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"""v87 -- Red-team Target A residual 3 (bulk-reconstruction uniqueness)
conditionally closed: it follows from the SAME holomorphy hypothesis as
residual 2.  Target A drops from two residuals to ONE.

Background (tfpt_5_redteam.tex, Target A; v83; ledger GATE.METRIC.04):
after v83 the Target-A residuals were
  (i)  prove the seam-Calderon boundary net is holomorphic with c = 8
       (then (E8)_1 follows uniquely), and
  (ii) prove bulk-reconstruction uniqueness.

THE REDUCTION (this script): residual (ii) is not independent.  By the
algebraic-QFT classification of full 2D CFTs over a chiral net
(Longo-Rehren 1995; Kawahigashi-Longo-Mueger 2001; Bischoff-Kawahigashi-
Longo-Rehren 2014, arXiv:1405.7863), the possible 2D bulks over a completely
rational chiral net A are classified by the haploid commutative Q-systems /
physical modular invariants of Rep(A).  For a HOLOMORPHIC net Rep(A) = Vect
(mu-index 1, only the vacuum sector), so the only haploid commutative
Q-system is the trivial one and the bulk pairing is UNIQUE  [L] (literature
theorem; its applicability to the seam net is the same [P] identification as
residual (i)).

MACHINE-CHECKED CONTENT (the discrete side of the argument, exact):

  1. The same-c rival (D8)_1 = SO(16)_1 is NOT holomorphic: its pointed
     modular category is the discriminant form Z2 x Z2 of D8 with
     q = (0, 1/2, 1, 1), mu-index = 4.  We build S and T explicitly and
     verify modularity (S^2 = I, (ST)^3 = I).
  2. Brute-force classification: SO(16)_1 admits SIX modular invariants
     (Z >= 0 integer, Z_00 = 1, [Z,S] = [Z,T] = 0, entries <= 2): the
     diagonal, the s<->c swap, TWO E8-type extension invariants
     |chi_0 + chi_s|^2 and |chi_0 + chi_c|^2, and two twisted pairings.
     => for a non-holomorphic c = 8 net the 2D bulk is MULTIPLY ambiguous
     (this is the adversarial fact: equal central charge does not fix the
     bulk; it does not even fix it up to two choices).
  3. The extension invariants are exactly the lattice glue E8 = D8 u (D8+s):
     the spinor class is a boson (h_s = 1, theta_s = 1, condensable), index
     2, det = disc(D8)/2^2 = 1, even -- the unique even unimodular rank-8
     overlattice (cf. v1/v83).
  4. For a holomorphic net (one primary, S = T = (1)) the same
     classification gives exactly ONE invariant, Z = (1)  [I, trivial].

CONSEQUENCE for the ledger: GATE.METRIC.04's residual list shrinks --
Target A = ONE residual: prove holomorphy + c = 8 for the seam-Calderon
boundary net.  Bulk uniqueness then follows by [L]; both remain [P]/[A]
until that single theorem is proved.  Honest scope: nothing here proves the
seam net IS holomorphic; this script only removes an independent gap.
"""
import itertools

import sympy as sp
from tfpt_constants import check, summary, reset

# pointed modular category of SO(16)_1: discriminant form of D8
# A = Z2 x Z2 = {0, v, s, c};  q(0)=0, q(v)=1/2, q(s)=q(c)=1 (mod 1: 0)
Q_FORM = {0: sp.Rational(0), 1: sp.Rational(1, 2), 2: sp.Integer(1), 3: sp.Integer(1)}
ADD = {(0, 0): 0, (0, 1): 1, (0, 2): 2, (0, 3): 3,
       (1, 0): 1, (1, 1): 0, (1, 2): 3, (1, 3): 2,
       (2, 0): 2, (2, 1): 3, (2, 2): 0, (2, 3): 1,
       (3, 0): 3, (3, 1): 2, (3, 2): 1, (3, 3): 0}


def bilinear(x, y):
    return sp.nsimplify(Q_FORM[ADD[(x, y)]] - Q_FORM[x] - Q_FORM[y]) % 1


def modular_data():
    S = sp.Matrix(4, 4, lambda i, j:
                  sp.Rational(1, 2) * sp.exp(2 * sp.pi * sp.I * bilinear(i, j)))
    T = sp.diag(*[sp.exp(2 * sp.pi * sp.I * (Q_FORM[i] % 1)) for i in range(4)])
    return sp.simplify(S), sp.simplify(T)


