#!/usr/bin/env python3 """ verify_P073.py -- Addendum 73: theta12 PMNS Closure. This verifier checks the numerical claims in 73_Addendum_Theta12Closure.tex. The lower-level cubic corrections are mostly arithmetic-consistent, but the paper's final "closure" argument mixes several incompatible Jarlskog/back-reaction calculations. The main intentional failures are: * cos(8.934 deg) and therefore K_TOE are miscomputed in the Jarlskog section. * Direct Jarlskog inversion gives a lower theta12, not 33.38 deg. * The closed-form theta12 formula as written evaluates to about 34.36 deg. * The final "definitive" U12 calculation uses y1^2*y2/8 = 0.006498, but the actual value is about 0.002685. * The paper explicitly derives Delta_J ~= -1.41 deg after claiming +0.44 deg. """ 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("P073 -- theta12 PMNS Closure") DEG = math.pi / 180 LAM = math.sin(math.pi / 14) THETA_C = math.pi / 14 Y2 = math.sin(2 * math.pi / 14) TH12_2_DEG = 32.59 TH23_TOE_DEG = 41.245 TH13_TOE_DEG = 8.934 DELTA_TOE_DEG = 66.43 # absolute value; the paper drops the sign. TH12_PDG_DEG = 33.44 TH23_PDG_DEG = 42.20 TH13_PDG_DEG = 8.620 DELTA_PDG_DEG = 67.0 def sind(x_deg): return math.sin(x_deg * DEG) def cosd(x_deg): return math.cos(x_deg * DEG) def asind(x): return math.asin(x) / DEG theta12_2 = TH12_2_DEG * DEG theta23_toe = TH23_TOE_DEG * DEG theta13_toe = TH13_TOE_DEG * DEG delta_toe = DELTA_TOE_DEG * DEG v.check("lambda = sin(pi/14)", LAM, 0.22252, rel=5e-6) v.check("lambda^2", LAM**2, 0.04952, rel=1e-4) v.check("lambda^3", LAM**3, 0.011018, rel=5e-5) v.check("lambda^3 in degrees", (LAM**3) / DEG, 0.63, rel=1e-2) v.check("theta_C = pi/14 in degrees", THETA_C / DEG, 12.857, rel=5e-5) theta12_lo = 45 - THETA_C / DEG theta12_nlo = (math.pi / 4 - LAM * (1 - LAM**2 / 2)) / DEG v.check("LO QLC theta12 = pi/4 - theta_C", theta12_lo, 32.14, rel=2e-3) v.check("NLO theta12 = pi/4 - lambda*(1-lambda^2/2)", theta12_nlo, 32.59, rel=1e-3) v.check("P72 theta12 residual", TH12_PDG_DEG - TH12_2_DEG, 0.85, rel=1e-2) peirce_rad = LAM**2 * Y2 / (8 * math.cos(theta12_2)) qlc_rad = LAM**3 / 4 v.check("sin(2*pi/14)", Y2, 0.43388, rel=1e-4) v.check("cos(32.59 deg)", math.cos(theta12_2), 0.8424, rel=5e-4) v.check("Peirce angular correction rad", peirce_rad, 0.003189, rel=1e-3) v.check("Peirce angular correction deg", peirce_rad / DEG, 0.183, rel=3e-3) v.check("QLC cubic correction rad = lambda^3/4", qlc_rad, 0.002755, rel=5e-4) v.check("QLC cubic correction deg", qlc_rad / DEG, 0.158, rel=3e-3) v.check("Peirce + QLC correction", peirce_rad / DEG + qlc_rad / DEG, 0.341, rel=3e-3) v.check( "cos(8.934 deg) used in Jarlskog section", cosd(TH13_TOE_DEG), 0.99388, rel=2e-3, detail="Expected fail: cos(8.934 deg) is about 0.98787, not 0.99388.", ) v.check("sin(2*41.245 deg)", sind(2 * TH23_TOE_DEG), 0.99133, rel=2e-4) v.check("sin(2*8.934 deg)", sind(2 * TH13_TOE_DEG), 0.30704, rel=1e-3) v.check("sin(66.43 deg)", sind(DELTA_TOE_DEG), 0.91626, rel=5e-4) k_toe = ( math.sin(2 * theta23_toe) * math.sin(2 * theta13_toe) * math.cos(theta13_toe) * math.sin(delta_toe) ) v.check( "K_TOE Jarlskog product", k_toe, 0.27761, rel=2e-3, detail="Expected