"""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)