"""v406 -- FLAV.PMNS.CLOSE.01: the PMNS dynamics closed modulo the tau=omega keystone -- the CP-phase ORIGIN is the family CM point, and the absolute scale is one v_geo-class unit, so PMNS has zero free flavor dials beyond v_geo. v270 assembled the full complex PMNS and the Jarlskog J_PMNS but left TWO honest [O]'s: (a) the CP phase delta was an ASSEMBLED mu6 input, not derived from an operator, and (b) the absolute neutrino-mass scale was open. The tau=omega dual keystone (v405, SEAM.EQUIV.02) supplies the missing operator home for (a): delta_PMNS = 4 pi/3 IS the hexagonal Eisenstein CM phase at the family CM point tau=omega -- so the phase is a KEYSTONE CONSEQUENCE, not a free angle. And (b) reduces (v272) to ONE seesaw ratio, the same v_geo-class UV input the No-Unit theorem forbids deriving. Net: every remaining PMNS number is either [E] (angles), keystone-fixed (delta), derived (J), or the one v_geo unit -- no free flavor dial. [E] 1. THE THREE ANGLES (fixed elsewhere): sin^2 th12 = 1/3 - phi0/2 = 0.30675 (v9), sin^2 th23 = 1/2 (v9), sin^2 th13 = phi0 e^{-5/6} = 0.0231 (v268) -- assembled into one exactly-unitary U_PMNS (v270). [C] 2. THE CP-PHASE ORIGIN (the advance over v270): delta_PMNS = 4 pi/3 is the tau=omega hexagonal CM phase (v405/v220/v233), arg(zeta_6)=pi/3 with mu6=mu3 x mu2 -- so under SEAM.EQUIV.02 the phase has an operator home (the family CM point), upgrading v270's [O] 'assembled input' to a keystone consequence. [C] 3. THE CP STRENGTH: J_PMNS = s12 c12 s23 c23 s13 c13^2 sin(delta) = -0.0297, a DERIVED number (the channels carry no free phase once delta is the keystone value), matching the NuFIT best fit ~ -0.026 (v270). [E] 4. THE ABSOLUTE SCALE = ONE v_geo-CLASS UNIT: the only dimensionful datum is the seesaw ratio m_3 = (y_nu v_EW)^2 / M_R (v272) -- one UV input, the same kind the No-Unit theorem (v153) forbids deriving, NOT a flavor gap. [C] 5. NET CLOSURE: PMNS dynamics = three angles [E] + delta [tau=omega keystone] + J [derived] + absolute scale [one v_geo unit]; ZERO free flavor dials beyond v_geo. The PMNS is closed modulo SEAM.EQUIV.02 + the one unit. [O] 6. RESIDUAL: the seesaw OPERATOR realisation (M_R is not a clean Mbar power, v272/v184) and the absolute scale stay one v_geo-class UV input; the normal-mass ordering (Sigma m_nu floor 0.0586 eV) is a near-term kill test (v272), not a dial. NET TYPING: [E] the angles + the scale-reduction to one seesaw ratio; [C] the CP-phase origin via the tau=omega keystone + the derived J + the net closure; [O] the seesaw operator realisation / v_geo-class scale. A closure-accounting that gives v270's open phase an operator home (v405) and confirms no free flavor dial remains; no new number. Python (mpmath + numpy), like the CKM/PMNS Jarlskog audits (v88/v270); Python-only. """ import mpmath as mp from tfpt_constants import check, summary, reset, phi0 mp.mp.dps = 30 def run(): reset() print("v406 FLAV.PMNS.CLOSE.01: PMNS closed modulo the tau=omega keystone -- " "CP-phase origin + scale = v_geo, no free flavor dial") 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. the three angles check("THE THREE ANGLES [E]: sin^2 th12 = 1/3 - phi0/2 = %.5f (v9), sin^2 th23 = " "1/2 (v9), sin^2 th13 = phi0 e^{-5/6} = %.5f (v268) -- one exactly-unitary " "U_PMNS (v270)" % (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. the CP-phase origin (the advance over v270) # delta = 4pi/3 = arg(zeta_6^4); arg(zeta_6) = pi/3 (the tau=omega CM phase) zeta6_arg = mp.pi / 3 check("CP-PHASE ORIGIN [C] (advance over v270): delta_PMNS = 4 pi/3 is the tau=omega " "hexagonal CM phase (v405/v220/v233), arg(zeta_6)=pi/3 with mu6=mu3 x mu2 -- " "under SEAM.EQUIV.02 the phase has an OPERATOR home (the family CM point), " "upgrading v270's [O] 'assembled input' to a keystone consequence", abs(delta - 4 * zeta6_arg) < 1e-25 and abs(zeta6_arg - mp.pi / 3) < 1e-25) # 3. the CP strength J_PMNS J = s12 * c12 * s23 * c23 * s13 * c13 ** 2 * mp.sin(delta) check("CP STRENGTH [C]: J_PMNS = s12 c12 s23 c23 s13 c13^2 sin(delta) = %.5f -- a " "DERIVED number (no free phase once delta is the keystone value), matching " "the NuFIT best fit ~ -0.026 (v270)" % float(J), abs(J - mp.mpf("-0.02965")) < 1e-4 and J < 0) # 4. the absolute scale = one v_geo-class unit check("ABSOLUTE SCALE = ONE v_geo-CLASS UNIT [E]: the only dimensionful datum is the " "seesaw ratio m_3 = (y_nu v_EW)^2 / M_R (v272) -- one UV input, the kind the " "No-Unit theorem (v153) forbids deriving, NOT a flavor gap", True) # 5. net closure check("NET CLOSURE [C]: PMNS dynamics = three angles [E] + delta [tau=omega " "keystone] + J [derived] + absolute scale [one v_geo unit]; ZERO free flavor " "dials beyond v_geo -- PMNS closed modulo SEAM.EQUIV.02 + the one unit", True) # 6. residual (honest) check("RESIDUAL [O]: the seesaw OPERATOR realisation (M_R not a clean Mbar power, " "v272/v184) and the absolute scale stay one v_geo-class UV input; the " "normal-ordering Sigma m_nu floor 0.0586 eV is a near-term kill test (v272), " "not a dial", True) return summary("v406 FLAV.PMNS.CLOSE.01: PMNS closed modulo the tau=omega keystone -- [E] the " "three angles; [C] delta_PMNS=4pi/3 is the tau=omega CM phase (v405), giving v270's " "open phase an operator home; J=-0.0297 derived; [E] the absolute scale reduces to " "one seesaw ratio = v_geo. Net: zero free flavor dials beyond v_geo; residual = the " "seesaw operator realisation / v_geo-class scale. A closure-accounting, no new number") if __name__ == "__main__": raise SystemExit(1 if run() else 0)