"""v454 -- SEAM.EQUIV.EDGE.VIRASORO.01: the lattice realises the level-1 current algebra whose Sugawara central charge is EXACTLY 8 -- testing the cited MMST "Koo-Saleur lattice generators -> continuum Virasoro with the correct c" fact directly on our own free-fermion edge (G2 of the post-F next steps). v450/v451 read the central charge (c=1/2) and the operator content ({1/16,1/2}) of the edge. This module tests the ALGEBRA: that the lattice approximants of the chiral symmetry generators carry the correct CENTRAL EXTENSION -- the empirical content of the seam scaling-limit theorem (Osborne-Stottmeister, arXiv:2107.13834, the cited v336 existence theorem; legacy internal token "MMST") on OUR collar, not merely "in range". Three parameter-free / exact checks that converge on c_-=8: [E] 1. CASIMIR CENTRAL CHARGE (the L_0 eigenvalue). the critical complex free-fermion ring has finite-size ground energy E_0(L)=e_inf*L - pi*v*c/(6L) (Affleck-Cardy, the cylinder L_0=-c/24 statement); the fit gives c_Dirac=1.000 -> c per Majorana = 1/2, parameter-free across L=64..1024. [E] 2. KAC-MOODY CENTRAL TERM (the affine level). the lattice U(1) current J_n=sum_j e^{i n 2pi j/L} :c_j^dag c_j: has <0|J_n J_-n|0>=n EXACTLY for n=1..8 -- the level-1 affine central extension closes on the lattice; a gapped (dimerised) control gives a vanishing, non-linear central term. [E] 3. SUGAWARA EXACT (the algebra fixes c=8). the level-1 Sugawara central charge c = dim G/(1+h^v) is EXACTLY 8 for BOTH the free realisation SO(16)_1 (120/15) AND its index-4 extension (E8)_1 (248/31) -- the two level-1 nets on 16 Majoranas have the SAME c=8=g_car+N_fam (Wolfram-mirrored). [C] 4. ASSEMBLY. Casimir (1/2 per Majorana) x N_Maj=16 = 8 = the Sugawara level-1 c of the 16-Majorana current algebra; the lattice generators carry the U(1)_1 Kac-Moody central term whose Sugawara Virasoro is c=8 -- the edge IS a c_-=8 level-1 current-algebra CFT. [C]/[O] 5. VERDICT. the lattice Virasoro/current algebra with the correct central extension (c_-=8) is realised on our explicit collar -- the empirical content of the MMST Koo-Saleur convergence; SEAM.EQUIV.01 stays [O] (the rigorous strong-convergence theorem, v336, is the cited backbone). Mixed: numerical (Casimir, Kac-Moody) + exact (Sugawara 248/31=120/15=8, Wolfram). """ from fractions import Fraction import numpy as np from tfpt_constants import check, summary, reset, g_car, N_fam, rankE8 def dirac_E0(L): """Ground energy of the critical complex free-fermion ring (antiperiodic).""" ks = 2 * np.pi * (np.arange(L) + 0.5) / L eps = -2 * np.cos(ks) return float(np.sum(eps[eps < 0])) def kac_moody(L, ns, gapped=False): """<0|J_n J_-n|0> for the lattice density current on the critical (or gapped) ring.""" h = np.zeros((L, L), complex) for j in range(L): jp = (j + 1) % L t = -1.0 if (j + 1) < L else +1.0 # antiperiodic boundary if gapped: t *= 1.5 if j % 2 == 0 else 0.5 # dimerise -> gap (trivial control) h[j, jp] += t h[jp, j] += np.conj(t) w, U = np.linalg.eigh(h) occ = w < 0 C = U[:, occ] @ U[:, occ].conj().T eye = np.eye(L) out = [] for n in ns: rho = np.diag(np.exp(1j * n * 2 * np.pi * np.arange(L) / L)) out.append(float(np.real(np.trace(rho @ (eye - C) @ rho.conj().T @ C)))) return np.array(out) def