"""v443 -- SEAM.E8.DISCRIMINATOR.01: the empirical/structural DISCRIMINATOR bundle (mode D of the v347 closure-mode classification) -- concrete FALSIFIABLE kill tests that separate the claimed seam = (E8)_1 from every nearby c=8 alternative. v347 lists four closure modes for the one open arrow; mode D is "test the consequences." This module realises mode D as a bundle of sharp discriminators against the nearest rival, SO(16)_1 = (D8)_1 (same central charge c=8, the v277 ambiguity). Each is an [X] kill test: if the physical seam ever shows the rival's signature, (E8)_1 is falsified. [E]/[X] 1. PRIMARY / GSD DISCRIMINATOR. |det Cartan(E8)|=1 (ONE primary, holomorphic) vs |det Cartan(D8)=SO(16)|=4 (FOUR primaries 1,v,s,c); the torus ground-state degeneracy = #primaries = 1 (E8) vs 4 (SO(16)). KILL TEST: a seam with GSD>1 / >1 superselection sector falsifies (E8)_1. [E]/[X] 2. CURRENT / THETA DISCRIMINATOR. level-1 current count dim E8=248 vs dim SO(16)=120; the E8 lattice theta has 240 norm-2 roots (=112 integer + 128 spinor, computed) vs D8's 112 -- the affine current spectrum and the lattice theta DIFFER. KILL TEST: the edge current multiplicity (248 vs 120) is observable. [E] 3. c=8 IS NOT A DISCRIMINATOR. both nets have c=8 (the v277 ambiguity), so the central charge alone CANNOT decide; only the order-4 mu4 condensation + det K=1 (the genus-1 statement, SEAM.S3.MODULAR.01/v90) selects E8. [E] 4. NON-VACUITY. the discriminators all fire on the rival (det 4!=1, currents 120!=248, roots 112!=240) -- the kill tests are non-vacuous. [X]/[C] 5. VERDICT. the (E8)_1 seam carries sharp falsifiable signatures distinct from every nearby c=8 net (mode D of v347 realised as concrete kill tests). SEAM.EQUIV.01 stays [O]: the discriminators distinguish the TARGET but do not by themselves PROVE the seam realises it (continuum existence, residual [O]); they make the claim FALSIFIABLE. Python-only (numpy; the det/root-count facts are Wolfram-mirrored via v89/v281/v422). """ import itertools import numpy as np from tfpt_constants import check, summary, reset, g_car, N_fam def cartan_E8(): return np.array([ [2, 0, -1, 0, 0, 0, 0, 0], [0, 2, 0, -1, 0, 0, 0, 0], [-1, 0, 2, -1, 0, 0, 0, 0], [0, -1, -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, 0, 0, 0, 0, 0, -1, 2]], dtype=float) def cartan_D8(): C = 2 * np.eye(8) for i in range(7): C[i, i + 1] = C[i + 1, i] = -1 C[6, 7] = C[7, 6] = 0 C[5, 7] = C[7, 5] = -1 # D8 fork return C def count_norm2_integer(): """norm-2 vectors of the D8 (integer, even-sum) lattice: (+-1,+-1,0^6).""" n = 0 for i, j in itertools.combinations(range(8), 2): for si in (1, -1): for sj in (1, -1): n += 1 # vector with +-1 at i,j return n # 28*4 = 112 def count_norm2_spinor(): """norm-2 vectors in the (1/2)^8 spinor coset with even # of minus signs.""" n = 0 for signs in itertools.product((0.5, -0.5), repeat=8): if sum(1 for s in signs if s < 0) % 2 == 0: # even minus signs if abs(sum(s * s for s in signs) - 2.0) < 1e-9: # norm^2 = 8*1/4 = 2 n += 1 return n # 128 def run(): reset() print("v443 SEAM.E8.DISCRIMINATOR: the mode-D empirical kill-test bundle " "(E8)_1 vs SO(16)_1") det_E8 = int(round(np.linalg.det(cartan_E8()))) det_D8 = int(round(np.linalg.det(cartan_D8()))) # ---- 1. primary / GSD discriminator ---- gsd_E8, gsd_D8 = abs(det_E8), abs(det_D8) # torus GSD = #primaries check("PRIMARY / GSD DISCRIMINATOR [E]/[X]: |det Cartan(E8)|=%d (ONE primary, " "holomorphic) vs |det Cartan(D8)=SO(16)|=%d (FOUR primaries 1,v,s,c); " "torus GSD = #primaries = %d vs %d. KILL TEST: a seam with GSD>1 / >1 " "superselection sector falsifies (E8)_1" % (det_E8, det_D8, gsd_E8, gsd_D8), det_E8 == 1 and det_D8 == 4) # ---- 2. current / theta discriminator ---- dim_E8, dim_SO16 = 248, 120 # level-1 current counts roots_int = count_norm2_integer() # 112 (D8 roots) roots_spin = count_norm2_spinor() # 128 (spinor coset) roots_E8 = roots_int + roots_spin # 240 check("CURRENT / THETA DISCRIMINATOR [E]/[X]: level-1 currents dim E8=%d vs " "dim SO(16)=%d; E8 lattice theta has %d norm-2 roots (=%d integer + %d " "spinor, computed) vs D8's %d -- current spectrum and lattice theta DIFFER. " "KILL TEST: the edge current multiplicity (248 vs 120) is observable" % (dim_E8, dim_SO16, roots_E8, roots_int, roots_spin, roots_int), roots_int == 112 and roots_spin == 128 and roots_E8 == 240 and dim_E8 == 240 + 8) # ---- 3. c=8 is NOT a discriminator ---- c = g_car + N_fam check("c=8 IS NOT A DISCRIMINATOR [E]: both (E8)_1 and SO(16)_1 have c=%d (the " "v277 ambiguity), so the central charge ALONE cannot decide; only the " "order-4 mu4 condensation + det K=1 (the genus-1 statement, " "SEAM.S3.MODULAR.01/v90) selects E8" % c, c == 8) # ---- 4. non-vacuity ---- fires = (det_D8 != det_E8) and (dim_SO16 != dim_E8) and (roots_int != roots_E8) check("NON-VACUITY [E]: the discriminators all fire on the rival (det %d!=%d, " "currents %d!=%d, roots %d!=%d) -- the kill tests are non-vacuous" % (det_D8, det_E8, dim_SO16, dim_E8, roots_int, roots_E8), fires) # ---- 5. verdict (typed [X]/[C]) ---- check("VERDICT [X]/[C]: the (E8)_1 seam carries sharp falsifiable signatures " "distinct from every nearby c=8 net (mode D of v347 as concrete kill " "tests). SEAM.EQUIV.01 stays [O]: the discriminators distinguish the " "TARGET but do not by themselves PROVE the seam realises it (continuum " "existence, residual [O]); they make the claim FALSIFIABLE", det_E8 == 1 and det_D8 == 4 and roots_E8 == 240 and fires and g_car == 5) return summary("v443 SEAM.E8.DISCRIMINATOR (mode-D kill tests (E8)_1 vs SO(16)_1; " "falsifiable; SEAM.EQUIV.01 stays [O])") if __name__ == "__main__": raise SystemExit(1 if run() else 0)