#!/usr/bin/env python3 """ verify_P098.py -- Addendum 98: Bryant-Fano NLO loop and Jarlskog protection. This verifier checks the two-channel cancellation arithmetic, the updated Wolfenstein/Jarlskog table, and selected proof-status claims in 98_Addendum_BryantFanoNLO.tex. The high-level difference-of-squares arithmetic is reproducible if one assumes opposite +/-4 lambda^2/5 loop factors. The flagged issues are that the paper's own Moufang-loop correction formula first gives only +lambda^2/N, so N=5 would not cancel the G2 loop; the proof obtains -1/4 on the CP direction before reinterpreting it as a positive correction; the abstract uses a sqrt-style 1-8 lambda^4/25 factor for A while the body/table use 1-16 lambda^4/25; and the exact Phi/J route remains stale relative to the printed trigonometric formula. """ import math import sys from pathlib import Path sys.path.insert(0, str(Path(__file__).resolve().parent)) PASS = FAIL = 0 _N = 0 def check(n, desc, cond): global PASS, FAIL ok = bool(cond) PASS += ok FAIL += not ok print(f" [{'PASS' if ok else 'FAIL'}] {n:>2}. {desc}") return ok class Verifier: """Output adapter: identical check semantics, modern [PASS]/[FAIL] format.""" def __init__(self, name): print(name) def check(self, label, computed, claimed, *, rel=1e-3, abs_tol=None, detail=""): if abs_tol is not None: ok = abs(computed - claimed) <= abs_tol err_detail = f"abs err={abs(computed - claimed):.6g}, tol={abs_tol:.6g}" else: if claimed == 0: ok = abs(computed) <= (rel or 1e-12) err_detail = f"abs value={abs(computed):.6g}, tol={rel:.6g}" else: err = (computed - claimed) / abs(claimed) ok = abs(err) <= (rel or 0) err_detail = f"rel err={100 * err:+.6g}%, tol={100 * (rel or 0):.6g}%" return self.record(label, ok, computed, claimed, detail, err_detail) def record(self, label, ok, computed="", claimed="", detail="", err_detail=""): global _N _N += 1 desc = label + (f" -- {detail}" if detail else "") check(_N, desc, ok) if computed != "" or claimed != "": print(f" computed: {computed}") print(f" claimed : {claimed}") if err_detail: print(f" {err_detail}") return ok def summary(self): print(f"\n{'='*60}\nRESULT: {PASS} PASS / {FAIL} FAIL") return 1 if FAIL else 0 v = Verifier("P098 -- Bryant-Fano NLO Jarlskog") PI = math.pi LAM = math.sin(PI / 14) LAM2 = LAM**2 LAM4 = LAM2**2 LAM6 = LAM**6 A_LO = math.cos(PI / 14) ** 2 * math.cos(2 * PI / 14) PHI_EXACT = math.cos(PI / 14) * math.cos(2 * PI / 14) * math.cos(5 * PI / 28) DELTA = 31 * PI / 84 SIN_DELTA = math.sin(DELTA) COS_DELTA = math.cos(DELTA) ETA_EXACT = 0.5 * PHI_EXACT * SIN_DELTA RHO_EXACT = 0.5 * PHI_EXACT * COS_DELTA J_LO_EXACT = A_LO**2 * LAM6 * ETA_EXACT J_LO_ROUNDED = 3.064e-5 J_PDG = 3.08e-5 J_SIGMA = 0.15e-5 PHI_STALE = 0.7429 ETA_STALE = 0.3439 RHO_STALE = 0.1484 A_STALE = 0.8563 def residual_pct(value: float, target: float) -> float: return 100 * (value - target) / target v.check("lambda = sin(pi/14)", LAM, 0.222521, rel=5e-6) v.check("lambda^2", LAM2, 0.04951, rel=2e-4) v.check("lambda^4", LAM4, 0.002451, rel=5e-4) v.check("lambda^6", LAM6, 1.2148e-4, rel=8e-4) v.check("A_LO", A_LO, 0.8563, rel=1e-4) v.check( "Phi_LO from exact formula", PHI_EXACT, 0.7429, rel=5e-4, detail="Expected fail: the printed exact formula gives about 0.743745.", ) v.check("sin(delta_0)", SIN_DELTA, 0.9162, rel=5e-4) v.check("cos(delta_0)", COS_DELTA, 0.4008, rel=3e-3) v.check( "J_LO from exact stated formulas", J_LO_EXACT, 3.064e-5, rel=5e-4, detail="Expected fail: exact Phi-derived eta gives about 3.035e-5; 3.064e-5 is the stale rounded path.", ) g2_factor = 1 - 4 * LAM2 / 5 fano_factor_assumed = 1 + 4 * LAM2 / 5 combined_block = g2_factor * fano_factor_assumed j_ratio_exact = combined_block**3 j_ratio_first_order = 1 - 48 * LAM4 / 25 v.check("G2 loop factor 1 - 4 lambda^2/5", g2_factor, 0.96039, rel=5e-6) v.check("assumed Fano loop factor 1 + 4 lambda^2/5", fano_factor_assumed, 1.03961, rel=5e-6) v.check("combined block factor", combined_block, 1 - 16 * LAM4 / 25, rel=1e-12) v.check("16 lambda^4 / 25", 16 * LAM4 / 25, 0.0015685, rel=5e-4) v.check("48 lambda^4 / 25", 48 * LAM4 / 25, 0.0047059, rel=5e-4) v.check("J ratio exact (1 - 16 lambda^4/25)^3", j_ratio_exact, 0.99529, rel=5e-5) v.check("J ratio first-order lambda^4", j_ratio_first_order, 0.995294, rel=5e-5) v.check("rounded-path J_NLO two-channel", J_LO_ROUNDED * j_ratio_exact, 3.050e-5, rel=2e-4) v.check("rounded-path J residual percent", residual_pct(J_LO_ROUNDED * j_ratio_exact, J_PDG), -0.97, rel=3e-2) v.check("rounded-path J sigma", abs(J_LO_ROUNDED * j_ratio_exact - J_PDG) / J_SIGMA, 0.20, rel=2e-2) v.check( "exact-formula J_NLO two-channel vs claimed", J_LO_EXACT * j_ratio_exact, 3.050e-5, rel=5e-3, detail="Expected fail: exact Phi-derived J_LO gives about 3.020e-5 after the same ratio.", ) v.check( "exact-formula J residual percent", residual_pct(J_LO_EXACT * j_ratio_exact, J_PDG), -0.97, rel=5e-2, detail="Expected fail: exact formula path gives about -1.94%, not -0.97%.", ) # Internal formula consistency for the Fano correction. L_FANO = 1 / 4 SIGMA = 4 * LAM2 N_FANO = 5 fano_delta_from_displayed_formula = L_FANO * SIGMA / N_FANO fano_delta_needed = 4 * LAM2 / 5 v.check( "displayed Fano correction formula gives +4 lambda^2/5 with N=5", fano_delta_from_displayed_formula, fano_delta_needed, rel=1e-6, detail="Expected fail: (1/4)*(4 lambda^2)/5 = lambda^2/5, a factor of four too small.", ) v.check( "N_Fano required by displayed +lambda^2/N formula", LAM2 / fano_delta_needed, 5, rel=1e-6, detail="Expected fail: the displayed +lambda^2/N formula would require N=5/4 to cancel -4 lambda^2/5.", ) v.record( "Moufang eigenvalue is +1/4 on the CP e7 direction", False, "The proof computes M_a(E23(e7)) = -1/4 E23(e7) before reinterpreting the correction sign.", "+1/4 eigenvalue", "Expected proof-audit fail: the positive correction is an extra sign convention, not the eigenvalue just computed on the CP direction.", ) v.record( "Moufang and Peirce maps are simultaneously equal and independent", False, "Definition states a o (X o a) = (a o X) o a, while the two-channel proof says these maps are distinct in J3(O).", "consistent map identities", "Expected proof-audit fail: the equality in the definition conflicts with the independence proof.", ) v.record( "N_Fano=5 is derived from first principles", False, "The paper derives N=5 from the cancellation condition and marks the root-system derivation as OP-P98-1.", "first-principles derivation", "Expected proof-audit fail: the value is conditional/self-consistency based, not first-principles proved.", ) # Wolfenstein table checks. body_block = 1 - 16 * LAM4 / 25 sqrt_style = 1 - 8 * LAM4 / 25 v.check( "abstract A_NLO factor 1 - 8 lambda^4/25 matches body/table factor", sqrt_style, body_block, rel=1e-6, detail="Expected fail: the abstract uses the square-root expansion; the body/table use 1 - 16 lambda^4/25.", ) v.check("two-channel A table", A_STALE * body_block, 0.8550, rel=1e-4) v.check("two-channel Phi table", PHI_STALE * body_block, 0.7417, rel=5e-5) v.check("two-channel eta table", ETA_STALE * body_block, 0.3434, rel=2e-4) v.check("two-channel rho table", RHO_STALE * body_block, 0.1482, rel=3e-4) v.check("G2-only rounded J table", J_LO_ROUNDED * g2_factor**3, 2.714e-5, rel=5e-4) v.check( "G2-only exact-formula J table", J_LO_EXACT * g2_factor**3, 2.714e-5, rel=5e-3, detail="Expected fail: exact formula path gives about 2.688e-5.", ) eta_lo_resid = abs(residual_pct(ETA_STALE, 0.3441)) eta_two_resid = abs(residual_pct(ETA_STALE * body_block, 0.3441)) v.record( "two-channel NLO parameters are closer to PDG for eta", eta_two_resid < eta_lo_resid, f"LO eta residual={eta_lo_resid:.4f}%, two-channel={eta_two_resid:.4f}%", "two-channel closer", "Expected fail: eta moves slightly farther from the PDG central value.", ) v.record( "16 lambda^4/25 is sub-permil", 16 * LAM4 / 25 < 0.001, f"{1000 * 16 * LAM4 / 25:.3f} per mille", "<1 per mille", "Expected wording fail: the correction is about 1.57 per mille, i.e. sub-percent but not sub-permil.", ) sys.exit(v.summary())