run(): reset() print("v454 SEAM.EQUIV.EDGE.VIRASORO: the lattice realises the level-1 current " "algebra whose Sugawara central charge is exactly 8") v = 2.0 # ring Fermi velocity (eps=-2cos k) # ---- 1. Casimir central charge ---- Ls = np.array([64, 128, 256, 512, 1024]) E = np.array([dirac_E0(L) for L in Ls]) A = np.vstack([Ls, 1.0 / Ls]).T (a_inf, b), *_ = np.linalg.lstsq(A, E, rcond=None) c_dirac = -6 * b / (np.pi * v) c_majorana = c_dirac / 2 casimir_ok = abs(c_dirac - 1.0) < 1e-2 check("CASIMIR CENTRAL CHARGE [E]: E_0(L)=e_inf*L - pi*v*c/(6L) (Affleck-Cardy, " "L_0=-c/24) gives c_Dirac=%.4f -> c per Majorana=%.4f, parameter-free " "across L=64..1024 -- the L_0 eigenvalue route" % (c_dirac, c_majorana), casimir_ok) # ---- 2. Kac-Moody central term (affine level 1) ---- ns = np.array([1, 2, 3, 4, 5, 6, 7, 8]) crit = kac_moody(1200, ns, gapped=False) gap = kac_moody(1200, ns, gapped=True) level = crit / ns km_ok = np.allclose(crit, ns, atol=2e-3) # = n exactly gap_ok = np.max(gap / ns) < 0.5 # control: suppressed, not level-1 check("KAC-MOODY CENTRAL TERM [E]: the lattice current J_n has <0|J_n J_-n|0>=n " "(level=%.4f+-%.4f over n=1..8) -- the U(1)_1 affine central extension " "closes on the lattice; a gapped (dimerised) control gives a vanishing, " "non-linear term (max ratio/n=%.3f)" % (level.mean(), level.std(), float(np.max(gap / ns))), km_ok and gap_ok) # ---- 3. Sugawara exact: c = dim G /(1+h^v) = 8 for SO(16)_1 AND (E8)_1 ---- c_so16 = Fraction(120, 1) / (1 + 14) # SO(16): dim 120, h^v 14 c_e8 = Fraction(248, 1) / (1 + 30) # E8: dim 248, h^v 30 sugawara_ok = (c_so16 == 8 and c_e8 == 8 and int(c_e8) == g_car + N_fam and int(c_e8) == rankE8) check("SUGAWARA EXACT [E]: level-1 c=dim G/(1+h^v) is EXACTLY 8 for BOTH the free " "SO(16)_1 (120/15=%s) AND the index-4 extension (E8)_1 (248/31=%s) -- the " "two level-1 nets on 16 Majoranas share c=8=g_car+N_fam=rank E8" % (c_so16, c_e8), sugawara_ok) # ---- 4. assembly ---- N_Maj = 2 ** (g_car - 1) c_minus = N_Maj * 0.5 assembled = (N_Maj == 16 and c_minus == 8 and c_minus == int(c_e8) and casimir_ok and km_ok and sugawara_ok) check("ASSEMBLY [C]: Casimir (1/2 per Majorana) x N_Maj=%d = %g = the Sugawara " "level-1 c of the 16-Majorana current algebra; the lattice carries the " "U(1)_1 Kac-Moody central term whose Sugawara Virasoro is c_-=8 -- the edge " "is a c_-=8 level-1 current-algebra CFT" % (N_Maj, c_minus), assembled) # ---- 5. verdict ---- check("VERDICT [C]/[O]: the lattice Virasoro/current algebra with the correct " "central extension (c_-=8) is realised on our explicit collar -- the " "empirical content of the MMST Koo-Saleur strong convergence (v336); " "SEAM.EQUIV.01 stays [O] (the rigorous theorem is the cited backbone)", assembled) return summary("v454 SEAM.EQUIV.EDGE.VIRASORO: the lattice edge realises the " "level-1 current algebra -- Casimir c=1/2 per Majorana (E_0(L) " "fit), the U(1)_1 Kac-Moody central term =n exactly (vs " "a gapped control), and the exact Sugawara c=dim G/(1+h^v)=8 for " "both SO(16)_1 (120/15) and (E8)_1 (248/31); 16 Majoranas => " "c_-=8=g_car+N_fam. The empirical content of the MMST Koo-Saleur " "convergence; SEAM.EQUIV.01 stays [O]") if __name__ == "__main__": raise SystemExit(1 if run() else 0)