"""v233 -- the CP phase is the universal family/triality phase, only sheet-split. v231 showed both CP phases are mu6 powers of one hexagonal unit rho=e^{i pi/3}, split by the Z2 sheet (rho^3=-1). This pushes the reduction one step further, into the canonical mu6 = mu3 (family/triality) x mu2 (sheet) factorisation: the FAMILY part of the CP phase is the SAME triality-centre element for both sectors -- it is not even a free choice of power. Only the Z2 sheet distinguishes quark from lepton. mu6 = mu3 x mu2, rho = omega^2 * (-1) with omega = e^{2 pi i/3} (the Z3 family triality centre = the 2/3 cusp weight), and -1 = rho^3 the Z2 sheet. quark delta_CKM,lead = arg(rho^1): family class = [rho] in mu6/mu2, sheet = -1 lepton delta_PMNS = arg(rho^4): family class = [rho] in mu6/mu2, sheet = +1 since rho^4 = rho^1 * rho^3 (rho^3 in mu2), the two phases have the SAME image in the family quotient mu6/mu2 ~ mu3 and OPPOSITE sheet -- so the CP phase IS the family triality phase (the cusp-weight Z3 centre), read on the two sheets. [E] 1. mu6 = mu3 x mu2 (gcd(3,2)=1): the family triality centre {1,omega,omega^2} = the cusp weights {0,1/3,2/3} x 2pi; the sheet is mu2 = {1,-1}. [E] 2. rho = omega^2 * (-1): the hexagonal CM unit factors as (family triality omega^2) x (sheet -1). [E] 3. SAME FAMILY CLASS: rho^4 = rho^1 * rho^3, and rho^3 in mu2, so rho^1 and rho^4 have the SAME image in the family quotient mu6/mu2 (= the same triality centre element); their mu2 (sheet) images are opposite ((-1)^1 vs (-1)^4). [E] 4. So the CP phase = the universal family triality phase (the 2/3 cusp-weight Z3 centre), sheet-split: quark = (triality) x (sheet -), lepton = (triality) x (sheet +). NOT a free choice of power -- one universal phase + the sheet. [E] 5. omega (the triality centre) sits in the (2,3,5) Milnor/Coxeter monodromy spectrum (e^{2 pi i m/30}, m the E8 exponents): the mu3 family factor of the order-30 cycle. So CP lives inside the same monodromy that builds the hull. [E] 6. NEG: a mu2-only phase carries no family triality; a mu3-only phase carries no sheet; only the full mu6=mu3 x mu2 gives 'one triality phase, two sheets'. Status: [E] for the mu3 x mu2 factorisation and the same-family/opposite-sheet statement; the physical CP derivation stays [C] (the quark seam correction 3 lambda_C^2 is the v88 misalignment). Reduces Target D's CP residual from 'one hexagonal unit + sheet' (v231) to 'the universal triality phase + sheet' -- the power choice is removed. Mirrored in wolfram/tfpt_readouts_extension.wl. """ import sympy as sp from tfpt_constants import check, summary, reset, N_fam def run(): reset() print("v233 CP = the universal family/triality phase, sheet-split (mu6 = mu3 x mu2)") rho = sp.exp(sp.I * sp.pi / 3) # hexagonal CM unit, order 6 omega = sp.exp(2 * sp.I * sp.pi / 3) # Z3 family / triality centre, order 3 sheet = sp.Integer(-1) # Z2 sheet = rho^3 check("mu6 = mu3 x mu2 (gcd(3,2)=1): family triality centre {1,omega,omega^2} " "= cusp weights {0,1/3,2/3} x 2pi; sheet mu2 = {1,-1}", sp.gcd(3, 2) == 1 and sp.simplify(omega**3) == 1 and sp.simplify(sheet**2) == 1 and sp.simplify(rho**3 - sheet) == 0) check("rho = omega^2 * (-1): the hexagonal CM unit = (family triality omega^2) x " "(sheet -1)", sp.simplify(rho - omega**2 * sheet) == 0) # cusp weights {0,1/3,2/3} -> triality phases {1, omega, omega^2} cusp = [sp.exp(2 * sp.I * sp.pi * w) for w in (sp.Integer(0), sp.Rational(1, 3), sp.Rational(2, 3))] check("the three cusp weights {0,1/3,2/3} give the triality centre {1,omega,omega^2}", set(sp.simplify(c) for c in cusp) == {sp.simplify(omega**k) for k in range(3)}) # quark rho^1, lepton rho^4: same family class (mu6/mu2), opposite sheet # family quotient: reduce exponent mod 2 is the SHEET; the family class is exponent mod 3 q_fam, q_sheet = 1 % 3, (-1)**1 l_fam, l_sheet = 4 % 3, (-1)**4 check("SAME FAMILY CLASS: rho^4 = rho^1 * rho^3 with rho^3 in mu2, so rho^1 and " "rho^4 have the SAME image in mu6/mu2 ~ mu3 (family class %d == %d); " "OPPOSITE sheet ((-1)^1=%d vs (-1)^4=%d)" % (q_fam, l_fam, q_sheet, l_sheet), sp.simplify(rho**4 - rho**1 * rho**3) == 0 and q_fam == l_fam and q_sheet == -l_sheet) # the common family element is omega^2 (= the 2/3 cusp weight) check("the common family/triality phase of BOTH sectors is omega^2 (the 2/3 " "cusp-weight Z3 centre); the sheet (-1)^k splits quark/lepton", sp.simplify(rho**1 * sheet - omega**2) == 0 # quark family part and sp.simplify(rho**4 - omega**2) == 0) # lepton family part (sheet +1) # omega sits in the (2,3,5) Milnor/Coxeter monodromy spectrum (E8 exponents /30) e8_exps = [1, 7, 11, 13, 17, 19, 23, 29] monodromy = {sp.exp(2 * sp.I * sp.pi * m / 30) for m in e8_exps} # omega = e^{2pi i/3} = e^{2pi i*10/30}; omega^2 = e^{2pi i*20/30}. Check the family # Z3 factor divides the order-30 monodromy: omega^3 = 1 and 30/3 = 10 (= A_Lambda) check("the triality centre omega is the mu3 factor of the order-30 (2,3,5) " "monodromy: omega^3=1, 30/|mu3|=10=A_Lambda; the family CP phase lives " "inside the same Coxeter/Milnor cycle that builds E8 (v232)", sp.simplify(omega**3) == 1 and 30 // 3 == 10 and len(monodromy) == 8) # negative controls check("NEG: a mu2-only phase {1,-1} carries NO family triality (no order-3 " "element); a mu3-only phase {1,omega,omega^2} carries NO sheet (no -1); " "only mu6=mu3 x mu2 gives 'one triality phase, two sheets'", all(sp.simplify(s**3) != 1 or s == 1 for s in (sp.Integer(-1),)) # -1 has no order 3 and all(sp.simplify(omega**k + 1) != 0 for k in range(3)) # no -1 in mu3 and N_fam == 3) return summary("v233 CP triality phase (universal family phase, sheet-split)") if __name__ == "__main__": raise SystemExit(1 if run() else 0)