TFPT scientific source excerpt
Original repository path: verification/v118_hexagon_family_dictionary.py
Original file SHA-256: 48b7a05bcd877afe13f0a92cce4087e86c23172823e605cda21c6a8ceafa6e37
Inclusive original lines: 54-138
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000054 | def run():
000055 |     reset()
000056 |     print("v118 hexagon-family dictionary (lepton c's = resolvent dets)")
000057 | 
000058 |     # 1. sign-twist lemma
000059 |     z6 = sp.exp(II * sp.pi / 3)
000060 |     spec = list(M0.eigenvals().keys())
000061 |     union = sorted([sp.nsimplify(sp.simplify(x)) for x in spec]
000062 |                    + [sp.nsimplify(sp.simplify(-x)) for x in spec], key=str)
000063 |     mu6 = sorted([sp.nsimplify(sp.simplify(z6 ** k)) for k in range(6)],
000064 |                  key=str)
000065 |     check("SIGN-TWIST LEMMA: -M0 has order 6 ((-M0)^3 = -1) and "
000066 |           "spec(M0) u spec(-M0) = mu_6 -- the hypercharge hexagon "
000067 |           "spectrum IS the family spectrum plus its sheet twist "
000068 |           "(Z6 = Z2 x Z3 = (-1) x <M0>)",
000069 |           sp.simplify((-M0) ** 6) == sp.eye(3)
000070 |           and sp.simplify((-M0) ** 3) == -sp.eye(3)
000071 |           and all(sp.simplify(a - b) == 0 for a, b in zip(union, mu6)))
000072 | 
000073 |     check("HEXAGON DENOMINATORS: for every r, 5/4 - cos(r pi/3) = "
000074 |           "|1 - zeta_6^r / 2|^2 -- v20's resolvent denominators are the "
000075 |           "eigen-denominators of the two family resolvents "
000076 |           "(1 -+ M0/2)^{-1}",
000077 |           all(sp.simplify(sp.expand_complex(
000078 |               (sp.Rational(5, 4) - sp.cos(r * sp.pi / 3))
000079 |               - (1 - z6 ** r / 2) * (1 - sp.conjugate(z6 ** r) / 2))) == 0
000080 |               for r in range(6)))
000081 | 
000082 |     # 2. cyclotomic determinant
000083 |     t = sp.Symbol('t')
000084 |     detm = sp.nsimplify(sp.det(sp.eye(3) - M0 / 2))
000085 |     detp = sp.nsimplify(sp.det(sp.eye(3) + M0 / 2))
000086 |     check("CYCLOTOMIC DETERMINANT: det(1 - t M0) = 1 - t^3 = "
000087 |           "(1 - t) Phi_3(t) (the N_fam cyclotomic) => det(1 - M0/2) = "
000088 |           "7/8 and det(1 + M0/2) = 9/8",
000089 |           sp.simplify(sp.det(sp.eye(3) - t * M0) - (1 - t ** 3)) == 0
000090 |           and detm == sp.Rational(7, 8) and detp == sp.Rational(9, 8))
000091 | 
000092 |     # 3. lepton coefficients as resolvent determinants
000093 |     mu4, nfam = 4, 3
000094 |     c_e = sp.Rational(mu4) / (2 * detm)
000095 |     c_mu = sp.Rational(nfam) / (2 * detp)
000096 |     c_tau = mu4 * detm / nfam
000097 |     check("THE LEPTON COEFFICIENTS ARE RESOLVENT DETERMINANTS: "
000098 |           "c_e = |mu_4|/(2 det-) = 16/7, c_mu = N_fam/(2 det+) = 4/3, "
000099 |           "c_tau = |mu_4| det-/N_fam = 7/6, product = |mu_4|/det+ = "
000100 |           "32/9 -- the lepton 7 and 9 ARE the family-resolvent "
000101 |           "determinants (x 8); delta = 1/2 = |(M0)_22| is supplied by "
000102 |           "the matrix itself (v117)",
000103 |           c_e == sp.Rational(16, 7) and c_mu == sp.Rational(4, 3)
000104 |           and c_tau == sp.Rational(7, 6)
000105 |           and sp.Rational(mu4) / detp == sp.Rational(32, 9)
000106 |           and sp.simplify(c_e * c_mu * c_tau
000107 |                           - sp.Rational(mu4) / detp) == 0
000108 |           and sp.Abs(M0[1, 1]) == sp.Rational(1, 2))
000109 | 
000110 |     # 4. sheet-extended group
000111 |     def freeze(m):
000112 |         return tuple(sp.simplify(x) for x in m)
000113 | 
000114 |     elems = {freeze(sp.eye(3)): True}
000115 |     frontier = [sp.eye(3)]
000116 |     while frontier:
000117 |         new = []
000118 |         for e in frontier:
000119 |             for g in (U, M0, -sp.eye(3)):
000120 |                 x = sp.expand(g * e)
000121 |                 kx = freeze(x)
000122 |                 if kx not in elems:
000123 |                     elems[kx] = True
000124 |                     new.append(x)
000125 |         frontier = new
000126 |     check("SHEET-EXTENDED GROUP: <U, M0, -1> has order 48 = Omega_adm "
000127 |           "= N_fam x dim S+ (the seed's quartic coefficient, v106) = "
000128 |           "|S4 x Z2| (exact enumeration; audit cross-link)",
000129 |           len(elems) == 48 and 3 * 16 == 48 and 24 * 2 == 48)
000130 | 
000131 |     # 5. residue
000132 |     check("RESIDUE [P] (recorded): the dictionary's VALUES are exact "
000133 |           "(hexagon spectrum, denominators, determinants); its ADDRESS "
000134 |           "TABLE (which hexagon site each fermion occupies; the quark "
000135 |           "composition) remains v20's established input / open -- "
000136 |           "nothing re-fished", True)
000137 | 
000138 |     return summary("v118 hexagon-family dictionary")