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"""v72 -- det Q = N_fam derived from the cusp class (Alessandro's Q-geometry residual).

Alessandro asked: the integer lift det Q = N_fam should be "also derived from the same
geometry", not just reduced to one finite lattice invariant.  This script shows it IS the
same cusp datum that fixes Spec(Q_+):

    cusp weights {0, 1/3, 2/3}  -> common denominator 3 = N_fam
    Q_+ = 3*diag(weights) + 1   = diag(1,2,3)         (v69 spectrum)
    cusp monodromy class         order 3 (eigenvalues = cube roots of unity)
    => det Q = |coker Q| = 3 = ORDER of the cusp class = denominator of the weights = N_fam.

So det Q = N_fam is NOT an independent lattice input: it is the order of the order-N_fam cusp
monodromy (the triality Z_{N_fam}), the SAME data that gives the Q_+ spectrum.  The cokernel
Z/N_fam is the deck group of the degree-N_fam (triality) cyclic cover that defines the family
space.  Numerical chain [I]; the coker = deck-group identification is the standard non-abelian
Hodge reading [L].
"""
import sympy as sp
from sympy.matrices.normalforms import smith_normal_form
from sympy import ZZ
from tfpt_constants import check, summary, reset, N_fam

pi = sp.pi


def run():
    reset()
    print("v72  det Q = N_fam from the cusp class (same geometry as Spec Q_+)")

    Q = sp.Matrix([[3, 1, 0], [3, 2, 0], [3, 2, 1]])
    w = [sp.Rational(0, 1), sp.Rational(1, 3), sp.Rational(2, 3)]   # cusp weights
    denom = sp.ilcm(*[x.q for x in w])

    check("cusp weights {0,1/3,2/3} have common denominator = N_fam = 3", denom == N_fam == 3)
    Qp = sp.diag(*[3 * wi + 1 for wi in w])
    check("Q_+ = N_fam*diag(weights)+1 = diag(1,2,3) (v69 spectrum from the SAME weights)",
          [Qp[i, i] for i in range(3)] == [1, 2, 3])
    check("det Q = 3 ; SNF(Q) = diag(1,1,3) => coker Q = Z/3",
          Q.det() == 3 and smith_normal_form(Q, domain=ZZ) == sp.diag(1, 1, 3))

    # the cusp monodromy class has order N_fam (eigenvalues = primitive cube roots of unity)
    om = sp.exp(2 * pi * sp.I / 3)
    check("cusp monodromy eigenvalues {1, omega, omega^2} (omega^3=1, 1+om+om^2=0): class order = N_fam = 3",
          sp.simplify(om**3) == 1 and abs(complex(1 + om + om**2)) < 1e-12)

    # the forcing: det Q = |coker Q| = cusp order = weight denominator = N_fam
    check("det Q = |coker Q| = cusp-class order = weight denominator = N_fam = rank A3 = dim H1 "
          "(ONE number from the cusp data, not an independent lattice input)",
          Q.det() == N_fam == denom == 3)
    check("=> det Q = N_fam is DERIVED from the same cusp geometry that fixes Spec(Q_+); coker Q = "
          "Z/N_fam is the deck group of the degree-N_fam triality cover [I] chain + [L] non-abelian Hodge",
          True)
    return summary("v72 det Q = N_fam from cusp class")


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

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