#!/usr/bin/env python3 """ verify_P109.py -- Addendum 109: NNNLO Weinberg-angle 600-cell loop. This verifier checks the Weinberg-angle numerics in 109_Addendum_WeinbergNNNLO.tex: lambda powers, LO/NLO/NNLO values, the golden target 3/(8 phi), 600-cell combinatorics, c6=-24/5, the NLO+c6 path, and the estimated c8 residual. The rounded NLO+c6 arithmetic reproduces. The flagged issue is that the paper calls c6 an NNNLO coefficient in the same loop expansion that contains c4, but the advertised -0.006% gap is obtained only by skipping the NNLO c4 term. The full expansion through c6 gives a much worse gap. Exact, non-rounded values also shift the required c6 and c8 estimates. """ from __future__ import annotations import math import sys from pathlib import Path sys.path.insert(0, str(Path(__file__).resolve().parent)) class Verifier: """Output shim: identical tolerance semantics to verify_common.Verifier, modern [PASS]/[FAIL] check-line output format.""" def __init__(self, name): self.PASS = 0 self.FAIL = 0 self.n = 0 print(name) def _mark(self, ok, desc): self.n += 1 if ok: self.PASS += 1 else: self.FAIL += 1 print(f" [{'PASS' if ok else 'FAIL'}] {self.n:>2}. {desc}") 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}%" self._mark(ok, label + (f" -- {detail}" if detail else "")) print(f" computed: {computed} claimed: {claimed} ({err_detail})") return ok def record(self, label, ok, computed="", claimed="", detail=""): ok = bool(ok) self._mark(ok, label + (f" -- {detail}" if detail else "")) if computed != "" or claimed != "": print(f" computed: {computed} claimed: {claimed}") return ok def summary(self): print(f"\n{'='*60}\nRESULT: {self.PASS} PASS / {self.FAIL} FAIL") return 0 if self.FAIL == 0 else 1 v = Verifier("P109 -- Weinberg NNNLO 600-cell") ROOT = Path(__file__).resolve().parents[1] TEX = (ROOT / "109_Addendum_WeinbergNNNLO.tex").read_text() PI = math.pi PHI = (1.0 + math.sqrt(5.0)) / 2.0 LAM = math.sin(PI / 14.0) BASE = 1.0 / (1.0 + PI) TARGET = 3.0 / (8.0 * PHI) C4 = -74.0 / 25.0 C6 = -24.0 / 5.0 def gap_pct(value: float) -> float: return 100.0 * (value - TARGET) / TARGET lo = BASE nlo = BASE * (1.0 - 4.0 * LAM**2 / 5.0) nnlo = BASE * (1.0 - 4.0 * LAM**2 / 5.0 + C4 * LAM**4) nlo_plus_c6 = BASE * (1.0 - 4.0 * LAM**2 / 5.0 + C6 * LAM**6) full_through_c6 = BASE * (1.0 - 4.0 * LAM**2 / 5.0 + C4 * LAM**4 + C6 * LAM**6) delta_c6 = C6 * LAM**6 / (1.0 + PI) v.check("lambda", LAM, 0.22252, rel=5e-5) v.check("lambda^2", LAM**2, 0.04952, rel=9e-5) v.check("lambda^4", LAM**4, 2.452e-3, rel=9e-5) v.check("lambda^6", LAM**6, 1.214e-4, rel=2e-4) v.check("lambda^8", LAM**8, 6.01e-6, rel=3e-4) v.check("golden target 3/(8phi)", TARGET, 0.23176, rel=2e-5) v.check("LO sin^2 theta_W", lo, 0.24145, rel=2e-5) v.check("LO gap percent", gap_pct(lo), 4.22, rel=1e-2) v.check("NLO sin^2 theta_W", nlo, 0.23189, rel=1e-5) v.check("NLO gap percent", gap_pct(nlo), 0.0542, rel=8e-4) v.check("NNLO sin^2 theta_W", nnlo, 0.23014, rel=2e-5) v.check("NNLO gap percent", gap_pct(nnlo), -0.70, rel=4e-3) v.check("600-cell cells per vertex", 600.0 / 120.0, 5.0, rel=1e-12) v.check("600-cell edges per vertex", 720.0 / 120.0, 6.0, rel=1e-12) v.check("G2 Weyl half-order", 12.0 / 2.0, 6.0, rel=1e-12) v.check("600-cell edge length 1/phi", 1.0 / PHI, 0.6180, rel=6e-5) v.check("c6 = -hvee*(E/V)/N_Fano", -(4.0 * 6.0) / 5.0, C6, rel=1e-12) v.check("NNNLO c6 shift", delta_c6, -1.408e-4, rel=8e-4) v.check("NLO plus c6 value", nlo_plus_c6, 0.23175, rel=2e-5) v.check("NLO plus c6 gap percent", gap_pct(nlo_plus_c6), -0.0065, rel=6e-3) v.check("NLO plus c6 improvement factor", abs((nlo - TARGET) / (nlo_plus_c6 - TARGET)), 8.3, rel=2e-2) required_c6_exact = -(nlo - TARGET) * (1.0 + PI) / LAM**6 v.check( "required c6 from exact NLO gap", required_c6_exact, -4.43, rel=5e-3, detail="Expected fail: -4.43 follows from rounded delta=0.00013; exact formulas give about -4.29.", ) v.check( "c6 discrepancy using exact required c6", 100.0 * (C6 - required_c6_exact) / abs(required_c6_exact), -8.4, rel=2e-2, detail="Expected fail: exact values give about -11.9%; -8.4% uses the rounded -4.43 required coefficient.", ) v.check( "full expansion through c6 value", full_through_c6, 0.23175, rel=2e-5, detail="Expected fail: including the NNLO c4 term as the expansion formula says gives about 0.229995, not the advertised NLO+c6 near-hit.", ) v.check( "full expansion through c6 gap percent", gap_pct(full_through_c6), -0.0065, rel=6e-3, detail="Expected fail: the full c2+c4+c6 path has about -0.76% gap.", ) c8_exact = -(nlo_plus_c6 - TARGET) * (1.0 + PI) / LAM**8 v.check( "c8 needed from exact NLO+c6 residual", c8_exact, 9.6, rel=3e-2, detail="Expected fail: using exact residual gives about 10.33; 9.6 follows from the rounded 1.4e-5 residual.", ) v.check("face-density c8 prediction", 4.0 * (1200.0 / 120.0) / 5.0, 8.0, rel=1e-12) v.record( "c6 is derived by an explicit Jordan-loop enumeration", False, computed="paper multiplies hvee(G2), E600/V600, and 1/N_Fano, but does not enumerate loop diagrams or prove this weighting rule", claimed="600-cell Jordan loop derives c6", detail="Expected proof-status fail.", ) v.record( "P105-1 is closed by P109", "Conjecture~P105-1 remains open" not in TEX, computed="paper explicitly leaves all-orders closure open", claimed="first NNNLO step toward proving P105-1", detail="Expected status note: the near-hit is not an exact closure.", ) sys.exit(v.summary())