#!/usr/bin/env python3 """ verify_P003.py -- Paper 03: Mathematical Foundations. This verifier checks the numerical and algebraic claims in toe/03_Paper_MathematicalFoundations.tex. Paper 3 repeats much of Paper 2's cubic-density framework, but adds some explicit proof statements. The main reproducible values are m0, m1, E_self, kappa=alpha^(5/4), and the final geometric-series mass-log approximation. The audit flags the theorem that still states E_norm=1 despite the proof computing 13.177, the wave equation/Neumann equilibrium problem, an overbroad moment-ratio theorem, an alpha-source mismatch in beta_QED, and an algebraic replacement of (1+x) by 1/(1-x) in the mass-log derivation. """ import math import sys from pathlib import Path sys.path.insert(0, str(Path(__file__).resolve().parent)) from verify_common import Verifier as _BaseVerifier, CheckResult class Verifier(_BaseVerifier): """Output-layer normalization only: same checks, modern [PASS]/[FAIL] format.""" def __init__(self, name: str): self.name = name self.results = [] print(name) def record(self, label, ok, computed="", claimed="", detail=""): self.results.append(CheckResult(label, ok, computed, claimed, detail)) n = len(self.results) note, info = "", detail if not ok and "Expected" in detail: i = detail.find("Expected") note = " -- " + detail[i:] info = detail[:i].rstrip().rstrip(";") print(f" [{'PASS' if ok else 'FAIL'}] {n:>2}. {label}{note}") if computed != "" or claimed != "": print(f" computed: {computed}") print(f" claimed : {claimed}") if info: print(f" {info}") return ok def summary(self): passed = sum(r.ok for r in self.results) failed = len(self.results) - passed print(f"\n{'='*60}\nRESULT: {passed} PASS / {failed} FAIL") return 1 if failed else 0 v = Verifier("P003 -- Mathematical Foundations") PI = math.pi A3 = 16 * PI**3 A2 = 3 * PI**2 A1 = 2 * PI MU0 = 4 * PI**3 + PI**2 + PI MU1 = A3 / 5 + A2 / 4 + A1 / 3 ALPHA = 1 / MU0 ALPHA_EXP = 1 / 137.035999084 def mu(n: int) -> float: return A3 / (n + 4) + A2 / (n + 3) + A1 / (n + 2) def rho_prime(x: float) -> float: return 48 * PI**3 * x**2 + 6 * PI**2 * x + 2 * PI def rho_second(x: float) -> float: return 96 * PI**3 * x + 6 * PI**2 energy = 0.5 * ( (48 * PI**3) ** 2 / 5 + (6 * PI**2) ** 2 / 3 + (2 * PI) ** 2 + 2 * (48 * PI**3) * (6 * PI**2) / 4 + 2 * (48 * PI**3) * (2 * PI) / 3 + 2 * (6 * PI**2) * (2 * PI) / 2 ) e_self = energy / MU0**2 delta_e = e_self - 4 * PI kappa = ALPHA ** (5 / 4) v.check("mu0 = 4*pi^3 + pi^2 + pi", MU0, 137.036303776, rel=1e-12) v.check("mu1", MU1, 108.716684, rel=3e-9) v.check("beta_geom = mu1/mu0", MU1 / MU0, 0.793342208, rel=5e-10) v.check("Dirichlet energy E[rho]", energy, 247444.809832, rel=2e-12) v.check("m0^2", MU0**2, 18778.952536, abs_tol=0.005) v.check("E_self = E/m0^2", e_self, 13.176712972, rel=5e-8) v.check( "Theorem 3.1 E_norm[rho] = 1", e_self, 1.0, rel=1e-10, detail="Expected fail: the proof itself computes E_self ~= 13.177 and says it is not 1.", ) v.check("Delta E = E_self - 4*pi", delta_e, 0.610342358, rel=5e-7) v.check("23*kappa*4*pi gap approximation", 23 * kappa * 4 * PI, delta_e, rel=1.1e-2) v.check("gap-to-amplitude safety factor", delta_e / (0.002 * e_self), 23.0, rel=2e-2) v.check("E_min(t) for A=kappa approx 0.002", e_self * (1 - 0.002), 13.151, rel=2e-4) v.check("E_max(t) for A=kappa approx 0.002", e_self * (1 + 0.002), 13.203, rel=2e-4) v.check("kappa = alpha^(5/4)", kappa, 0.002132826, rel=1e-7) v.check("kappa_obs vs alpha^(5/4) relative error", abs(0.0022 - kappa) / 0.0022, 0.0305, rel=2e-2) n_opt = math.log(0.0022) / math.log(ALPHA) v.check("n_opt from ln(kappa_obs)/ln(alpha)", n_opt, 1.243698, rel=5e-6) v.check("5/4 exponent relative distance from n_opt percent", 100 * abs(1.25 - n_opt) / n_opt, 0.51, rel=2e-2) v.record( "5/4 