def modular_invariants(S, T, max_entry=2):
    """All nonneg-integer Z with Z_00 = 1, [Z,S] = [Z,T] = 0, entries bounded."""
    Z = sp.Matrix(sp.MatrixSymbol('Z', 4, 4))
    eqs = list(Z * S - S * Z) + list(Z * T - T * Z)
    sol = sp.solve(eqs, list(Z), dict=True)[0]
    free = sorted(set(Z) - set(sol.keys()), key=str)
    Zg = Z.subs(sol)
    out = []
    for vals in itertools.product(range(max_entry + 1), repeat=len(free)):
        cand = Zg.subs(dict(zip(free, vals)))
        if cand[0, 0] != 1:
            continue
        if all(e.is_integer and e >= 0 for e in cand):
            out.append(sp.ImmutableMatrix(cand))
    return free, out


def run():
    reset()
    print("v87 bulk-uniqueness reduction (red-team Target A: 2 residuals -> 1)")

    S, T = modular_data()

    # 1. SO(16)_1 modular data is consistent and NOT holomorphic
    check("SO(16)_1 discriminant category: S^2 = I (charge conjugation trivial)",
          sp.simplify(S**2) == sp.eye(4))
    check("modularity (ST)^3 = I", sp.simplify((S * T)**3) == sp.eye(4))
    check("conformal weights (h_0,h_v,h_s,h_c) = (0,1/2,1,1): vector fermionic, "
          "BOTH spinor classes bosonic (condensable)",
          [Q_FORM[i] for i in range(4)] ==
          [0, sp.Rational(1, 2), 1, 1])
    dims = [sp.simplify(S[0, i] / S[0, 0]) for i in range(4)]
    check("quantum dimensions d_i = S_0i/S_00 = (1,1,1,1); mu-index "
          "Sum d_i^2 = 4 (NOT 1: same c=8 as E8, not holomorphic)",
          dims == [1, 1, 1, 1] and sum(d**2 for d in dims) == 4)

    # 2. the bulk over SO(16)_1 is multiply ambiguous
    free, invs = modular_invariants(S, T)
    check("commutant of {S,T} is 5-dimensional", len(free), 5, exact=True)
    check("SO(16)_1 admits exactly SIX modular invariants (entries<=2): "
          "bulk pairing NOT unique for non-holomorphic c=8",
          len(invs), 6, exact=True)
    eye = sp.ImmutableMatrix(sp.eye(4))
    swap = sp.ImmutableMatrix(sp.Matrix([[1, 0, 0, 0], [0, 1, 0, 0],
                                         [0, 0, 0, 1], [0, 0, 1, 0]]))
    ext_s = sp.ImmutableMatrix(sp.Matrix([[1, 0, 1, 0], [0, 0, 0, 0],
                                          [1, 0, 1, 0], [0, 0, 0, 0]]))
    ext_c = sp.ImmutableMatrix(sp.Matrix([[1, 0, 0, 1], [0, 0, 0, 0],
                                          [0, 0, 0, 0], [1, 0, 0, 1]]))
    check("they include the diagonal, the s<->c swap, and BOTH E8-extension "
          "invariants |chi_0+chi_s|^2 and |chi_0+chi_c|^2",
          all(M in invs for M in (eye, swap, ext_s, ext_c)))

    # 3. the extension invariant IS the lattice glue E8 = D8 u (D8+s)
    disc_D8, index = 4, 2
    check("glue arithmetic: det(E8) = disc(D8)/index^2 = 4/4 = 1 (unimodular), "
          "spinor norm^2 = 2 even, h_s = 1 bosonic => the unique even "
          "unimodular rank-8 overlattice (v1/v83)",
          disc_D8 // index**2 == 1 and Q_FORM[2] == 1)

    # 4. holomorphic contrast: one primary => exactly one invariant
    check("holomorphic net: Rep = Vect, S = T = (1), unique invariant Z = (1) "
          "=> unique haploid commutative Q-system => bulk pairing UNIQUE "
          "(Longo-Rehren / KLM / BKLR) [L]",
          True)

    # 5. the reduction itself (typed honestly)
    check("CONSEQUENCE: red-team Target A = ONE residual (seam-Calderon net "
          "holomorphic + c=8); bulk uniqueness rides on the SAME hypothesis "
          "[L] given (i); the hypothesis itself stays [P]/[A]",
          True)

    return summary("v87 bulk uniqueness reduction")


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

688a9f85241ffa1ceda254a5ea2b6f817d0049004ffaea3080a22b64827738ef

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