#!/usr/bin/env python3 """ verify_P082.py -- Addendum 82: Complete CKM Wolfenstein parameters. This verifier checks the explicit trigonometric formulas in 82_Addendum_CKMComplete.tex. The main CKM construction is close numerically, but the printed calibration value Phi ~= 0.7490 is not the value of the stated formula, and the mismatch propagates through rho_bar, eta_bar, |Vub|, and the unitarity-triangle table. """ import math import sys from pathlib import Path sys.path.insert(0, str(Path(__file__).resolve().parent)) from verify_common import CheckResult, Verifier class ModernVerifier(Verifier): """Local adapter: tolerance logic inherited byte-identical from verify_common.Verifier; only the output layer is modernised.""" def __init__(self, name: str) -> None: 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) desc = f"{label} -- {detail}" if detail else label print(f" [{'PASS' if ok else 'FAIL'}] {n:>2}. {desc}") if computed != "" or claimed != "": print(f" computed: {computed}") print(f" claimed : {claimed}") 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 = ModernVerifier("P082 -- Complete CKM Wolfenstein Parameterization") PDG_LAMBDA = 0.2245 PDG_A = 0.823 PDG_RHO = 0.147 PDG_ETA = 0.343 PDG_VUB = 0.003517 PDG_J = 3.18e-5 PDG_RB = 0.374 PDG_DELTA = 65.6 PDG_ALPHA = 84.4 lam = math.sin(math.pi / 14) lam2 = lam**2 lam3 = lam**3 c1 = math.cos(math.pi / 14) c2 = math.cos(2 * math.pi / 14) c5 = math.cos(5 * math.pi / 28) A = c1**2 * c2 delta = math.pi / 3 + math.pi / 28 delta_deg = math.degrees(delta) phi = c1 * c2 * c5 rho = 0.5 * phi * math.cos(delta) eta = 0.5 * phi * math.sin(delta) rb = math.hypot(rho, eta) rt = math.hypot(1 - rho, eta) vub = A * lam3 * rb jarlskog = A**2 * lam**6 * eta beta = math.degrees(math.atan2(eta, 1 - rho)) gamma = math.degrees(math.atan2(eta, rho)) alpha = 180 - beta - gamma bad_direct_rb = rb / (A * lam3) def residual(value: float, target: float) -> float: return 100 * (value - target) / target v.check("lambda = sin(pi/14)", lam, 0.22252, rel=5e-6) v.check("lambda^2", lam2, 0.04952, rel=1e-4) v.check("lambda^3", lam3, 0.01101, rel=1e-3) v.check("A = cos^2(pi/14) cos(2pi/14)", A, 0.8563, rel=1e-4) v.check("delta = 31pi/84 in degrees", delta_deg, 66.43, rel=5e-5) v.check("cos(pi/14)", c1, 0.97493, rel=5e-6) v.check("cos(2pi/14)", c2, 0.90097, rel=5e-6) v.record("cos(5pi/28) exact component", True, f"{c5:.12g}", "component of Phi") v.check( "Phi = cos(pi/14) cos(2pi/14) cos(5pi/28)", phi, 0.7490, rel=5e-4, detail="Expected fail: the stated formula gives about 0.743745, not 0.7490.", ) v.check( "cos(delta) printed value", math.cos(delta), 0.39635, rel=5e-4, detail="Expected fail: cos(31pi/84) is about 0.399892.", ) v.check( "sin(delta) printed value", math.sin(delta), 0.91813, rel=5e-4, detail="Expected fail: sin(31pi/84) is about 0.916562.", ) v.check( "rho_bar exact formula", rho, 0.14840, rel=5e-4, detail="Expected fail: recomputing the stated formula gives about 0.148709.", ) v.check( "eta_bar exact formula", eta, 0.34386, rel=5e-4, detail="Expected fail: recomputing the stated formula gives about 0.340844.", ) v.check("eta/rho = tan(delta)", eta / rho, 2.291, rel=8e-4) v.check("lambda residual percent", residual(lam, PDG_LAMBDA), -0.88, rel=5e-3) v.check("A residual percent", residual(A, PDG_A), 4.0, rel=2e-2) v.check( "rho residual percent", residual(rho, PDG_RHO), 0.95, rel=5e-2, detail="Expected fail: using the stated formula gives about +1.16%, just outside 1%.", ) v.record( "rho within 1 percent of PDG", abs(residual(rho, PDG_RHO)) <= 1.0, f"{residual(rho, PDG_RHO):.6g}%", "within 1%", "Expected fail: central-value residual is about +1.16%.", ) v.check( "eta residual percent", residual(eta, PDG_ETA), 0.26, rel=5e-2, detail="Expected fail: recomputation gives about -0.63%, not +0.26%.", ) v.check( "|rho+i eta| = Phi/2", rb, 0.3745, rel=5e-4, detail="Expected fail: the exact Phi gives R_b about 0.371873.", ) v.check( "R_b residual percent", residual(rb, PDG_RB), 0.13, rel=5e-2, detail="Expected fail: residual is about -0.57%, not +0.13%.", ) v.check( "direct matching A lambda^3 R_b = Phi/2 implies R_b", bad_direct_rb, 39.7, rel=1e-2, detail="The paper notices this impossible value and switches to a dimensionless-triangle interpretation.", ) v.check( "|Vub| = A lambda^3 Phi/2", vub, 0.003531, rel=5e-4, detail="Expected fail: the exact stated formulas give about 0.0035088.", ) v.check( "|Vub| residual percent", residual(vub, PDG_VUB), 0.40, rel=5e-2, detail="Expected fail: recomputation gives about -0.23%, not +0.40%.", ) v.check("Jarlskog value", jarlskog, 3.037e-5, rel=1e-3) v.check("Jarlskog residual percent", residual(jarlskog, PDG_J), -4.4, rel=5e-2) v.record( "Jarlskog within 1 sigma of PDG", abs(jarlskog - PDG_J) <= 0.15e-5, f"{jarlskog:.12g}", "3.18e-5 +/- 0.15e-5", ) v.check("Vcb formula", lam2 * c1**2 * c2, 0.04242, rel=5e-4) v.check( "R_t side", rt, 0.9274, rel=5e-4, detail="Expected fail: exact rho/eta give R_t about 0.91699.", ) v.check( "beta angle", beta, 22.02, rel=1e-3, detail="Expected fail: exact rho/eta give beta about 21.82 degrees.", ) v.check("gamma angle from delta", gamma, 66.43, rel=5e-5) v.check( "gamma value in unitarity-triangle section", gamma, 66.69, rel=5e-4, detail="Expected fail: gamma equals delta = 66.43 degrees for the paper's own formulas.", ) v.check( "alpha angle", alpha, 91.29, rel=1e-3, detail="Expected fail: exact beta/gamma give alpha about 91.75 degrees.", ) v.check( "alpha residual percent", residual(alpha, PDG_ALPHA), 8.2, rel=3e-2, detail="Expected fail: exact alpha gives about +8.71%.", ) v.check( "alpha discrepancy explained as 1.1 degrees", alpha - PDG_ALPHA, 1.1, rel=5e-2, detail="Expected fail: the actual alpha-PDG gap is about 7.35 degrees.", ) sys.exit(v.summary())