"""v301 -- SEAM.EQUIV.A.INV.01: discharge Route A's single open hypothesis -- the v297 LIT-A input "the raw quasi-free seam bulk is invertible / short-range-entangled (SRE)" -- via the free-fermion classification. Since v300 showed Route B collapses into Route A's rationality, this is the LAST unconditional gap of SEAM.EQUIV.01. THE MOVE. A GAPPED free-fermion (Gaussian / quasi-free) 2+1D bulk cannot carry intrinsic topological order: free fermions give the integer (IQHE / Chern-insulator) phases, ALL of which are INVERTIBLE (no fractional anyons) -- intrinsic topological order requires interactions. The only invariant of a gapped free-fermion phase is the chiral central charge c (Kitaev's "periodic table", class D, Z-classified by the chiral Majorana number). The carrier is 16 Majoranas (v148), so c = 16/2 = 8 = g_car + N_fam: the canonical c=8 invertible phase (Kitaev's E8 state). Hence "invertible" is AUTOMATIC for the gapped quasi-free seam, and LIT-A's hypothesis reduces from a TOPOLOGICAL-ORDER statement to the SPECTRAL input "the bulk is gapped + quasi-free" (v160 + the mass gap). [E] 1. CARRIER = 16 MAJORANAS, c = 8. g_car + N_fam = 5 + 3 = 8 = 16/2 (the v148 NS/R census of the 16 carrier Majoranas). [E] 2. SO(16)_1 ⊃ (E8)_1 AT c = 8. dim so(16) = 120 currents + 128 spinor = 248 = dim E8, the level-1 currents -- the 16-Majorana theory sits at c=8 with the E8 current content (the v154 simple-current extension). [C] 3. FREE-FERMION INVERTIBILITY (Kitaev periodic table). gapped Gaussian fermion phases (class D, no symmetry) are Z-classified by the chiral Majorana number and are ALL invertible -- no intrinsic anyons; intrinsic topological order needs interactions. So a gapped quasi-free bulk is invertible, automatically. [E] 4. INVERTIBILITY INVARIANT |det K| = #anyons = 1. the E8 K-matrix (Cartan E8) has det = 1 (a single primary => invertible); the interacting/gauged contrast D8 = SO(16) has det(Cartan) = 4 (four anyons), which a free-fermion bulk cannot produce -- output-consistency with (3). [O] 5. THE REDUCTION. LIT-A's "invertible bulk" now FOLLOWS from "gapped 16-Majorana quasi-free bulk" (v148 carrier + v160 quasi-free fixed point + the mass gap); the SEAM.EQUIV.01 residual changes character -- from "is the bulk invertible (could it hide anyons)?" (now: no, a gapped free-fermion bulk cannot) to "is the quasi-free bulk gapped?" (a spectral input). Combined with v300 (Route B = Route A's rationality), the WHOLE remaining open content of SEAM.EQUIV.01 is the single spectral statement "the quasi-free seam bulk is gapped". Status: [E] the carrier/c=8/embedding/det invariants; [C] the free-fermion invertibility import (Kitaev); [O] the reduced residual (gapped + quasi-free). Discharges the TOPOLOGICAL-ORDER part of Route A's last hypothesis, reducing it to a spectral input; it does NOT manufacture the gap. Python (numpy + sympy). """ import numpy as np from tfpt_constants import check, summary, reset, g_car, N_fam def _cartan(edges, n): """Cartan matrix C = 2I - A for a simply-laced Dynkin diagram (0-indexed edges).""" A = np.zeros((n, n), int) for i, j in edges: A[i, j] = A[j, i] = 1 return 2 * np.eye(n, dtype=int) - A # E8 (Bourbaki): chain 1-3-4-5-6-7-8 with node 2 forked off node 4 E8_EDGES = [(0, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7), (1, 3)] # D8 = SO(16): chain 1-2-3-4-5-6 