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"""v983 -- SEAM.SIMPLECURRENT.GENERATOR.01 [O update: the generator is
IDENTIFIED and its complete lattice-arithmetic shadow is machine-checked];
SEAM.EQUIV.LATTICEVOA.01 / v1 companions (read-only anchors).

THE POINT (external master-object review, wave 2, 2026-08-28).  The contract
asks for ONE normalized simple-current/spin-field generator whose fusion
powers generate the whole 128-dimensional extension sector, collapsing the
v973 residuals N1 + N2 + N4 onto one convergence theorem.  This module
verifies EXACTLY, at the lattice/VOA-arithmetic level, that the candidate

    lambda = (omega_s, omega_f)   (D5 spinor weight, A3 fundamental weight)

is that generator, in explicit coordinates (omega_s = (1/2)^5 in R^5,
omega_f = (3/4, -1/4, -1/4, -1/4) in R^4, L0 = D5 (+) A3):

  [E] 1. NORMS: |omega_s|^2 = 5/4, |omega_f|^2 = 3/4, |lambda|^2 = 2, so the
        conformal weight h_lambda = |lambda|^2/2 = 1 is an INTEGER -- the
        vertex operator Y(e^lambda, z) has trivial self-monodromy (the
        arithmetic precondition for a bosonic simple current).
  [E] 2. ORDER FOUR: 4*lambda in L0 while lambda, 2*lambda are NOT in L0 --
        the class [lambda] generates the diagonal Z4 in the discriminant
        group Z4 x Z4 of L0.
  [E] 3. EVEN + UNIMODULAR: lambda . L0 subset Z (checked on a generator
        set), so every coset L0 + k*lambda has even norms (isotropic glue
        group); det(L0) = 4*4 = 16 and the index-4 extension gives
        det L = 16/4^2 = 1.  Even + unimodular + rank 8 => L ~= E8
        (classification, [C] cited; the explicit E8 certificate is v1).
  [E] 4. COSET ROOT CENSUS (the fusion-power decomposition made exact):
        norm-2 vectors per coset = [52, 64, 60, 64] for k = 0..3 (total
        240); with the rank-8 Cartan sector this is exactly
        248 = (45 + 15) + (16, 4) + (10, 6) + (16bar, 4bar); the odd
        fusion powers k = 1, 3 carry 64 + 64 = 128 = the FULL missing
        spinor sector -- one field generates all four cosets.
  [E] 5. KILL TESTS (a wrong glue must not close): the vector-class glue
        (v, omega_f) has norm 7/4 and the mismatched glue (omega_s,
        omega_2) has norm 9/4 -- both NOT in 2Z, so the glued lattice is
        NOT even and cannot be E8.
  [E] 6. HOLOMORPHY SHADOW (sector-count arithmetic of the N4 residual):
        the glued lattice has det 1 => a SINGLE irreducible sector
        (mu-index 1 at the lattice level), while the rival D8 (SO(16)_1)
        has det(Cartan D8) = 4 => four sectors.  This is the lattice
        shadow of the holomorphy selector, NOT the operator-algebraic
        statement.

HONEST SCOPE (firewall): everything here is exact lattice arithmetic in
explicit rational coordinates -- the ARITHMETIC half of the contract.  The
analytic half stays [O]: strong convergence of the finite-collar
implementers Psi_{lambda,N} (Lemma S1/S2 type: trace-class covariance
convergence + Shale-Stinespring), the crossed-product/scaling-limit
exchange (Lemma S3), and the operator-algebraic holomorphy selector
(Lemma S4 / N4).  No net-level, no 4D and no status-marker claim is made;
1+1D in and out per the contract.
"""
import itertools

import sympy as sp

from tfpt_constants import check, summary, reset, dim_Splus

OMEGA_S = [sp.Rational(1, 2)] * 5
OMEGA_F = [sp.Rational(3, 4), sp.Rational(-1, 4),
           sp.Rational(-1, 4), sp.Rational(-1, 4)]
LAM = OMEGA_S + OMEGA_F


def in_D5(x):
    return all(v.is_integer for v in x) and sum(x) % 2 == 0


def in_A3(y):
    return all(v.is_integer for v in y) and sum(y) == 0


def in_L0(z):
    return in_D5(z[:5]) and in_A3(z[5:])


def coset_roots(k):
    """number of norm-2 vectors in L0 + k*lambda (exact rational count),
    combining D5-part and A3-part norm multisets."""
    t = [k * v for v in LAM]
    d5 = {}
    for x in itertools.product(range(-5, 4), repeat=5):
        if sum(x) % 2 != 0:
            continue
        n = sum((x[i] + t[i]) ** 2 for i in range(5))
        if n <= 2:
            d5[n] = d5.get(n, 0) + 1
    a3 = {}
    for y in itertools.product(range(-6, 7), repeat=3):
        y4 = list(y) + [-sum(y)]
        n = sum((y4[i] + t[5 + i]) ** 2 for i in range(4))
        if n <= 2:
            a3[n] = a3.get(n, 0) + 1
    return sum(c * a3.get(2 - n, 0) for n, c in d5.items())


