#!/usr/bin/env python3 """ verify_P065.py -- Addendum 65: PMNS-Yukawa compatibility. This verifier checks the PMNS compatibility arithmetic and the proof-status claims in 65_Addendum_PMNSCompatibility.tex. The headline leading-order angle numerics reproduce: theta12=5*pi/28, theta23=pi/4, and the quoted percent residuals are mostly consistent. The flagged issues are proof-level and internal consistency problems: the displacement-angle equivalence is miswritten, the moduli-count statements alternate between two and three Yukawa constraints, Z3 symmetry is used as if it uniquely forced QLC/TBM, and the theta23 proof contains a self-correction whose displayed complement formula would give about 87.7 degrees rather than 45 degrees. """ from __future__ import annotations 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: identical tolerance logic, modern output 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) desc, info = label, detail i = detail.find("Expected") if i >= 0: desc = f"{label} -- {detail[i:]}" info = detail[:i].rstrip().rstrip(";") print(f" [{'PASS' if ok else 'FAIL'}] {n:>2}. {desc}") if computed != "" or claimed != "": print(f" computed: {computed}") print(f" claimed : {claimed}") if info: print(f" {info}") return bool(ok) def summary(self) -> int: passed = sum(bool(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("P065 -- PMNS-Yukawa Compatibility") ROOT = Path(__file__).resolve().parents[1] TEX = (ROOT / "65_Addendum_PMNSCompatibility.tex").read_text() PI = math.pi THETA_C = PI / 14 THETA12 = PI / 4 - THETA_C THETA23 = PI / 4 PDG_THETA_C = 12.80 PDG_THETA12 = 33.44 PDG_THETA23 = 42.20 PDG_THETA13 = 8.62 PDG_DELTA = -67.0 def deg(rad: float) -> float: return math.degrees(rad) def pct(value: float, target: float) -> float: return 100 * (value - target) / abs(target) def matrix_toe(theta: float): c = math.cos(theta) s = math.sin(theta) r2 = math.sqrt(2) return [ [c, s, 0.0], [-s / r2, c / r2, 1 / r2], [s / r2, -c / r2, 1 / r2], ] def dot(row_a, row_b): return sum(a * b for a, b in zip(row_a, row_b)) y1 = math.sin(PI / 14) y2 = math.sin(2 * PI / 14) phi12 = PI / 2 - PI / 14 phi13 = PI / 2 - 2 * PI / 14 qlc_sum_pdg = PDG_THETA_C + PDG_THETA12 theta23_ckm = 2.3 theta23_complement = 90 - theta23_ckm u_toe = matrix_toe(THETA12) v.check("y1 = sin(pi/14)", y1, 0.2225, rel=2e-4) v.check("y2 = sin(2pi/14)", y2, 0.4339, rel=2e-4) v.check("theta_C = pi/14 degrees", deg(THETA_C), 12.857, rel=3e-5) v.check("theta_C residual percent", pct(deg(THETA_C), PDG_THETA_C), 0.4, rel=2e-1) v.check("theta12 PMNS = 5pi/28", deg(THETA12), 32.14, rel=1e-4) v.check("theta12 discrepancy degrees", PDG_THETA12 - deg(THETA12), 1.3, rel=3e-2) v.check("theta12 residual percent", pct(deg(THETA12), PDG_THETA12), -3.9, rel=2e-2) v.check("QLC TOE sum", deg(THETA_C + THETA12), 45.0, rel=1e-12) v.check("PDG QLC sum", qlc_sum_pdg, 46.3, rel=2e-3) v.check("QLC sum residual percent vs PDG sum", pct(45.0, qlc_sum_pdg), -2.9, rel=8e-2) v.check("theta23 leading value", deg(THETA23), 45.0, rel=1e-12) v.check("theta23 residual percent", pct(deg(THETA23), PDG_THETA23), 6.6, rel=1e-2) v.check( "PDG theta23 as fraction of pi/4", PDG_THETA23 / 45.0, 0.949, rel=1e-3, detail="Expected fail: 42.2/45 is about 0.938, not 0.949.", ) v.check("TBM theta12", deg(math.asin(1 / math.sqrt(3))), 35.26, rel=2e-4) v.check("TBM theta12 residual percent", pct(deg(math.asin(1 / math.sqrt(3))), PDG_THETA12), 5.4, rel=2e-2) v.check("theta13 rough 2/3 candidate", y1 * 2 / 3, 0.148, rel=3e-3) v.check("theta13 candidate angle", deg(math.asin(y1 * 2 / 3)), PDG_THETA13, rel=2e-2) v.check("rough J estimate using listed PDG s12/c12", (0.551 * 0.834) / 13, 0.035, rel=2e-2) v.check("varphi12 displacement", deg(phi12), deg(6 * PI / 14), rel=1e-12) v.check("varphi13 displacement", deg(phi13), deg(5 * PI / 14), rel=1e-12) v.check("cos(varphi12) equals y1", math.cos(phi12), y1, rel=1e-12) v.check("cos(varphi13) equals y2", math.cos(phi13), y2, rel=1e-12) v.record( "printed sine/cosine equivalence for displacement angles is well formed", "sin\\varphi_{ij} = \\cos(\\langle\\hat{u}_{ij}, e_7\\rangle^{-1})" not in TEX, "paper writes sin(varphi_ij)=cos(^{-1})", "sin(varphi_ij)=sqrt(1-^2) or cos(k*pi/14)", "Expected fail: the inverse-inner-product expression is not the trigonometric inverse needed here.", ) v.record( "moduli-count statements consistently say Yukawa fixes two independent orbit moduli", "fix three scalar inner products (three of the four invariants)" not in TEX, "abstract/body says three Yukawa scalar products fix three of four invariants; later proof says two of four", "consistent two-modulus count after G2 anchoring", "Expected proof-audit fail: y3=1 is the chosen anchor, not an independent remaining quotient modulus in the later parameter count.", ) v.record( "Yukawa overlaps directly determine theta12 and theta23", False, "the independence lemma says Yukawa eigenvalues and PMNS angles are independent observables; theta12 is added through QLC/Z3", "direct determination by the three inner-product constraints", "Expected internal-consistency fail.", ) v.record( "QLC relation is derived from Z3 geometry rather than assumed", False, "the proof assumes a pi/4 tribimaximal axis/maximal-mixing principle; Z3 alone does not uniquely fix QLC", "structural identity proved", "Expected proof-audit fail: this is a conditional structural ansatz, not a derivation from the displayed equations.", ) v.record( "Z3 symmetry alone proves theta23 maximality", False, "the proof appeals to TBM/S3 fixed-point structure and then notes the CKM formula needs care", "theta23=pi/4 proved from Z3", "Expected proof-audit fail: Z3 cyclic symmetry alone does not uniquely determine the TBM matrix or maximal 2-3 mixing.", ) v.check( "theta23 complement formula in proof", theta23_complement, 45.0, rel=1e-3, detail="Expected fail: pi/2 - theta23_CKM with theta23_CKM≈2.3 deg is about 87.7 deg, not pi/4.", ) v.record( "theta23 proof has no self-correction marker", "Wait -- that formula applies to the CKM parametrisation" not in TEX, "paper includes a 'Wait --' correction inside the theorem proof", "clean proof", "Expected proof-audit fail: the theorem proof itself marks the key relation as needing care.", ) v.record( "sector-to-off-diagonal PMNS rotation is explicitly constructed", False, "the text defines sector and off-diagonal bases but does not give an explicit unitary map between diagonal E_ii and imaginary-octonion directions", "PMNS rotation derived in J3(O)", "Expected proof-audit fail: the compatibility picture is plausible but the actual rotation is not constructed.", ) v.record( "Phase 5a closure status is unconditional", "Proved (cond.)" not in TEX, "status table says several PMNS statements are proved conditionally and theta13/delta_CP remain open", "unconditional closure", "Expected status fail: closure is conditional on A57/QLC/Z3 assumptions and leaves Phase 5b-ii open.", ) for i, row_i in enumerate(u_toe): v.check(f"PMNS TOE row {i + 1} norm", dot(row_i, row_i), 1.0, rel=1e-12) v.check("PMNS TOE row1-row2 dot", dot(u_toe[0], u_toe[1]), 0.0, abs_tol=1e-12) v.check("PMNS TOE row1-row3 dot", dot(u_toe[0], u_toe[2]), 0.0, abs_tol=1e-12) v.check("PMNS TOE row2-row3 dot", dot(u_toe[1], u_toe[2]), 0.0, abs_tol=1e-12) sys.exit(v.summary())