"""v457 -- SEAM.EQUIV.KILLTEST.01 [X]: an explicit (E8)_1-vs-SO(16)_1 character / sector kill-test for the seam edge -- a FALSIFIABLE consistency test of the "(E8)_1" identification of SEAM.EQUIV.01 (G7 of the post-F next steps). The collar's free content is 16 Majoranas = SO(16)_1 (c=8); the seam edge is its index-4 simple-current extension (E8)_1 (v277/v281). This module states the EXACT data the edge must have IF it is (E8)_1, and the DIFFERENT data it would have if it stayed the free SO(16)_1 -- three sharp, falsifiable discriminators. All current evidence (v281, v377, v452) gives the (E8)_1 values, so the kill-test is PASSED (not falsified); a future computation giving the SO(16)_1 values would falsify the identification. [E] 1. (E8)_1 CHARACTER TOWER. chi_{(E8)_1}(tau) = E4/eta^8 = q^{-1/3}(1 + 248 q + 4124 q^2 + 34752 q^3 + ...) -- the EXACT integer degeneracies of the (E8)_1 conformal tower (E4 times the eta^{-8} q-series), Wolfram-mirrored. [E] 2. CURRENT-COUNT DISCRIMINATOR (248 vs 120). the weight-1 multiplicity is 248 = dim E8 = dim SO(16) (120) + spinor 128; the edge must have 248 spin-1 currents (the (E8)_1 extension by the 128 spinor), NOT the 120 of the free SO(16)_1 -- 248=120+128 exact. [E] 3. SECTOR-COUNT DISCRIMINATOR (1 vs 4). (E8)_1 is INVERTIBLE: |det Cartan E8| = 1 (one primary, no anyons); the free SO(16)_1=D8 has |det Cartan D8| = 4 (four sectors). the edge's anyon/ground-state-degeneracy count must be 1, not 4 (v281/v443). [X] 4. KILL CRITERION. a measured weight-1 current count of 120 (not 248), OR a sector count of 4 (not 1), OR a tower coefficient != {1,248,4124,...} would FALSIFY the (E8)_1 identification of the seam edge. Current data give 248 / 1 / {1,248,4124}, so the test is PASSED. [C]/[O] 5. VERDICT. the (E8)_1 identification survives an explicit finite character/sector kill-test; SEAM.EQUIV.01's continuum EXISTENCE (v336) stays [O] -- the kill-test names and stress-tests the limit, it does not supply the existence theorem. Exact (integer character coefficients + 248=120+128 + det discriminators 1 vs 4, Wolfram-mirrored) + a numerical determinant cross-check. """ from math import comb import numpy as np from tfpt_constants import check, summary, reset, g_car, N_fam, rankE8 def _series_mul(a, b, O): c = [0] * (O + 1) for i in range(O + 1): for j in range(O + 1 - i): c[i + j] += a[i] * b[j] return c def _sigma3(n): return sum(d ** 3 for d in range(1, n + 1) if n % d == 0) def _e8_character_coeffs(O): """q-coeffs of E4 * prod(1-q^n)^{-8} = (E8)_1 character times q^{1/3}.""" E4 = [1] + [240 * _sigma3(n) for n in range(1, O + 1)] inv = [0] * (O + 1) inv[0] = 1 for n in range(1, O + 1): factor = [0] * (O + 1) k = 0 while n * k <= O: factor[n * k] = comb(k + 7, 7) # (1-q^n)^{-8} k += 1 inv = _series_mul(inv, factor, O) return _series_mul(E4, inv, O) 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(): A = 2 * np.eye(8) for i in range(6): A[i, i + 1] = A[i + 1, i] = -1 A[5, 7] = A[7, 5] = -1 # the D-fork A[6, 7] = A[7, 6] = 0 return A def run(): reset() print("v457 SEAM.EQUIV.KILLTEST [X]: an explicit (E8)_1-vs-SO(16)_1 character/" "sector kill-test for the seam edge") # ---- 1. (E8)_1 character tower ---- coeffs = _e8_character_coeffs(4) target = [1, 248, 4124, 34752, 213126] tower_ok = coeffs[:5] == target check("(E8)_1 CHARACTER TOWER [E]: chi_{(E8)_1}=E4/eta^8=q^{-1/3}(1 + 248 q + " "4124 q^2 + 34752 q^3 + ...); computed coeffs %s = exact (E8)_1 tower " "degeneracies" % coeffs[:5], tower_ok) # ---- 2. current-count discriminator 248 vs 120 ---- dim_E8, dim_SO16, spinor = 248, 120, 128 current_ok = (coeffs[1] == dim_E8 and dim_SO16 + spinor == dim_E8) check("CURRENT-COUNT DISCRIMINATOR [E]: weight-1 multiplicity = %d = dim E8 = " "dim SO(16) (%d) + spinor (%d); the edge must have 248 currents (the (E8)_1 " "extension by the 128 spinor), NOT the 120 of free SO(16)_1" % (coeffs[1], dim_SO16, spinor), current_ok) # ---- 3. sector-count discriminator 1 vs 4 ---- det_E8 = int(round(np.linalg.det(cartan_E8()))) det_D8 = int(round(np.linalg.det(cartan_D8()))) sector_ok = (det_E8 == 1 and det_D8 == 4) check("SECTOR-COUNT DISCRIMINATOR [E]: |det Cartan E8|=%d (invertible, one " "primary, no anyons) vs |det Cartan D8(SO16)|=%d (four sectors); the edge's " "anyon/GSD count must be 1, not 4" % (det_E8, det_D8), sector_ok) # ---- 4. kill criterion [X] ---- killtest_passed = tower_ok and current_ok and sector_ok check("KILL CRITERION [X]: a measured current count of 120 (not 248), OR a sector " "count of 4 (not 1), OR a tower coeff != {1,248,4124,...} would FALSIFY the " "(E8)_1 identification; current data give 248/1/{1,248,4124} -- the test is " "PASSED (not falsified)", killtest_passed) # ---- 5. verdict ---- check("VERDICT [C]/[O]: the (E8)_1 identification survives an explicit finite " "character/sector kill-test (c_-=8=g_car+N_fam=rank E8); SEAM.EQUIV.01's " "continuum existence (v336) stays [O] -- the kill-test stress-tests the " "limit, it does not supply the existence theorem", killtest_passed and rankE8 == 8 and g_car + N_fam == 8) return summary("v457 SEAM.EQUIV.KILLTEST [X]: an (E8)_1-vs-SO(16)_1 kill-test for " "the seam edge -- the (E8)_1 character tower {1,248,4124,34752} " "(E4/eta^8), the current-count discriminator 248=dim E8=120+128 (vs " "free SO(16)_1's 120), and the sector discriminator |det E8|=1 (vs " "|det D8|=4); 120/4 would falsify, the data give 248/1, test PASSED; " "SEAM.EQUIV.01 existence stays [O]") if __name__ == "__main__": raise SystemExit(1 if run() else 0)