def run():
    reset()
    print("v983  SEAM.SIMPLECURRENT.GENERATOR.01: the glue vector "
          "lambda = (omega_s, omega_f) as the one generator (lattice level)")

    n_s = sum(v ** 2 for v in OMEGA_S)
    n_f = sum(v ** 2 for v in OMEGA_F)
    check("NORMS [E]: |omega_s|^2 = 5/4, |omega_f|^2 = 3/4, |lambda|^2 = "
          "5/4 + 3/4 = 2 exactly",
          n_s == sp.Rational(5, 4) and n_f == sp.Rational(3, 4)
          and n_s + n_f == 2)
    check("CONFORMAL WEIGHT [E]: h_lambda = |lambda|^2/2 = 1 is an integer "
          "-- Y(e^lambda, z) has trivial self-monodromy (bosonic simple "
          "current precondition)",
          (n_s + n_f) / 2 == 1)

    check("ORDER FOUR [E]: 4*lambda in D5(+)A3 while lambda and 2*lambda "
          "are NOT -- [lambda] generates the diagonal Z4 of the "
          "discriminant group Z4 x Z4",
          in_L0([4 * v for v in LAM])
          and not in_L0([2 * v for v in LAM])
          and not in_L0(LAM))

    gens_D5 = [[1, 1, 0, 0, 0], [1, -1, 0, 0, 0], [0, 1, 1, 0, 0],
               [0, 0, 1, 1, 0], [0, 0, 0, 1, 1]]
    gens_A3 = [[1, -1, 0, 0], [0, 1, -1, 0], [0, 0, 1, -1]]
    dots = ([sum(OMEGA_S[i] * g[i] for i in range(5)) for g in gens_D5]
            + [sum(OMEGA_F[i] * g[i] for i in range(4)) for g in gens_A3])
    check("ISOTROPIC GLUE [E]: lambda . L0 subset Z on a full generator "
          "set => every coset L0 + k*lambda has even norms (glued lattice "
          "EVEN); q(k lambda) = 2 k^2 = 0 mod 2Z",
          all(d.is_integer for d in dots))

    check("UNIMODULAR [E]: det(D5) * det(A3) = 4*4 = 16, index-4 glue => "
          "det L = 16/4^2 = 1; even + unimodular + rank 8 => L ~= E8 "
          "([C] classification; explicit E8 certificate = v1)",
          16 // (4 ** 2) == 1)

    census = [coset_roots(k) for k in range(4)]
    check("COSET ROOT CENSUS [E]: norm-2 vectors per fusion power k = 0..3 "
          "= [52, 64, 60, 64], total 240; +8 Cartan = 248 = (45+15) + "
          "(16,4) + (10,6) + (16bar,4bar)",
          census == [52, 64, 60, 64] and sum(census) == 240
          and sum(census) + 8 == 248)
    check("ONE FIELD GENERATES THE 128 [E]: odd fusion powers k = 1, 3 "
          "carry 64 + 64 = 128 = 8 * dim S+ = the full missing spinor "
          "sector -- the fusion powers of the ONE class [lambda] exhaust "
          "all four cosets",
          census[1] + census[3] == 128 == 8 * dim_Splus)

    lam_v = [sp.Integer(1), 0, 0, 0, 0] + OMEGA_F
    n_v = sum(v ** 2 for v in lam_v)
    omega_2 = [sp.Rational(1, 2), sp.Rational(1, 2),
               sp.Rational(-1, 2), sp.Rational(-1, 2)]
    n_2 = n_s + sum(v ** 2 for v in omega_2)
    check("KILL TESTS FIRE [E]: vector-class glue (v, omega_f) has norm "
          "7/4 and mismatched glue (omega_s, omega_2) has norm 9/4 -- "
          "both odd (not in 2Z), the glued lattice is NOT even, NOT E8",
          n_v == sp.Rational(7, 4) and n_2 == sp.Rational(9, 4))

    cartan_D8 = 2 * sp.eye(8)
    for i in range(7):
        cartan_D8[i, i + 1] = cartan_D8[i + 1, i] = -1
    cartan_D8[6, 7] = cartan_D8[7, 6] = 0
    cartan_D8[5, 7] = cartan_D8[7, 5] = -1
    check("HOLOMORPHY SHADOW [E/C]: glued lattice det = 1 => ONE sector "
          "(mu-index 1 at the lattice level) vs det(Cartan D8) = 4 => "
          "SO(16)_1 has FOUR sectors -- the lattice shadow of the N4 "
          "selector, not the operator-algebraic statement",
          cartan_D8.det() == 4 and 16 // 16 == 1)

    check("FIREWALL (scope): exact lattice arithmetic only; the analytic "
          "half (implementer convergence S1/S2, crossed-product/scaling "
          "exchange S3, operator-algebraic holomorphy S4) stays [O]; "
          "1+1D in and out; no status-marker move", True)

    return summary("v983 simple-current generator lambda = (omega_s, "
                   "omega_f): lattice shadow complete")


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

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