minimizes alpha^n over all exponents", False, computed=f"real optimum n={n_opt:.6f}", claimed="n=5/4 minimizes among power laws", detail="Expected fail: 5/4 is a good simple fraction, not the unconstrained minimizer.", ) v.check("characteristic action 1/alpha^(5/2)", ALPHA ** (-5 / 2), 2.2e5, rel=1e-2) v.check( "rho_cubic satisfies Neumann boundary at x=0", rho_prime(0.0), 0.0, abs_tol=1e-12, detail="Expected fail: rho'(0)=2*pi.", ) v.check( "rho_cubic satisfies Neumann boundary at x=1", rho_prime(1.0), 0.0, abs_tol=1e-12, detail="Expected fail: rho'(1) is nonzero.", ) v.check( "rho_cubic is static equilibrium of stated wave equation at x=1/2", rho_second(0.5), 0.0, abs_tol=1e-12, detail="Expected fail: the written equation leaves rho_cubic'' at zeroth order.", ) v.record( "linearization drops the eta*rho_cubic'' term correctly", False, computed="eta_tt = eta_xx - kappa*eta*rho_cubic'' plus residual terms", claimed="eta_tt = (1-kappa) eta_xx", detail="Expected fail: the displayed first-order equation does not imply the next line.", ) v.check("fundamental frequency omega_1", PI * math.sqrt(1 - kappa), 3.136, rel=1e-3) v.check("log-energy period 2*pi/omega_1", 2 * PI / (PI * math.sqrt(1 - kappa)), 2.0, rel=2e-3) v.check("mu2", mu(2), 90.175963, rel=5e-9) v.check("mu3", mu(3), 77.062931, rel=5e-6) v.check("mu1/mu0", mu(1) / mu(0), 0.7933, rel=1e-4) v.check("mu2/mu1", mu(2) / mu(1), 0.8295, rel=1e-4) v.check("mu3/mu2", mu(3) / mu(2), 0.8546, rel=1e-4) v.check( "moment-ratio theorem lower bound at n=0", mu(1) / mu(0), 0.83, abs_tol=0.03, detail="Expected fail: the theorem says ratios are about 0.83-0.86 for n=0,1,2,... but n=0 is 0.7933.", ) v.check( "moment-ratio theorem upper bound at n=3", mu(4) / mu(3), 0.86, abs_tol=0.0, detail="Expected fail: ratios soon exceed 0.86 and tend to 1.", ) beta_geom = MU1 / MU0 beta_qed_geom = 2 * ALPHA**2 / (3 * PI) beta_qed_exp = 2 * ALPHA_EXP**2 / (3 * PI) ratio_geom_alpha = beta_geom / beta_qed_geom ratio_exp_alpha = beta_geom / beta_qed_exp log_ratio = math.log(1.221e19 / (5.110e-4)) c_emp = ratio_exp_alpha / log_ratio c_formula = 10 * MU0**3 / (MU0**2 + MU1) v.check( "beta_QED from displayed 2/(3*pi*mu0^2)", beta_qed_geom, 1.130029e-5, rel=1e-6, detail="Expected fail: 1.130029e-5 is the experimental-alpha value, not the geometric-mu0 formula.", ) v.check( "beta ratio from displayed geometric beta_QED", ratio_geom_alpha, 70205.483, rel=1e-6, detail="Expected fail: the quoted ratio again uses experimental alpha.", ) v.check("beta ratio using experimental alpha", ratio_exp_alpha, 70205.483, rel=1e-6) v.check("mass hierarchy log using rounded P3 masses", log_ratio, 51.527840, rel=2e-6) v.check("C extracted from rounded P3 masses", c_emp, 1362.477, rel=5e-5) v.check("C_formula exact", c_formula, 1362.475, rel=5e-7) v.check( "C/C_formula table value", c_emp / c_formula, 1.000015, rel=2e-6, detail="Expected fail: the ratio is about 1.000001 with the stated formulas.", ) v.check( "C relative error percent", 100 * abs(c_emp - c_formula) / c_emp, 0.015, rel=1e-2, detail="Expected fail: the percent error is about 0.0001%, not 0.015%.", ) x = MU1 * ALPHA**2 mass_log_from_c = (3 * PI * MU1 / 20) * (1 + x) mass_log_geometric_series = (3 * PI * MU1 / 20) / (1 - x) v.check("mass-log formula from substituting C gives (1+x)", mass_log_from_c, 51.528124, rel=5e-8) v.check( "algebraic step (1+x) = 1/(1-x)", 1 + x, 1 / (1 - x), rel=1e-12, detail="Expected fail: this is only an approximation; it changes the mass-log by about 0.00335%.", ) v.check("geometric-series mass-log RHS", mass_log_geometric_series, 51.529851, rel=5e-8) v.check("geometric-series relative error percent", 100 * abs(mass_log_geometric_series - log_ratio) / log_ratio, 0.0039, rel=5e-2) sys.exit(v.summary())