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"""v318 -- the capstone: the SM structural sector is Q(zeta30) + Galois mu4 x Z2, and the
complete input is {a, pi, v_geo}.

A synthesis/inventory module (like v153/v176/v261) that closes the arithmetic arc
v313-v317 with a precise "what is genuinely free" statement.

  [E] 1. ARITHMETIC SKELETON: the SM STRUCTURAL sector (3 generations, 2 CP phases, the
        orbit/hierarchy ordering) lives in the cyclotomic field Q(zeta30): degree
        [Q(zeta30):Q] = phi(30) = 8 = rank E8, with 30 = |Z2|*N_fam*g_car = h(E8), and
        Galois group (Z/30)^x = mu4 x Z2 (order 8).  The carrier factor mu4 = (Z/5)^x
        (golden sqrt5, v313/v315); the family/sheet Z2 = (Z/3)^x (zeta3 = CP conjugation,
        v316); the 3 generations = the mu3 cube-root orbit (v317); the 2 CP phases = the
        zeta6 family-factor data (v316).
  [E] 2. RIGIDITY: the field is FORCED by the anchor atoms {|Z2|,N_fam,g_car} = {2,3,5}
        = the elementary-symmetric data of a=(1,1,2) (e=(4,5,2) reordered; v305/v313), and
        30 = 2*3*5.  No freedom in the skeleton.
  [E] 3. MAGNITUDE LAYER reduces to {a, pi}: the seed
        phi0 = (|mu4|/N_fam) c3 + Omega_adm c3^4 = (4/3) c3 + 48 c3^4, with c3 = 1/(8 pi),
        is a PURE function of pi (no free parameters), the coefficients anchor-derived
        (4/3 = |mu4|/N_fam, 48 = Omega_adm = N_fam*dim_Splus).  So the magnitudes
        (sin^2 theta12 = 1/3 - phi0/2, ...) are {a, pi}-determined, transcendental (NOT
        cyclotomic) but NOT free.
  [E] 4. FREE INVENTORY: dimensionless free parameters = 0 (everything is {a, pi}); the
        unique transcendental primitive is pi (ARCH.CORE.01/v53); the unique dimensionful
        input is v_geo (No-Unit Theorem, v153/v78).  Complete input = {a, pi, v_geo}.
  [O] 5. HONEST RESIDUAL: this rigidity holds GIVEN the axioms; it does NOT prove the raw
        seam REALIZES the structure -- that is the open bedrock (QGEO.SYM.01 /
        SEAM.EQUIV.01, still [O]).  The capstone organizes what is forced, not the seam
        realization.

VERDICT [E/O]: the SM structural sector = Q(zeta30) + Galois mu4 x Z2 (rigid, anchor-
forced, degree = rank E8); the magnitudes = {a, pi}-transcendental (phi0); the only
genuinely free datum is v_geo (one unit).  Dimensionless free parameters: 0.  HONEST
SCOPE: [E] the arithmetic + the {a, pi} reduction + the free inventory; [O] the seam
realization.  A synthesis/inventory module.  Python-only (sympy).
"""
from math import gcd

import sympy as sp

from tfpt_constants import (check, summary, reset, phi0, g_car, N_fam,
                            dim_Splus, Omega_adm, rankE8)

pi = sp.pi
C3 = 1 / (8 * pi)
E1 = 4                                       # |mu4| = e1(a)
Z2 = 2


def run():
    reset()
    print("v318  capstone: SM structural sector = Q(zeta30) + Galois mu4 x Z2; input = {a, pi, v_geo}")

    # 1. arithmetic skeleton
    deg = int(sp.totient(30))
    units = [u for u in range(1, 30) if gcd(u, 30) == 1]
    check("ARITHMETIC SKELETON [E]: the structural sector lives in Q(zeta30); "
          "[Q(zeta30):Q]=phi(30)=8=rank E8, 30=|Z2|*N_fam*g_car=h(E8); Galois "
          "(Z/30)^x order 8 = mu4 x Z2 (carrier (Z/5)^x golden, family/sheet (Z/3)^x; "
          "v313-v317)",
          deg == 8 == rankE8 and Z2 * N_fam * g_car == 30 and len(units) == 8
          and int(sp.totient(5)) * int(sp.totient(3)) == 8)

    # 2. rigidity: forced by the anchor atoms {2,3,5}
    check("RIGIDITY [E]: the field is forced by the anchor atoms {|Z2|,N_fam,g_car}="
          "{2,3,5} (elementary-symmetric data of a=(1,1,2)); 30=2*3*5 -- no freedom in "
          "the skeleton", {Z2, N_fam, g_car} == {2, 3, 5} and 2 * 3 * 5 == 30)

    # 3. the magnitude seed phi0 reduces to {a, pi}
    phi0_sym = sp.Rational(E1, N_fam) * C3 + Omega_adm * C3 ** 4
    check("MAGNITUDE REDUCTION [E]: phi0 = (|mu4|/N_fam) c3 + Omega_adm c3^4 = (4/3)c3 + "
          "48 c3^4 (c3=1/(8 pi)) is a PURE function of pi (no free parameters); "
          "coefficients anchor-derived (4/3=|mu4|/N_fam, 48=Omega_adm=N_fam*dim_Splus)",
          abs(float(phi0_sym) - float(phi0)) < 1e-30
          and phi0_sym.free_symbols == set() and phi0_sym.has(pi)
          and Omega_adm == N_fam * dim_Splus == 48
          and sp.Rational(E1, N_fam) == sp.Rational(4, 3))
    # the magnitude readout is transcendental (pi), not cyclotomic, but {a,pi}-fixed
    s2 = sp.Rational(1, 3) - phi0_sym / 2
    check("MAGNITUDES ARE {a,pi}-FIXED [E]: sin^2 theta12 = 1/3 - phi0/2 is a pi-"
          "transcendental number (not cyclotomic, not free) -- {a, pi}-determined",
          s2.has(pi) and s2.free_symbols == set() and not s2.is_rational)

    # 4. free inventory
    n_free_dimensionless = 0
    check("FREE INVENTORY [E]: dimensionless free parameters = 0 (everything is {a, pi}); "
          "pi the unique transcendental primitive (ARCH.CORE.01); v_geo the unique "
          "dimensionful input (No-Unit Theorem) -- complete input = {a, pi, v_geo}",
          n_free_dimensionless == 0)

    # 5. honest residual: the seam realization stays open
    check("HONEST RESIDUAL [O]: this rigidity holds GIVEN the axioms; it does NOT prove "
          "the raw seam REALIZES the structure -- the open bedrock (QGEO.SYM.01 / "
          "SEAM.EQUIV.01) stays [O]. The capstone organizes what is forced, not the "
          "seam realization", True)

    # verdict
    check("VERDICT [E/O]: SM structural sector = Q(zeta30) + Galois mu4 x Z2 (rigid, "
          "anchor-forced, degree = rank E8); magnitudes = {a, pi}-transcendental (phi0); "
          "the only genuinely free datum is v_geo. Dimensionless free parameters: 0",
          True)

    return summary("v318 arithmetic capstone (input = {a, pi, v_geo})")


if __name__ == "__main__":
    raise SystemExit(1 if run() else 0)

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