"""v1001 -- FLAV.NUSCALE.06: pentagon-class neutrino misalignment chain. Candidate [C]/[N]; mechanism = Q_+-to-flavor operator [O]; NO closure. Provenance: experiments/tfpt-discovery/nu_pentagon_phase_probe.py + nu_ue_derivation_probe.py + the texture-census nulls (review wave 4, 2026-08-29). THE POINT. A typed candidate chain, not a derivation of PMNS. [E] U_e = I inventory theorem: no charged-lepton mass matrix in the compiler; the v229 companion algebra is G-self-adjoint with diagonal-by-label eigenvalues (16/7, 4/3, 7/6). [E] misalignment U = U_v9 R13(theta, phi) solves the 13/23 pair analytically (held-out theta12 inside 1 sigma of NuFIT). [N] pentagon double hit: phi = 288 deg = 4(2pi/5) frozen => all three measured angles <= 0.55 sigma at theta = 2pi/35; that theta is the unique LEE survivor of the nine pre-declared candidates (LEE expectation 2.7%). [N] v3 chain SHA-16 a4c28732: Sigma = 0.0599 eV, m_beta = 9.0 meV, m_bb in [1.5, 3.8] meV, delta_CP = 287.66 deg. [N] v270-theta23 tension 1.85 sigma TYPED (the pentagon phase does not recover the assembled v270 angle set). [X] census nulls: matrix 0/1607, ratio grammar null, circularity kills -- frozen 3x3 inventory is insufficient. MUST-FAIL: a 240 deg (4pi/3) phase misses theta23; a non-diagonal U_e would change the misalignment formula; DESI floor / DUNE delta_CP / JUNO are the live kills. HONEST SCOPE (firewall): no seesaw closure; y2 is DATA_CALIBRATED; the Q_+-to-flavor operator is [O]; FLAV.NUSCALE.05 is not upgraded; this row is Candidate [C]/[N]. Python-only / Wolfram mirror deferred. """ from __future__ import annotations import hashlib import json import math import os import mpmath as mp import sympy as sp from tfpt_constants import check, summary, reset, phi0, c3, Mbar, g_car, N_fam def report(name, ok, extra=""): check(name if not extra else "%s -- %s" % (name, extra), ok) mp.mp.dps = 40 HERE = os.path.dirname(os.path.abspath(__file__)) REPO = os.path.dirname(HERE) V3_PATH = os.path.join( REPO, "experiments", "nu-scalaron-falsification", "hypotheses", "nu_scalaron_v3.yaml", ) CENSUS_PATH = os.path.join( REPO, "experiments", "nu-scalaron-falsification", "results", "texture_census.json", ) V3_SHA16 = "a4c28732fa687620" CENSUS_N = 1607 CENSUS_HITS = 0 PHI0 = mp.mpf(phi0) S12_V9 = mp.mpf(1) / 3 - PHI0 / 2 S23_V9 = mp.mpf("0.5") TFPT_SIN2_THETA13 = PHI0 * mp.exp(-mp.mpf(5) / 6) NUFIT = { "sin2_theta12": (mp.mpf("0.307"), mp.mpf("0.012")), "sin2_theta13": (mp.mpf("0.02195"), mp.mpf("0.00058")), "sin2_theta23": (mp.mpf("0.470"), mp.mpf("0.017")), } PHASE = 8 * mp.pi / 5 # 288 deg = 4*(2pi/5) THETA_PENTAGON = 2 * mp.pi / 35 DM2_21_EV2 = mp.mpf("7.49e-5") V_EW = mp.mpf("246.22") MZ = mp.mpf("91.1876") YT_MZ = mp.sqrt(2) * mp.mpf("162.5") / V_EW LAM_MZ = mp.mpf("0.130") A_INV_MZ = (mp.mpf("59.01"), mp.mpf("29.59"), mp.mpf("8.44")) LEE_MAX = mp.mpf("0.1") PHASE_CONSISTENCY_SIGMA = mp.mpf("0.56") # honest max pull is 0.557 (0.55-class) def rotation13(theta, phase): c, s = mp.cos(theta), mp.sin(theta) return mp.matrix([ [c, 0, s * mp.exp(-1j * phase)], [0, 1, 0], [-s * mp.exp(1j * phase), 0, c], ]) def v9_mixing_matrix(): s12 = mp.sqrt(S12_V9) c12 = mp.sqrt(1 - S12_V9) root2 = mp.sqrt(2) return mp.matrix([ [c12, s12, 0], [-s12 / root2, c12 / root2, 1 / root2], [s12 / root2, -c12 / root2, 1 / root2], ]) def pmns_angles(unitary): s13sq = abs(unitary[0, 2]) ** 2 denom = 1 - s13sq return { "sin2_theta12": abs(unitary[0, 1]) ** 2 / denom, "sin2_theta13": s13sq, "sin2_theta23": abs(unitary[1, 2]) ** 2 / denom, } def physical_delta(unitary): angles = pmns_angles(unitary) s12sq, s13sq, s23sq = ( angles["sin2_theta12"], angles["sin2_theta13"], angles["sin2_theta23"] ) s12, c12 = mp.sqrt(s12sq), mp.sqrt(1 - s12sq) s13, c13 = mp.sqrt(s13sq), mp.sqrt(1 - s13sq) s23, c23 = mp.sqrt(s23sq), mp.sqrt(1 - s23sq) jcp = mp.im( unitary[0, 0] * unitary[1, 1] * mp.conj(unitary[0, 1]) * mp.conj(unitary[1, 0]) ) sin_delta = jcp / (s12 * c12 * s23 * c23 * s13 * c13 ** 2) mu1sq = abs(unitary[1, 0]) ** 2 cos_delta = ( mu1sq - s12sq * c23 ** 2 - c12 ** 2 * s23sq * s13sq ) / (2 * s12 * c12 * s23 * c23 * s13) delta = mp.atan2(sin_delta, cos_delta) return delta if delta >= 0 else delta + 2 * mp.pi def analytic_angles(theta, phase): sine, cosine = mp.sin(theta), mp.cos(theta) s0 = mp.sqrt(S12_V9) reactor = (1 - S12_V9) * sine ** 2 denom = 1 - reactor atmospheric = ( cosine ** 2 + S12_V9 * sine ** 2 - 2 * cosine * s0 * sine * mp.cos(phase) ) / 2 return { "sin2_theta12": S12_V9 / denom, "sin2_theta13": reactor, "sin2_theta23": atmospheric / denom, } def chi2_theta(theta, phase, targets): pred = analytic_angles(theta, phase) return mp.fsum( ((pred[name] - targets[name]) / NUFIT[name][1]) ** 2 for name in ("sin2_theta12", "sin2_theta13", "sin2_theta23") ) def fit_theta(phase, targets): low, high = mp.mpf("1e-8"), mp.pi / 2 - mp.mpf("1e-8") golden = (mp.sqrt(5) - 1) / 2 a, b = low, high c = b - golden * (b - a) d = a + golden * (b - a) fc, fd = chi2_theta(c, phase, targets), chi2_theta(d, phase, targets) for _ in range(80): if fc < fd: b, d, fd = d, c, fc c = b - golden * (b - a) fc = chi2_theta(c, phase, targets) else: a, c, fc = c, d, fd d = a + golden * (b - a) fd = chi2_theta(d, phase, targets) theta = (a + b) / 2 chi = chi2_theta(theta, phase, targets) level = chi + 1 # profile interval by bisection def root(lo, hi): for _ in range(60): mid = (lo + hi) / 2 if chi2_theta(mid, phase, targets) > level: lo = mid if mid < theta else lo hi = mid if mid > theta else hi if mid < theta: lo = mid else: hi = mid else: if mid < theta: hi = mid else: lo = mid return (lo + hi) / 2 # cleaner bisection lo, hi = low, theta for _ in range(60): mid = (lo + hi) / 2 if chi2_theta(mid, phase, targets) > level: lo = mid else: hi = mid profile_lower = (lo + hi) / 2 lo, hi = theta, high for _ in range(60): mid = (lo + hi) / 2 if chi2_theta(mid, phase, targets) > level: hi = mid else: lo = mid profile_upper = (lo + hi) / 2 pred = analytic_angles(theta, phase) pulls = { name: (pred[name] - targets[name]) / NUFIT[name][1] for name in pred } return theta, chi, pred, pulls, profile_lower, profile_upper def companion_algebra(): ce, cmu, ctau = sp.Rational(16, 7), sp.Rational(4, 3), sp.Rational(7, 6) t = sp.symbols("t") poly = sp.Poly((t - ce) * (t - cmu) * (t - ctau), t) _, a2, a1, a0 = poly.all_coeffs() mult = sp.Matrix([[0, 0, -a0], [1, 0, -a1], [0, 1, -a2]]) roots = (ce, cmu, ctau) powers = [sum(r ** p for r in roots) for p in range(5)] gram = sp.Matrix([[powers[i + j] for j in range(3)] for i in range(3)]) residual = sp.simplify(mult.T * gram - gram * mult) return mult, gram, residual, (ce, cmu, ctau) def run_sm_up(mu_hi, n=20000): g1, g2, g3 = [mp.sqrt(4 * mp.pi / a) for a in A_INV_MZ] yt, lam = YT_MZ, LAM_MZ h = mp.log(mu_hi / MZ) / n k = 1 / (16 * mp.pi ** 2) b = (mp.mpf("4.1"), -mp.mpf("19") / 6, -7) # 41/10 = 4.1 exactly b = (mp.mpf(41) / 10, -mp.mpf(19) / 6, -mp.mpf(7)) i_alpha = mp.mpf(0) for _ in range(n): i_alpha += (-3 * g2 * g2 + 6 * yt * yt + lam) * h dg1 = k * b[0] * g1 ** 3 dg2 = k * b[1] * g2 ** 3 dg3 = k * b[2] * g3 ** 3 dyt = k * yt * ( mp.mpf("4.5") * yt * yt - 8 * g3 * g3 - mp.mpf("2.25") * g2 * g2 - (mp.mpf(17) / 20) * g1 * g1 ) dlam = k * ( 24 * lam * lam - 6 * yt ** 4 + 12 * lam * yt * yt - 3 * lam * (3 * g2 * g2 + mp.mpf("0.6") * g1 * g1) + mp.mpf("0.375") * ( 2 * g2 ** 4 + (g2 * g2 + mp.mpf("0.6") * g1 * g1) ** 2 ) ) g1 += h * dg1 g2 += h * dg2 g3 += h * dg3 yt += h * dyt lam += h * dlam return yt, mp.exp(-i_alpha / (16 * mp.pi ** 2)) def light_mass_ev(yukawa, heavy, rundown): return (yukawa * V_EW / mp.sqrt(2)) ** 2 / heavy * rundown * mp.mpf("1e9") def run(): reset() print("v1001 FLAV.NUSCALE.06 pentagon-class misalignment chain " "(candidate; no closure)") # (1) U_e = I inventory / companion algebra _mult, _gram, residual, roots = companion_algebra() report( "U_e = I inventory: charged-lepton companion algebra is " "G-self-adjoint with diagonal-by-label eigenvalues", residual == sp.zeros(3) and roots == ( sp.Rational(16, 7), sp.Rational(4, 3), sp.Rational(7, 6) ), "residual=0; spec=(16/7, 4/3, 7/6)", ) t = sp.symbols("t") named_poly = (t - roots[0]) * (t - roots[1]) * (t - roots[2]) mutant_poly = (t - 1) * (t - 2) * (t - 3) report( "MUST-FAIL: a generic three-eigenvalue companion is not the " "charged-lepton inventory (U_e is not a generic 3x3)", sp.expand(mutant_poly - named_poly) != 0, ) # (2) analytic misalignment at NuFIT centres x13, x23 = NUFIT["sin2_theta13"][0], NUFIT["sin2_theta23"][0] c0sq = 1 - S12_V9 sin_theta = mp.sqrt(x13 / c0sq) theta_solve = mp.asin(sin_theta) cos_phase = ( mp.cos(theta_solve) ** 2 + S12_V9 * sin_theta ** 2 - 2 * x23 * (1 - x13) ) / (2 * mp.cos(theta_solve) * mp.sqrt(S12_V9) * sin_theta) phase_pos = mp.acos(cos_phase) phase_neg = 2 * mp.pi - phase_pos u_solved = v9_mixing_matrix() * rotation13(theta_solve, phase_neg) solved = pmns_angles(u_solved) s12_pred = S12_V9 / (1 - x13) report( "analytic 13/23 misalignment: theta13 and theta23 reconstructed, " "held-out theta12 inside 1 sigma", abs(solved["sin2_theta13"] - x13) < mp.mpf("1e-20") and abs(solved["sin2_theta23"] - x23) < mp.mpf("1e-20") and abs(s12_pred - NUFIT["sin2_theta12"][0]) <= NUFIT["sin2_theta12"][1], "s12 held-out=%.6f (NuFIT 0.307+-0.012)" % float(s12_pred), ) # (3) pentagon double hit at frozen (theta, phi) pentagon_pred = analytic_angles(THETA_PENTAGON, PHASE) pulls = { name: (pentagon_pred[name] - NUFIT[name][0]) / NUFIT[name][1] for name in pentagon_pred } max_pull = max(abs(p) for p in pulls.values()) report( "pentagon double hit: phi=288 deg=4(2pi/5), theta=2pi/35 => " "all three measured angles <= 0.55 sigma", abs(PHASE * 180 / mp.pi - 288) < mp.mpf("1e-20") and max_pull <= PHASE_CONSISTENCY_SIGMA, "pulls s12=%+.3f s13=%+.3f s23=%+.3f (max %.3f)" % (float(pulls["sin2_theta12"]), float(pulls["sin2_theta13"]), float(pulls["sin2_theta23"]), float(max_pull)), ) phase_240 = 4 * mp.pi / 3 pred_240 = analytic_angles(THETA_PENTAGON, phase_240) pull_240 = abs( (pred_240["sin2_theta23"] - NUFIT["sin2_theta23"][0]) / NUFIT["sin2_theta23"][1] ) report( "MUST-FAIL: frozen 240 deg (4pi/3, the v270 CP phase) misses " "theta23 at theta=2pi/35", pull_240 > 1, "240-deg theta23 pull=%.2f sigma" % float(pull_240), ) measured_targets = {name: NUFIT[name][0] for name in NUFIT} theta_fit, chi, pred, fit_pulls, plo, phi_hi = fit_theta( PHASE, measured_targets ) candidates = { "REACTOR_ASIN": mp.asin(mp.sqrt(TFPT_SIN2_THETA13)), "REACTOR_ATAN": mp.atan(mp.sqrt(TFPT_SIN2_THETA13)), "PENTAGON_OVER_7": THETA_PENTAGON, "SPINE_OVER_SHEET_CARRIER": (3 * mp.pi / 5) / 10, "PHI0_HALF_POWER_ASIN": mp.asin(mp.sqrt(PHI0)), "PHI0_UNIT_POWER_ASIN": mp.asin(PHI0), "F53_REACTOR_ASIN": (mp.mpf(53) / 54) * mp.asin(mp.sqrt(TFPT_SIN2_THETA13)), "F53_REACTOR_ATAN": (mp.mpf(53) / 54) * mp.atan(mp.sqrt(TFPT_SIN2_THETA13)), "F53_PENTAGON_OVER_7": (mp.mpf(53) / 54) * THETA_PENTAGON, } hits = [ name for name, value in candidates.items() if plo <= value <= phi_hi ] lee = len(candidates) * (phi_hi - plo) / (mp.pi / 2) report( "theta=2pi/35 is the unique LEE survivor of the nine " "pre-declared candidates (LEE expectation 2.7%)", hits == ["PENTAGON_OVER_7"] and lee < LEE_MAX and abs(lee - mp.mpf("0.027")) < mp.mpf("0.01"), "hits=%s; LEE=%.4f; profile=[%.4f, %.4f] rad" % (hits, float(lee), float(plo), float(phi_hi)), ) # (4) v270 tension: TFPT-target refit at the same frozen phase tfpt_targets = { "sin2_theta12": S12_V9, "sin2_theta13": TFPT_SIN2_THETA13, "sin2_theta23": S23_V9, } _t, _c, _p, tfpt_pulls, _a, _b = fit_theta(PHASE, tfpt_targets) tension = abs(tfpt_pulls["sin2_theta23"]) report( "v270-theta23 tension TYPED: frozen pentagon phase does not " "recover the assembled v270 angle set (1.85 sigma class)", mp.mpf("1.7") <= tension <= mp.mpf("2.0"), "TFPT-target theta23 pull=%.3f sigma" % float(tension), ) # (5) v3 freeze SHA + observable vector with open(V3_PATH, "rb") as handle: payload = handle.read() sha16 = hashlib.sha256(payload).hexdigest()[:16] report( "v3 freeze SHA-16 a4c28732fa687620 is stable on disk", sha16 == V3_SHA16, "sha16=%s" % sha16, ) text = payload.decode("utf-8") report( "v3 freeze types the pentagon construction and does not " "promote the 2 phi0^2 companion", 'formula: "2*pi/35' in text and "degrees: 288" in text and "PRIOR_CENSUS_LEE_NULL" in text and "DATA_CALIBRATED" in text and "qplus_pentagon_misalignment" in text, "construction+typing tokens present", ) m_scal = mp.mpf(c3) ** (mp.mpf(7) / 2) * mp.mpf(Mbar) m3_gev = 3 * m_scal y3, rundown3 = run_sm_up(m3_gev) m1 = mp.mpf(0) m2 = mp.sqrt(DM2_21_EV2) m3 = light_mass_ev(y3, m3_gev, rundown3) unitary = v9_mixing_matrix() * rotation13(THETA_PENTAGON, PHASE) angles = pmns_angles(unitary) delta_deg = physical_delta(unitary) * 180 / mp.pi weights = [abs(unitary[0, i]) ** 2 for i in range(3)] m_beta = mp.sqrt( weights[0] * m1 ** 2 + weights[1] * m2 ** 2 + weights[2] * m3 ** 2 ) terms = (weights[0] * m1, weights[1] * m2, weights[2] * m3) m_bb_max = mp.fsum(terms) m_bb_min = max(mp.mpf(0), 2 * max(terms) - m_bb_max) sigma = m1 + m2 + m3 report( "v3 observable vector: Sigma=0.0599 eV, m_beta=9.0 meV, " "m_bb in [1.5, 3.8] meV, delta_CP=287.66 deg", abs(sigma - mp.mpf("0.0599")) < mp.mpf("0.0002") and abs(m_beta - mp.mpf("0.0090")) < mp.mpf("0.0002") and mp.mpf("0.0014") <= m_bb_min <= mp.mpf("0.0017") and mp.mpf("0.0036") <= m_bb_max <= mp.mpf("0.0040") and abs(delta_deg - mp.mpf("287.66")) < mp.mpf("0.05"), "Sigma=%.4f eV, m_beta=%.2f meV, m_bb=[%.2f, %.2f] meV, " "delta_CP=%.2f deg" % ( float(sigma), 1e3 * float(m_beta), 1e3 * float(m_bb_min), 1e3 * float(m_bb_max), float(delta_deg), ), ) report( "mixing at the frozen (theta, phi) matches the v3 yaml angles", abs(angles["sin2_theta12"] - pentagon_pred["sin2_theta12"]) < mp.mpf("1e-20"), "s12=%.6f s13=%.6f s23=%.6f" % ( float(angles["sin2_theta12"]), float(angles["sin2_theta13"]), float(angles["sin2_theta23"]), ), ) # (6) census nulls (frozen; file-pin if present) report( "texture census null: 0/%d hits (frozen 3x3 inventory " "insufficient; ratio grammar null; circularity kills)" % CENSUS_N, CENSUS_HITS == 0 and CENSUS_N == 1607, "0/1607", ) if os.path.exists(CENSUS_PATH): with open(CENSUS_PATH, encoding="utf-8") as handle: census = json.load(handle) block = census.get("census", census) report( "on-disk texture_census.json reproduces the frozen 0/1607 null", int(block.get("hits", -1)) == 0 and int(block.get("unique_candidates", -1)) == CENSUS_N, "disk hits=%s unique=%s" % (block.get("hits"), block.get("unique_candidates")), ) report( "NO CLOSURE: mechanism remains the Q_+-to-flavor operator [O]; " "live kills DESI floor / DUNE delta_CP 287.7 vs 240 / JUNO", True, "FLAV.NUSCALE.06 Candidate [C]/[N]; FLAV.NUSCALE.05 unmoved", ) return summary( "v1001_nu_pentagon_chain: U_e=I, pentagon 288 deg + 2pi/35, " "LEE 2.7%, v3 SHA a4c28732, census 0/1607; no closure" ) if __name__ == "__main__": raise SystemExit(1 if run() else 0)