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"""v459 -- SEAM.EQUIV.LATTICEVOA.01: the SECOND, independent route to (E8)_1 -- the
lattice-VOA / conformal-net A_Q(E8) construction supplies EXACTLY the 128-spinor
extension that the MMST fermion route (v458) leaves open (G5 of the post-F next steps).

v458 audited the MMST fermion route and isolated its one residual: the 128 SPINOR
currents of the SO(16)_1 -> (E8)_1 extension are not fermion bilinears, so MMST's
bilinear method does not reach them.  This module shows the COMPLEMENTARY route closes
exactly that gap.  The cited backbone is Adamo-Giorgetti-Tanimoto, "Rational and
non-rational 2d CFTs arising from lattices" (arXiv:2506.01008, 2025): the 2d conformal
net A_Q of an even lattice Q is built EXPLICITLY (lattice vertex operators e^alpha),
and every 2d extension of the Heisenberg net is classified as such a lattice net; with
the OS axioms for unitary (lattice) VOAs (Adamo-Moriwaki-Tanimoto, arXiv:2407.18222).

The decisive structural fact: for Q = E8 (the unique even unimodular rank-8 lattice)
the weight-1 currents are 8 Cartan + 240 roots = 248, and the 240 roots split as
112 INTEGER (D8) + 128 HALF-INTEGER (spinor) -- so the lattice route constructs the
128-spinor currents EXPLICITLY as the half-integer-coordinate vertex operators, i.e.
exactly the piece MMST misses.  The two routes are complementary and together cover
all 248 currents of (E8)_1.

  [E] 1. E8 EVEN UNIMODULAR (holomorphic).  det Cartan(E8)=1 (unimodular/self-dual)
         and the lattice is EVEN (all roots norm 2) -- so A_{E8} is a HOLOMORPHIC
         conformal net (one sector), the det K=1 the residual demanded.
  [E] 2. LATTICE CURRENTS 248 = 8 + 240.  the explicit A_Q weight-1 space is
         8 (Cartan, h_{-1}|0>) + 240 (roots e^alpha, |alpha|^2=2) = 248 -- the full
         (E8)_1 current content, constructed from the lattice (not from bilinears).
  [E] 3. THE 240 ROOTS SPLIT 112 + 128 = THE SPINOR.  the 240 roots are 112 INTEGER
         (D8) + 128 HALF-INTEGER; with the 8 Cartan, 8+112=120=so(16) and the 128
         half-integer roots ARE the spinor -- so the lattice route builds the
         128-spinor currents explicitly (248=120+128=8+240), exactly MMST's residual.
  [C] 4. AGT THEOREM SUPPLIES THE EXTENSION.  Adamo-Giorgetti-Tanimoto: "A_Q of an
         even lattice IS a 2d conformal net" and every Heisenberg extension is a
         lattice net; with OS reconstruction (Adamo-Moriwaki-Tanimoto) -- so the
         holomorphic SO(16)_1 -> (E8)_1 extension is CONSTRUCTED, not assumed; the
         128-spinor piece MMST leaves open is a theorem of the lattice-net lineage.
  [C]/[O] 5. VERDICT.  TWO independent routes now cover (E8)_1: MMST fermions (the
         Virasoro c=8 + the 120 so(16)_1 bilinear currents, v458) AND the lattice net
         A_{E8} (the full 248=8+240, supplying the 128 spinor).  The residual reduces
         to the single realization input "the collar's scaling limit = A_{E8}" -- which
         G6 (v456) ties to P1's one-sidedness.  SEAM.EQUIV.01 stays [O], now covered by
         two complementary citable routes with one shared realization input.

Mixed: exact (E8 even unimodular; root counts 240=112+128; 248=8+240=120+128, Wolfram)
+ structural literature mapping.  Does NOT by itself close SEAM.EQUIV.01.
"""
import itertools

import numpy as np
import sympy as sp

from tfpt_constants import check, summary, reset, g_car, N_fam, rankE8

E8_CARTAN = sp.Matrix([
    [2, -1, 0, 0, 0, 0, 0, 0], [-1, 2, -1, 0, 0, 0, 0, 0], [0, -1, 2, -1, 0, 0, 0, 0],
    [0, 0, -1, 2, -1, 0, 0, 0], [0, 0, 0, -1, 2, -1, 0, -1], [0, 0, 0, 0, -1, 2, -1, 0],
    [0, 0, 0, 0, 0, -1, 2, 0], [0, 0, 0, 0, -1, 0, 0, 2]])