with the two prongs 7,8 forked off node 6 D8_EDGES = [(0, 1), (1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (5, 7)] def run(): reset() print("v301 SEAM.EQUIV.A.INV.01: discharge Route A's 'invertible bulk' via the free-fermion classification") # 1. carrier = 16 Majoranas, c = 8 n_majorana = 16 c = n_majorana / 2 check("CARRIER = 16 MAJORANAS, c=8 [E]: g_car + N_fam = %d + %d = %d = 16/2 = c " "(the v148 NS/R census of the 16 carrier Majoranas) -- the central charge is " "fixed by the carrier count, not assumed" % (g_car, N_fam, g_car + N_fam), g_car + N_fam == 8 and c == 8.0) # 2. SO(16)_1 ⊃ (E8)_1: 120 currents + 128 spinor = 248 dim_so16 = n_majorana * (n_majorana - 1) // 2 # adjoint so(16) spinor = 2 ** (n_majorana // 2 - 1) # one chirality dim_e8 = 248 check("SO(16)_1 ⊃ (E8)_1 AT c=8 [E]: dim so(16)=%d currents + %d spinor (one " "chirality) = %d = dim E8 = the level-1 currents (the v154 simple-current " "extension) -- the 16-Majorana theory carries the E8 current content" % (dim_so16, spinor, dim_so16 + spinor), dim_so16 == 120 and spinor == 128 and dim_so16 + spinor == dim_e8) # 3. free-fermion invertibility (Kitaev): gapped Gaussian => invertible check("FREE-FERMION INVERTIBILITY [C]: a gapped Gaussian (quasi-free) 2+1D fermion " "phase (class D, no symmetry) is Z-classified by the chiral Majorana number " "and is ALWAYS invertible -- no intrinsic anyons; intrinsic topological order " "requires interactions (Kitaev periodic table). So a gapped quasi-free bulk is " "invertible automatically", True) # 4. invertibility invariant: #anyons = |det K| ; E8 -> 1, D8 -> 4 (contrast) det_e8 = int(round(np.linalg.det(_cartan(E8_EDGES, 8)))) det_d8 = int(round(np.linalg.det(_cartan(D8_EDGES, 8)))) check("INVERTIBILITY INVARIANT |det K| = #anyons [E]: the E8 K-matrix (Cartan E8) has " "det = %d (one primary => invertible); the gauged/interacting contrast " "D8=SO(16) has det(Cartan) = %d (four anyons), which a free-fermion bulk " "CANNOT produce -- output-consistency with the free-fermion classification" % (det_e8, det_d8), det_e8 == 1 and det_d8 == 4) # 5. the reduction check("THE REDUCTION [O]: LIT-A's 'invertible bulk' now FOLLOWS from 'gapped " "16-Majorana quasi-free bulk' (v148 carrier + v160 quasi-free fixed point + " "the mass gap); the SEAM.EQUIV.01 residual changes character -- from 'is the " "bulk invertible (could it hide anyons)?' (now: a gapped free-fermion bulk " "cannot) to 'is the quasi-free bulk gapped?' (a spectral input)", True) # 6. verdict check("VERDICT [O]: with v300 (Route B = Route A's rationality) and this discharge of " "Route A's topological-order hypothesis, the WHOLE remaining open content of " "SEAM.EQUIV.01 is the single SPECTRAL statement 'the quasi-free seam bulk is " "gapped'. The topological-order obstruction (hidden anyons) is removed; the gap " "is NOT manufactured", det_e8 == 1 and (g_car + N_fam) == 8) return summary("v301 SEAM.EQUIV.A.INV.01: Route A's 'invertible bulk' hypothesis is " "discharged by the free-fermion classification (gapped 16-Majorana c=8 " "quasi-free bulk is automatically invertible, #anyons=|det K_E8|=1); " "with v300 the whole SEAM.EQUIV.01 open content reduces to the single " "spectral input 'the quasi-free seam bulk is gapped' -- no " "topological-order obstruction remains") if __name__ == "__main__": raise SystemExit(1 if run() else 0)