fail: using the correct cos(8.934 deg) gives about 0.27543.", ) v.check( "J_TOE coefficient K_TOE/8", k_toe / 8, 0.034701, rel=2e-3, detail="Expected fail: follows from the K_TOE arithmetic error.", ) j_toe_2 = math.sin(2 * theta12_2) * k_toe / 8 j_pdg = ( sind(2 * TH12_PDG_DEG) * sind(2 * TH23_PDG_DEG) * sind(2 * TH13_PDG_DEG) * cosd(TH13_PDG_DEG) * sind(DELTA_PDG_DEG) ) / 8 v.check( "J_TOE at theta12=32.59 deg", j_toe_2, 0.031500, rel=3e-3, detail="Expected fail at strict precision: corrected K_TOE gives about 0.03125.", ) v.check("PDG absolute Jarlskog value", j_pdg, 0.030973, rel=5e-3) theta_from_pdg_j_with_toe_k = 0.5 * asind(j_pdg / (k_toe / 8)) v.check( "Jarlskog inversion using PDG J and TOE K", theta_from_pdg_j_with_toe_k, 33.38, abs_tol=0.1, detail="Expected fail: direct inversion gives about 31.8 deg, i.e. lower theta12.", ) v.record( "Jarlskog section gives mutually incompatible theta12 values", False, "31.62 deg, 32.59 deg, 33.62 deg, 33.19 deg, and 33.38 deg all appear", "one independent determination", "Expected fail: this is a logical consistency check, not a numeric tolerance.", ) delta_j_claim_deg = 0.44 j0_claim = 0.032220 k_claim = 0.27761 delta_j_from_paper_formula_deg = ((0.031500 - j0_claim) / ((k_claim / 4) * math.cos(2 * theta12_2))) / DEG v.check( "Delta_J from the paper's explicit derivative formula", delta_j_from_paper_formula_deg, delta_j_claim_deg, abs_tol=0.1, detail="Expected fail: the explicit formula gives about -1.41 deg, not +0.44 deg.", ) theta_full_as_written_deg = ( math.pi / 4 - THETA_C + LAM**2 / 2 + peirce_rad + LAM**3 / (4 * math.sin(2 * theta12_2)) + delta_j_claim_deg * DEG ) / DEG v.check( "closed-form theta12 formula as written", theta_full_as_written_deg, 33.38, abs_tol=0.1, detail="Expected fail: using +lambda^2/2 in radians evaluates near 34.36 deg.", ) peirce_u12_scalar = LAM**2 * Y2 / 8 v.check( "final calculation scalar y1^2*y2/8", peirce_u12_scalar, 0.006498, rel=1e-2, detail="Expected fail: y1^2*y2/8 is about 0.002685.", ) v.check( "final Peirce delta |U12|", peirce_u12_scalar * sind(TH12_2_DEG), 0.003502, rel=1e-2, detail="Expected fail: corrected scalar times sin(32.59 deg) is about 0.00145.", ) d_u12_c = math.cos(theta12_2) * LAM**3 / 2 v.check("final Cabibbo delta |U12|", d_u12_c, 0.004641, rel=5e-4) paper_u12_3 = 0.54689 theta_from_paper_u12_correct_cos = asind(paper_u12_3 / cosd(TH13_TOE_DEG)) v.check( "theta12 extracted from paper's |U12| with correct cos(theta13)", theta_from_paper_u12_correct_cos, 33.38, abs_tol=0.1, detail="Expected fail: corrected cos(8.934 deg) gives about 33.6 deg.", ) u12_corrected = sind(TH12_2_DEG) + peirce_u12_scalar * sind(TH12_2_DEG) + d_u12_c theta_corrected_definitive = asind(u12_corrected / cosd(TH13_TOE_DEG)) v.record( "corrected definitive calculation still closes within 1 sigma", abs(theta_corrected_definitive - TH12_PDG_DEG) < 0.77, f"{theta_corrected_definitive:.4f} deg", "within 0.77 deg of 33.44 deg", ) v.check("final stated residual 33.44 - 33.38", TH12_PDG_DEG - 33.38, 0.06, abs_tol=1e-12) v.check("final stated sigma residual", (TH12_PDG_DEG - 33.38) / 0.77, 0.08, rel=3e-2) sys.exit(v.summary())