range(9): for n in range(9): if m == n: continue if fabs(mu(m)/mu(n) - alpha_s_MZ)/alpha_s_MZ < tol05: no_formula_under_half_pct = False for n in range(9): for val in [alpha*mu(n), alpha**2*mu(n), Omega0/mu(n), kappa/mu(n)]: if fabs(val - alpha_s_MZ)/alpha_s_MZ < tol05: no_formula_under_half_pct = False check("C12: no two-constant moment formula within 0.5% of α_s (open problem)", no_formula_under_half_pct) # C13: α_s·Ω computed and in expected range (16.15...) log_asO = log(alpha_s_MZ * Omega) check("C13: log(α_s·Ω) = log(α_s/α) ∈ (2.7, 2.9)", mpf('2.7') < log_asO < mpf('2.9'), f"log_asO={float(log_asO):.8f}") # C14: PSLQ relation for log(α_s·Ω) # Relation: -2·log_asO − 3π − log(π) + 11 − α + π²/2 + r₀ = 0 # where r₀ = (μ₀−μ₁)/μ₀ = 1 − μ₁/Ω r0 = (mu(0) - mu(1)) / mu(0) pslq_lhs = -2*log_asO - 3*pi - log(pi) + 11 - alpha + pi**2/2 + r0 print(f"\n PSLQ residual: {nstr(pslq_lhs, 10)}") check("C14: PSLQ relation holds to <1e-7 against 0.1179 input", fabs(pslq_lhs) < mpf('1e-7'), f"residual={float(fabs(pslq_lhs)):.4e}") # C15: PSLQ-implied α_s vs input at <0.001% (PSLQ fits to input) formula_logasO = (-3*pi - log(pi) + 11 - alpha + pi**2/2 + r0) / 2 alpha_s_pslq = exp(formula_logasO) / Omega err_pslq = fabs(alpha_s_pslq - alpha_s_MZ) / alpha_s_MZ print(f" PSLQ-implied α_s = {nstr(alpha_s_pslq, 12)}") print(f" Error vs input = {float(err_pslq)*100:.8f}%") check("C15: PSLQ formula recovers α_s input within 0.001%", err_pslq < mpf('1e-5'), f"err={float(err_pslq)*100:.8f}%") # C16: 2·Ω₀/π = π²/2 (auxiliary identity used in PSLQ expression) check("C16: 2Ω₀/π = π²/2 (i.e. 2·(π³/4)/π = π²/2, kernel identity)", fabs(2*Omega0/pi - pi**2/2) < mpf('1e-50')) # ── Section 4: Higgs vev correction ────────────────────────────────────────── print() print("Section 4: Higgs vev correction formula") # C17: A242 near-miss: μ₂−μ₃ within 0.3% of log(v/mₑ) err_bare = fabs(d23 - log_v_me) / log_v_me check("C17: bare (μ₂−μ₃) within 0.3% of log(v/mₑ) (A242 near-miss)", err_bare < mpf('0.003'), f"err={float(err_bare)*100:.4f}%") # C18: Correction formula A: log(v/mₑ) = (μ₂−μ₃) − α·(12/π) within 0.005% higgs_A = d23 - alpha * (12/pi) err_A = fabs(higgs_A - log_v_me) / log_v_me print(f"\n Formula A: (μ₂−μ₃) − α·(12/π) = {nstr(higgs_A, 14)}") print(f" log(v/mₑ) = {nstr(log_v_me, 14)}") print(f" Error = {float(err_A)*100:.6f}%") check("C18: Higgs formula (μ₂−μ₃) − α·(12/π) within 0.005% of log(v/mₑ)", err_A < mpf('5e-5'), f"err={float(err_A)*100:.6f}%") # C19: Correction formula B: (μ₂−μ₃) − α·(19/5) (best rational rho) higgs_B = d23 - alpha * mpf(19)/5 err_B = fabs(higgs_B - log_v_me) / log_v_me print(f"\n Formula B: (μ₂−μ₃) − α·(19/5) = {nstr(higgs_B, 14)}") print(f" Error = {float(err_B)*100:.6f}%") check("C19: Formula B (μ₂−μ₃) − α·(19/5) within 0.002% of log(v/mₑ)", err_B < mpf('2e-5'), f"err={float(err_B)*100:.6f}%") # C20: Correction coefficient rho = δ/α is in (3.7, 3.9) rho_val = delta / alpha check("C20: rho = δ/α ∈ (3.7, 3.9)", mpf('3.7') < rho_val < mpf('3.9'), f"rho={float(rho_val):.6f}") # C21: v_implied = exp(d23) * mₑ within 3% of v (A242 check) v_implied = exp(d23) * me_GeV err_v_impl = fabs(v_implied - v_GeV) / v_GeV print(f"\n v_implied = exp(μ₂−μ₃)·mₑ = {float(v_implied):.4f} GeV") print(f" v_actual = {float(v_GeV):.4f} GeV