"""v270 -- FLAV.PMNS.03: the complete complex PMNS matrix and the leptonic Jarlskog invariant J_PMNS (the CP STRENGTH), assembled from the four TFPT channels into one unitary object. The three angle channels and the CP phase are each fixed elsewhere; what is NEW here is that they assemble into a single unitary U_PMNS and that the CP STRENGTH J_PMNS = Im(U_e1 U_mu2 U*_e2 U*_mu1) is then a DERIVED number (no new freedom) that confronts the measured leptonic CP observable. This is the lepton- sector analogue of the CKM Jarlskog audit (v88), which existed only for quarks. The four TFPT channels (each derived/fixed elsewhere, assembled here): sin^2 theta12 = 1/3 - phi0/2 solar, mu-tau seesaw (v9 / v263) sin^2 theta23 = 1/2 (theta23=45) atmospheric, mu-tau sym (v9 / v263) sin^2 theta13 = phi0 e^{-5/6} reactor, carrier trace (v16 / v268) delta_PMNS = 4 pi/3 = 240 deg mu6 / triality phase (v231 / v233) [E] 1. ANGLE CHANNELS. the three mixing angles are the TFPT readouts: sin^2 th12 = 1/3 - phi0/2 = 0.30675, sin^2 th23 = 1/2, sin^2 th13 = phi0 e^{-5/6} = 0.02311 -- each from its own (separately established) channel. [E] 2. UNITARY ASSEMBLY. U = R23 . U13(delta) . R12 (PDG standard parametrisation) with these angles and delta = 4 pi/3 is exactly unitary (||U^dag U - I|| < 1e-12) -- one complex PMNS matrix, not three loose angles. [C] 3. JARLSKOG / CP STRENGTH. J_PMNS = Im(U_e1 U_mu2 U*_e2 U*_mu1) = s12 c12 s23 c23 s13 c13^2 sin(delta) = -0.02965, with maximal amplitude J_max = 0.03424 -- a DERIVED CP strength (the channels carry no free phase once delta is the mu6 value), the leptonic analogue of the CKM J (v88). [C] 4. DATA. J_max = 0.0342 vs NuFIT 5.2 J_max = 0.0332 (~3%); J_PMNS = -0.0297 vs the NuFIT best fit J ~ -0.026 (delta_data ~ 232 deg, TFPT delta = 240 deg within ~1 sigma); the negative sign (CP-violating, delta in (180,360)) is reproduced. A conditional readout, falsifiable by sharper delta_CP. [O] 5. PROVENANCE + RESIDUAL. delta is the mu6/triality phase (v231/v233), an ASSEMBLED input, NOT derived from the M_nu/D_F seesaw; and the absolute neutrino-mass scale (Sigma m_nu, the seesaw M_R) stays open. So this closes the CP-STRENGTH readout, not the phase's operator origin nor the scale. Status: [E] the angle channels + the exact unitary assembly; [C] the derived Jarlskog CP strength + its data confrontation; [O] the phase's operator origin and the absolute scale. Assembles the full PMNS matrix + CP strength; does not derive delta from D_F nor fix the scale. Python-only (mpmath + numpy; numerical readout, like the CKM J audit v88 -- no Wolfram mirror). """ import mpmath as mp import numpy as np from tfpt_constants import check, summary, reset, phi0 mp.mp.dps = 30 # NuFIT 5.2 (2022, normal ordering) data snapshot, frozen for comparison J_MAX_DATA = mp.mpf("0.0332") J_BEST_DATA = mp.mpf("-0.026") DELTA_DATA_DEG = mp.mpf("232") def pmns_matrix(s12, s23, s13, delta): """full complex PMNS in the PDG standard parametrisation U = R23.U13(delta).R12.""" c12, c23, c13 = (mp.sqrt(1 - s12 ** 2), mp.sqrt(1 - s23 ** 2), mp.sqrt(1 - s13 ** 2)) e = mp.e ** (-1j * delta) U = mp.matrix([ [c12 * c13, s12 * c13, s13 * e], [-s12 * c23 - c12 * s23 * s13 * mp.conj(e), c12 * c23 - s12 * s23 * s13 * mp.conj(e), s23 * c13], [s12 * s23 - c12 * c23 * s13 * mp.conj(e), -c12 * s23 - s12 * c23 * s13 * mp.conj(e), c23 * c13], ]) return U def run(): reset() print("v270 FLAV.PMNS.03: the complete complex PMNS matrix + the leptonic Jarlskog J_PMNS (CP strength)") p0 = mp.mpf(phi0) s12sq = mp.mpf(1) / 3 - p0 / 2 s23sq = mp.mpf(1) / 2 s13sq = p0 * mp.e ** (-mp.mpf(5) / 6) delta = mp.mpf(4) * mp.pi / 3 s12, s23, s13 = mp.sqrt(s12sq), mp.sqrt(s23sq), mp.sqrt(s13sq) c12, c23, c13 = mp.sqrt(1 - s12sq), mp.sqrt(1 - s23sq), mp.sqrt(1 - s13sq) # 1. angle channels check("ANGLE CHANNELS [E]: sin^2 th12 = 1/3 - phi0/2 = %.5f (v9/v263), " "sin^2 th23 = 1/2 (theta23 = 45 deg, v9/v263), sin^2 th13 = phi0 e^{-5/6} " "= %.5f (v16/v268) -- the three TFPT mixing-angle readouts" % (float(s12sq), float(s13sq)), abs(s12sq - mp.mpf("0.306747")) < 1e-5 and s23sq == mp.mpf(0.5) and abs(s13sq - mp.mpf("0.0231")) < 5e-4) # 2. unitary assembly U = pmns_matrix(s12, s23, s13, delta) Udag = U.transpose_conj() resid = mp.norm(Udag * U - mp.eye(3)) check("UNITARY ASSEMBLY [E]: U = R23 . U13(delta) . R12 with delta = 4 pi/3 = " "240 deg is exactly unitary (||U^dag U - I|| = %.1e) -- one complex PMNS " "matrix, not three loose angles" % float(resid), resid < 1e-12) # 3. Jarlskog / CP strength (two independent ways) J_formula = s12 * c12 * s23 * c23 * s13 * c13 ** 2 * mp.sin(delta) J_matrix = (U[0, 0] * U[1, 1] * mp.conj(U[0, 1]) * mp.conj(U[1, 0])).imag J_max = s12 * c12 * s23 * c23 * s13 * c13 ** 2 agree = abs(J_formula - J_matrix) < 1e-12 check("JARLSKOG / CP STRENGTH [C]: J_PMNS = Im(U_e1 U_mu2 U*_e2 U*_mu1) = " "%.5f (formula and full-matrix agree: |diff| = %.1e), amplitude J_max = " "%.5f -- a DERIVED CP strength, the leptonic analogue of the CKM J (v88)" % (float(J_matrix), float(abs(J_formula - J_matrix)), float(J_max)), agree and abs(J_matrix - mp.mpf("-0.02965")) < 1e-4) # 4. data dJmax = 100 * abs(J_max - J_MAX_DATA) / J_MAX_DATA check("DATA [C]: J_max = %.4f vs NuFIT 5.2 J_max = 0.0332 (~%.0f%%); J_PMNS = " "%.4f vs NuFIT best fit ~ -0.026 (delta_data ~ 232 deg, TFPT 240 deg within " "~1 sigma); the negative sign (CP-violating, delta in (pi,2pi)) is " "reproduced -- a conditional readout, falsifiable by sharper delta_CP" % (float(J_max), float(dJmax), float(J_matrix)), dJmax < 6 and J_matrix < 0 and abs(J_matrix - J_BEST_DATA) < 0.01) # 5. provenance + residual (honest) check("PROVENANCE + RESIDUAL [O]: delta = 4 pi/3 is the mu6/triality phase " "(v231/v233), an ASSEMBLED input -- NOT derived from the M_nu/D_F seesaw; " "and the absolute neutrino-mass scale (Sigma m_nu, the seesaw M_R) stays " "open. This closes the CP-STRENGTH readout, not the phase's operator origin " "nor the absolute scale -- a PMNS assembly, not a full operator closure", True) return summary("v270 full complex PMNS + Jarlskog CP strength J = -0.0297 (assembled, data-consistent); scale + phase-origin stay open (FLAV.PMNS.03)") if __name__ == "__main__": raise SystemExit(1 if run() else 0)