#!/usr/bin/env python3 """ verify_P056.py -- Addendum 56: Weyl geodesic derivation of Yukawa angles. This verifier checks the numerical sine/Cabibbo content and audits the proof status in 56_Addendum_WeylGeodesicYukawa.tex. The basic values sin(pi/14), sin(2*pi/14), their ratios, and the dimension identity 14=dim G2 all reproduce. The flagged issues are proof/internal-consistency problems: the paper says a Coxeter element of order 6 lifts to order 14, lists an 8-element eigenvalue set for a 7-dimensional representation, corrects its own "unique plane" claim to a 6-dimensional family, reverses the y=cos(theta) ordering inequality, and leaves the minimal-step and singlet-projection inputs open despite the headline closure language. """ from __future__ import annotations import cmath 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): """Output adapter: modern check-line format. Tolerance logic is inherited unchanged from verify_common.Verifier; only printing and the footer differ.""" 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) expected = (not ok) and ("Expected" in str(detail)) desc = f"{label} -- {detail}" if expected else label print(f" [{'PASS' if ok else 'FAIL'}] {n:>2}. {desc}") if computed != "" or claimed != "": print(f" computed: {computed}") print(f" claimed : {claimed}") if detail and not expected: print(f" {detail}") 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("P056 -- Weyl Geodesic Yukawa Angles") ROOT = Path(__file__).resolve().parents[1] TEX = (ROOT / "56_Addendum_WeylGeodesicYukawa.tex").read_text() PI = math.pi H_G2 = 6 DIM_G2 = 14 N_IMO = 7 THETA_C = PI / 14 WEYL_CHAMBER = PI / 6 PDG_VUS = 0.22431 y1 = math.sin(PI / 14) y2 = math.sin(2 * PI / 14) y3 = math.sin(7 * PI / 14) theta1 = PI / 2 - PI / 14 theta2 = PI / 2 - 2 * PI / 14 def deg(x: float) -> float: return math.degrees(x) def pct(value: float, target: float) -> float: return 100 * (value - target) / abs(target) def rounded_complex(z: complex) -> tuple[float, float]: return (round(z.real, 12), round(z.imag, 12)) v.check("Coxeter number h_G2", H_G2, 6, rel=1e-12) v.check("dimension identity dim G2 = 2(h+1)", DIM_G2, 2 * (H_G2 + 1), rel=1e-12) v.check("dimension identity dim G2 = 2|ImO|", DIM_G2, 2 * N_IMO, rel=1e-12) v.check("Cabibbo angle pi/14 degrees", deg(THETA_C), 12.857, rel=3e-5) v.check("theta_C / Weyl chamber width", THETA_C / WEYL_CHAMBER, 3 / 7, rel=1e-12) v.check("y1=sin(pi/14)", y1, 0.22252, rel=5e-5) v.check("y2=sin(2pi/14)", y2, 0.43388, rel=1e-5) v.check("y3=sin(7pi/14)", y3, 1.0, rel=1e-12) v.check("Cabibbo residual percent", pct(y1, PDG_VUS), -0.80, rel=5e-3) v.check("y1/y2", y1 / y2, 0.5129, rel=1e-4) v.check("y2/y1", y2 / y1, 1.9500, rel=1e-4) v.check("double-angle ratio 2cos(pi/14)", y2 / y1, 2 * math.cos(PI / 14), rel=1e-12) v.check("theta1 from e7 axis", theta1, 3 * PI / 7, rel=1e-12) v.check("theta2 from e7 axis", theta2, 5 * PI / 14, rel=1e-12) v.check("cos(theta1)=sin(pi/14)", math.cos(theta1), y1, rel=1e-12) v.check("cos(theta2)=sin(2pi/14)", math.cos(theta2), y2, rel=1e-12) v.check("moduli arithmetic 8-3-2+1", 8 - 3 - 2 + 1, 4, rel=1e-12) claimed_eigen_indices = [0, 1, -1, 2, -2, 5, -5, 7] claimed_eigenvalues = { rounded_complex(cmath.exp(2j * PI * k / 14)) for k in claimed_eigen_indices } v.record( "claimed order-14 eigenvalue list has dimension 7", len(claimed_eigenvalues) == 7, computed=f"{len(claimed_eigenvalues)} distinct eigenvalues from k={claimed_eigen_indices}", claimed="7-dimensional complexified representation", detail="Expected fail: the displayed 14th-root set has 8 distinct entries, not 7.", ) v.record( "Coxeter element of order 6 can lift to order 14 in the same representation", False, computed="a representation image of an order-6 element has order dividing 6", claimed="Coxeter element has order 6 on the real representation and lifts to order 14 on V7 tensor C", detail="Expected proof-audit fail: an order-14 lift needs an additional construction, not the same Coxeter element.", ) v.record( "14th-root eigenvalue proposition is consistent with the proof's 7th roots", "e^{2\\pi i k / 14}" not in TEX or "e^{2\\pi i k / 7}" not in TEX, computed="proposition displays 14th roots, proof says the eigenvalues are 7th roots", claimed="single eigenvalue description", detail="Expected internal-consistency fail.", ) v.record( "unique associative 3-plane through e7 is proved", "Uniqueness is \\emph{not} absolute" not in TEX, computed="paper states the family through e7 has dimension 6", claimed="fold-map anchor forces the unique associative 3-plane", detail="Expected proof-audit fail: the lemma title/claim is corrected inside its own proof.", ) v.record( "strict imaginary associativity gives nonzero y1,y2", "giving $y_1 = y_2 = 0$" not in TEX, computed="paper's corrected associativity derivation gives y1=y2=0", claimed="associativity plus anchor derives sin(pi/14), sin(2pi/14)", detail="Expected proof-audit fail: nonzero values require the later SU(3) singlet-projection reinterpretation.", ) v.record( "ordering condition for y=cos(theta) is written correctly", False, computed=f"chosen values have theta1={theta1:.6f} > theta2={theta2:.6f} while y1={y1:.6f} < y2={y2:.6f}", claimed="paper says y1