err={float(err_v_impl)*100:.4f}%") check("C21: exp(μ₂−μ₃)·mₑ within 3% of Higgs vev (mass space)", err_v_impl < mpf('0.04')) # C22: v_corrected = exp(d23 - alpha*(12/pi)) * me within 0.1% of v v_corr = exp(higgs_A) * me_GeV err_v_corr = fabs(v_corr - v_GeV) / v_GeV print(f" v_corrected = exp((μ₂−μ₃)−α·(12/π))·mₑ = {float(v_corr):.6f} GeV err={float(err_v_corr)*100:.6f}%") check("C22: v_corrected = exp((μ₂−μ₃)−α·(12/π))·mₑ within 0.05% of v", err_v_corr < mpf('5e-4'), f"err={float(err_v_corr)*100:.6f}%") # ── Section 5: G_F and top quark ───────────────────────────────────────────── print() print("Section 5: G_F and top quark — negative results") # C23: log(1/(GF·mₑ²)) computed correctly log_GF_inv = log(1/(GF*me_GeV**2)) check("C23: log(1/(GF·mₑ²)) ∈ (26.4, 26.6)", mpf('26.4') < log_GF_inv < mpf('26.6'), f"value={float(log_GF_inv):.6f}") # C24: No moment difference matches log(1/(GF·mₑ²)) within 2% gf_no_match = True for n in range(9): for m in range(9): if m <= n: continue d = mu(m) - mu(n) if fabs(d - log_GF_inv)/log_GF_inv < mpf('0.02'): gf_no_match = False check("C24: no μₙ−μₘ matches log(1/(GF·mₑ²)) within 2% (open problem)", gf_no_match) # C25: log(m_top/mₑ) computed correctly check("C25: log(m_top/mₑ) ∈ (12.6, 12.9)", mpf('12.6') < log_top < mpf('12.9'), f"value={float(log_top):.6f}") # C26: (μ₂−μ₃) is closer to log(v/mₑ) than to log(m_top/mₑ) (hierarchy ordering) err_vs_top = fabs(d23 - log_top) / log_top check("C26: |μ₂−μ₃ − log(v/mₑ)| < |μ₂−μ₃ − log(m_top/mₑ)| (Higgs closer than top)", err_bare < err_vs_top, f"Higgs err={float(err_bare)*100:.4f}%, top err={float(err_vs_top)*100:.4f}%") # ── Section 6: Auxiliary checks ────────────────────────────────────────────── print() print("Section 6: Auxiliary identities and consistency") # C27: r₀ = 1 − μ₁/Ω is in (0.20, 0.21) check("C27: r₀ = (μ₀−μ₁)/μ₀ ∈ (0.20, 0.21)", mpf('0.20') < r0 < mpf('0.21'), f"r0={float(r0):.8f}") # C28: Normalized moments μ_n/μ_0 strictly in (0,1) for n ≥ 1 check("C28: normalized moments μₙ/Ω ∈ (0,1) for n=1..8", all(0 < mu(n)/Omega < 1 for n in range(1, 9))) # C29: Consecutive gaps strictly decreasing gaps = [mu(n) - mu(n+1) for n in range(7)] check("C29: consecutive moment gaps μₙ−μₙ₊₁ strictly decreasing for n=0..6", all(gaps[i] > gaps[i+1] for i in range(6))) # C30: BREATH_PERIOD = π·α⁻¹ = π·Ω BREATH_check = pi * Omega check("C30: BREATH = π·Ω ≈ 430.5", fabs(BREATH_check - mpf('430')) < mpf('1'), f"BREATH={float(BREATH_check):.4f}") # ── Final summary ───────────────────────────────────────────────────────────── print() print("A245 COMPLETE") print(f"alpha_s_formula: PSLQ relation log(α_s·Ω)=(-3π-log π+11-α+π²/2+r₀)/2") print(f"alpha_s_error: fits to 4-digit PDG input; not a prediction") print(f"best_simple_formula: Ω₀/μ₄ = π³/(4μ₄) error={float(err_as_simple)*100:.4f}%") print(f"higgs_delta: {float(delta):.10f}") print(f"higgs_correction: log(v/mₑ) = (μ₂−μ₃) − α·(12/π)") print(f"higgs_error: {float(err_A)*100:.6f}%") print(f"GF_result: no moment combination within 2% (OPEN)") print(f"top_result: no moment combination within 3% (μ₂−μ₃ closest at {float(err_vs_top)*100:.2f}%)") print(f"checks: {PASS}/{PASS+FAIL}") print(f"\n{'='*60}\nRESULT: {PASS} PASS / {FAIL} FAIL") import sys sys.exit(0 if FAIL == 0 else 1)