"""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)