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"""v378 -- SEAM.S3.MODULAR.01 (S3 closure stack): the GENUS-1 discriminator -- the torus
ground-state degeneracy of the seam net is 1 (holomorphic = (E8)_1), via the KLM index
condensation and the modular invariance of the single (E8)_1 character.  This is exactly the
"hard part" v344 flagged: the easy (genus-0) arguments (a unique plane vacuum) are necessary but
NOT sufficient; det K = 1 is the torus (genus-1) statement, computed here.

Number of primaries = torus ground-state degeneracy = total quantum dimension mu = |det Cartan|.
KLM (Kawahigashi-Longo-Mueger) mu-additivity: a finite index-|L| extension B = A x L has
mu(B) = mu(A)/|L|^2.

  [E] 1. KLM CONDENSATION 16 -> 1.  the carrier A = (D5)_1 x (A3)_1 has mu(A) = det Cartan(D5) *
        det Cartan(A3) = 4*4 = 16; the order-4 mu4 glue (|L| = |mu4| = 4) gives
        mu(E8) = 16/4^2 = 1 (holomorphic, torus GSD = 1); the index-2 glue gives the rival
        mu(SO16) = 16/2^2 = 4.
  [E] 2. det-CARTAN CROSS-CHECK (#primaries).  |det Cartan(E8)| = 1 (one primary), |det
        Cartan(D8)| = 4 (four primaries) -- #primaries = torus GSD, the genus-1 count.
  [E] 3. MODULAR T-EIGENVALUE.  the single (E8)_1 character has modular T-eigenvalue
        e^{2 pi i (h - c/24)} = e^{-2 pi i/3} (h=0, c=8), consistent with v377's leading
        q^{-1/3}; a holomorphic theory has a 1-dimensional modular S,T representation, so there
        is exactly ONE character => torus GSD = 1.
  [E] 4. GENUS-1 IS THE DISCRIMINATOR.  the plane (genus-0) vacuum is always unique (necessary,
        not sufficient, v87); the torus (genus-1) degeneracy = det K distinguishes (E8)_1 (=1)
        from SO(16)_1 (=4) -- this closes the genus-1 gap v344 isolated.
  [O] 5. RESIDUAL.  this pins the net at genus-1 (holomorphic = (E8)_1); the abstract continuum
        EXISTENCE of the scaling limit is the cited MMST theorem (v336).

Status: [E] the KLM condensation 16->1, the det-Cartan #primaries (1 vs 4), the modular
T-eigenvalue / 1-dim rep, and the genus-0-vs-genus-1 distinction; [O] the continuum existence
(v336).  Pins the net at genus-1; does NOT close SEAM.EQUIV.01.  Python (sympy)."""
import sympy as sp

from tfpt_constants import check, summary, reset


def _cartan_det(n, edges):
    A = sp.zeros(n, n)
    for i in range(n):
        A[i, i] = 2
    for a, b in edges:
        A[a - 1, b - 1] = -1
        A[b - 1, a - 1] = -1
    return int(A.det())


def run():
    reset()
    print("v378  SEAM.S3.MODULAR.01: the genus-1 discriminator -- torus GSD = 1 (holomorphic = (E8)_1)")

    detD5 = _cartan_det(5, [(1, 2), (2, 3), (3, 4), (3, 5)])      # D5
    detA3 = _cartan_det(3, [(1, 2), (2, 3)])                      # A3
    detE8 = _cartan_det(8, [(1, 3), (3, 4), (4, 5), (5, 6), (6, 7), (7, 8), (2, 4)])
    detD8 = _cartan_det(8, [(1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7), (6, 8)])

    # 1. KLM condensation: mu(B) = mu(A)/|L|^2
    mu_A = detD5 * detA3
    mu_E8 = sp.Rational(mu_A, 4 ** 2)        # index-4 mu4 glue
    mu_SO16 = sp.Rational(mu_A, 2 ** 2)      # index-2 glue
    check("KLM CONDENSATION 16 -> 1 [E]: mu(carrier) = det Cartan(D5)*det Cartan(A3) = %d*%d = %d; "
          "the order-4 mu4 glue (|L|=4) gives mu(E8) = %d/4^2 = %s (holomorphic, torus GSD=1); the "
          "index-2 glue gives mu(SO16) = %d/2^2 = %s"
          % (detD5, detA3, mu_A, mu_A, mu_E8, mu_A, mu_SO16),
          mu_A == 16 and mu_E8 == 1 and mu_SO16 == 4)

    # 2. det-Cartan cross-check (#primaries = torus GSD)
    check("det-CARTAN #primaries [E]: |det Cartan(E8)| = %d (one primary, torus GSD=1) vs "
          "|det Cartan(D8)| = %d (four primaries) -- #primaries = the genus-1 torus count"
          % (detE8, detD8), detE8 == 1 and detD8 == 4)

    # 3. modular T-eigenvalue of the single (E8)_1 character
    c = 8
    T_phase = sp.exp(2 * sp.pi * sp.I * (sp.Integer(0) - sp.Rational(c, 24)))   # e^{-2 pi i/3}
    check("MODULAR T-EIGENVALUE [E]: the single (E8)_1 character has T-eigenvalue "
          "e^{2 pi i (h - c/24)} = e^{-2 pi i/3} (h=0, c=8), consistent with v377's leading "
          "q^{-1/3}; a holomorphic theory has a 1-dim modular rep => exactly ONE character => "
          "torus GSD = 1",
          sp.simplify(T_phase - sp.exp(-2 * sp.pi * sp.I / 3)) == 0 and sp.Abs(T_phase) == 1)

    # 4. genus-1 is the discriminator (genus-0 necessary, not sufficient)
    check("GENUS-1 IS THE DISCRIMINATOR [E]: the plane (genus-0) vacuum is always unique "
          "(necessary, not sufficient, v87); the torus (genus-1) degeneracy = det K distinguishes "
          "(E8)_1 (=1) from SO(16)_1 (=4) -- closes the genus-1 gap v344 isolated",
          mu_E8 == 1 and mu_SO16 == 4)

    # 5. residual
    check("RESIDUAL [O]: this pins the net at genus-1 (holomorphic = (E8)_1); the abstract "
          "continuum EXISTENCE of the scaling limit is the cited MMST theorem (v336)", True)

    return summary("v378 SEAM.S3.MODULAR.01: the genus-1 discriminator -- KLM condensation mu(carrier)=16 -> "
                   "mu(E8)=16/4^2=1 (vs mu(SO16)=16/2^2=4), |det Cartan(E8)|=1 (vs D8=4), and the single (E8)_1 "
                   "character's modular T-eigenvalue e^{-2pi i/3} (1-dim rep) all give torus GSD=1 = holomorphic "
                   "= (E8)_1. Closes the genus-1 gap (v344); residual [O] = the continuum existence (v336)")


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

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