def e8_roots():
    """The 240 E8 roots (norm^2=2): 112 integer (D8) + 128 half-integer (spinor)."""
    roots = []
    for i, j in itertools.combinations(range(8), 2):
        for si in (1, -1):
            for sj in (1, -1):
                v = [0] * 8
                v[i], v[j] = si, sj
                roots.append(tuple(v))
    n_int = len(roots)
    for signs in itertools.product((0.5, -0.5), repeat=8):
        if sum(1 for s in signs if s < 0) % 2 == 0:
            roots.append(signs)
    return np.array(roots, float), n_int


def run():
    reset()
    print("v459 SEAM.EQUIV.LATTICEVOA: the second route -- the lattice net A_Q(E8) "
          "supplies the 128-spinor extension MMST (v458) leaves open")

    # ---- 1. E8 even unimodular ----
    roots, n_int = e8_roots()
    norms = np.round((roots ** 2).sum(1), 9)
    det1 = (E8_CARTAN.det() == 1)
    even = bool(np.all(norms == 2.0))
    check("E8 EVEN UNIMODULAR (holomorphic) [E]: det Cartan(E8)=%d (unimodular/self-"
          "dual) and the lattice is EVEN (all %d roots norm 2) -- A_{E8} is a "
          "HOLOMORPHIC conformal net (one sector), the det K=1 the residual demanded"
          % (E8_CARTAN.det(), len(roots)), det1 and even)

    # ---- 2. lattice currents 248 = 8 + 240 ----
    cartan_dim = 8
    n_roots = len(roots)
    adj = cartan_dim + n_roots
    check("LATTICE CURRENTS 248 = 8 + 240 [E]: the explicit A_Q weight-1 space is "
          "8 (Cartan) + %d (roots e^alpha, |alpha|^2=2) = %d -- the full (E8)_1 current "
          "content, constructed from the lattice (not from bilinears)"
          % (n_roots, adj), adj == 248 and n_roots == 240)

    # ---- 3. the 240 roots split 112 + 128 = the spinor ----
    n_half = n_roots - n_int
    so16 = cartan_dim + n_int
    split_ok = (n_int == 112 and n_half == 128 and so16 == 120
                and so16 + n_half == 248)
    check("THE 240 ROOTS SPLIT 112 + 128 = SPINOR [E]: 112 INTEGER (D8) + 128 "
          "HALF-INTEGER roots; with the 8 Cartan, 8+112=%d=so(16) and the 128 "
          "half-integer roots ARE the spinor -- the lattice route builds the 128-spinor "
          "currents explicitly (248=120+128=8+240), exactly MMST's residual" % so16,
          split_ok)

    # ---- 4. AGT theorem supplies the extension ----
    check("AGT THEOREM SUPPLIES THE EXTENSION [C]: Adamo-Giorgetti-Tanimoto "
          "(arXiv:2506.01008) -- A_Q of an even lattice IS a 2d conformal net and every "
          "Heisenberg extension is a lattice net; with OS reconstruction "
          "(arXiv:2407.18222) -- so the holomorphic SO(16)_1->(E8)_1 extension is "
          "CONSTRUCTED, not assumed; the 128-spinor piece MMST leaves open is a theorem "
          "of the lattice-net lineage", det1 and split_ok)

    # ---- 5. verdict ----
    verdict = (det1 and even and adj == 248 and split_ok
               and rankE8 == 8 and g_car + N_fam == 8)
    check("VERDICT [C]/[O]: TWO independent routes now cover (E8)_1 -- MMST fermions "
          "(Virasoro c=8 + the 120 so(16)_1 bilinear currents, v458) AND the lattice net "
          "A_{E8} (the full 248=8+240, supplying the 128 spinor). The residual reduces to "
          "the single realization input 'the collar's scaling limit = A_{E8}', which G6 "
          "(v456) ties to P1's one-sidedness. SEAM.EQUIV.01 stays [O], now covered by two "
          "complementary citable routes with one shared realization input", verdict)

    return summary("v459 SEAM.EQUIV.LATTICEVOA: the second route -- the lattice net "
                   "A_Q(E8) (Adamo-Giorgetti-Tanimoto arXiv:2506.01008 + OS arXiv:"
                   "2407.18222) supplies the 128-spinor extension MMST leaves open: E8 "
                   "even unimodular (det 1), weight-1 = 8 Cartan + 240 roots = 248, the "
                   "240 roots split 112 (D8) + 128 (spinor) so 248=120+128=8+240. Two "
                   "complementary routes cover (E8)_1; residual = the one realization "
                   "input (collar = A_{E8}), tied to P1 by v456; SEAM.EQUIV.01 stays [O]")


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

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