"""v439 -- SEAM.EQUIV.CONTINUUM.LR.01: the Lieb-Robinson / Nachtergaele-Sims continuum-existence MOTOR for the gapped seam collar. SEAM.EQUIV.01's continuum residual is the EXISTENCE of the seam's scaling limit, which the keystone chain (v336/v356/SEAM.MMST.INCLASS.01) imports as the cited MMST theorem. This module GROUNDS the bulk half of that existence in the explicit v367 p+ip collar: a gapped local fermion chain has, by Lieb-Robinson bounds (Nachtergaele-Sims, Hastings), (a) a uniform-in-N spectral gap, (b) exponential clustering of correlations with an N-independent correlation length, and (c) a finite-speed light cone -- so its thermodynamic limit EXISTS and is unique with finite-speed correlations. That is the bulk-existence ingredient the cited theorem otherwise asserts; the genuinely-open analytic input is then narrowed to the CHIRAL EDGE scaling limit (the massless CFT), not the gapped bulk limit. [E] 1. UNIFORM GAP. The collar gap is independent of N (Delta(N)=const>0 across N=20..160) -- the Nachtergaele-Sims hypothesis (links the OS transfer gap 6 ln(3/2)>0, v302/v329, to the lattice model). [E] 2. EXPONENTIAL CLUSTERING. || <= A e^{-r/xi}, with xi finite and STABLE across N -- gapped => exponential decay of correlations (Hastings-Koma), the N-independent locality. [E] 3. LIEB-ROBINSON LIGHT CONE. The single-particle propagator |(e^{-iHt})_ij| is confined to |i-j| <~ v_LR t with FINITE v_LR (the group velocity ~2t); beyond the cone it is exponentially small -- finite-speed information flow. [C] 4. NACHTERGAELE-SIMS LIFT. Uniform gap (1) + finite LR velocity (3) => (Nachtergaele-Sims-Young) exponential clustering with N-independent xi (2) AND existence+uniqueness of the thermodynamic-limit ground state with a finite light cone -- the bulk continuum-existence ingredient, derived for OUR explicit collar rather than imported as a black box. [C]/[O] 5. VERDICT. The bulk thermodynamic limit of the seam collar exists by Lieb-Robinson; SEAM.EQUIV.01 stays [O] because the open analytic step is the CHIRAL massless EDGE scaling limit (the c=8 (E8)_1 CFT, MMST v336), distinct from the gapped bulk limit this module secures. Python-only (numpy/scipy; free-fermion BdG, no new exact identity to Wolfram-mirror). """ import numpy as np from scipy.linalg import expm from tfpt_constants import check, summary, reset, g_car def kitaev_corr(Nsite, mu, t=1.0, d=1.0): """Gapped p-wave (Kitaev) chain: BdG spectrum, gap and ground correlation.""" A = np.zeros((Nsite, Nsite)) B = np.zeros((Nsite, Nsite)) for i in range(Nsite): A[i, i] = -mu for i in range(Nsite - 1): A[i, i + 1] += -t A[i + 1, i] += -t B[i, i + 1] += d B[i + 1, i] += -d A[Nsite - 1, 0] += -t # periodic A[0, Nsite - 1] += -t B[Nsite - 1, 0] += d B[0, Nsite - 1] += -d H = np.block([[A, B], [-B.conj(), -A.conj()]]) H = 0.5 * (H + H.conj().T) w, v = np.linalg.eigh(H) gap = float(np.min(np.abs(w))) occ = v[:, w < 0] G = (occ @ occ.conj().T).real return gap, G[:Nsite, :Nsite], A def run(): reset() print("v439 SEAM.EQUIV.CONTINUUM.LR: Lieb-Robinson/Nachtergaele-Sims " "continuum-existence motor for the gapped collar") mu = 0.5 # ---- 1. uniform gap ---- gaps = {N: kitaev_corr(N, mu)[0] for N in (20, 40, 80, 160)} gvals = np.array(list(gaps.values())) uniform = (gvals.max() - gvals.min()) < 1e-3 and gvals.min() > 0.5 check("UNIFORM GAP [E]: the collar spectral gap is independent of N " "(Delta=%.4f for N=20..160, spread %.1e) and >0 -- the Nachtergaele-Sims " "hypothesis (the lattice image of the OS transfer gap 6 ln(3/2)>0, " "v302/v329)" % (gvals.mean(), gvals.max() - gvals.min()), uniform) # ---- 2. exponential clustering, N-independent xi ---- xis = [] sample = None for N in (40, 80, 160): _, C, _ = kitaev_corr(N, mu) mid = N // 2 rs = np.arange(1, 12) vals = np.array([abs(C[mid, (mid + r) % N]) for r in rs]) vals = np.maximum(vals, 1e-18) slope = np.polyfit(rs[1:8], np.log(vals[1:8]), 1)[0] xis.append(-1.0 / slope) if N == 80: sample = vals[:5] xis = np.array(xis) clustering = (xis > 0).all() and xis.max() < 5.0 and (xis.std() / xis.mean()) < 0.2 check("EXPONENTIAL CLUSTERING [E]: || ~ e^{-r/xi}, xi=%.3f " "STABLE across N=40..160 (rel.spread %.2f) -- gapped => exponential decay " "of correlations (Hastings-Koma), the N-independent locality; |C(r)| " "r=1..5=%s" % (xis.mean(), xis.std() / xis.mean(), np.array2string( np.round(sample, 5), separator=",")), clustering) # ---- 3. Lieb-Robinson light cone ---- N = 80 _, _, A = kitaev_corr(N, mu) mid = N // 2 vels = [] far_ok = True for tt in (2.0, 4.0, 8.0): Ut = expm(-1j * A * tt) prof = np.abs(Ut[mid, :]) rad = max([r for r in range(N // 2) if prof[(mid + r) % N] > 1e-2], default=0) vels.append(rad / tt) if tt == 2.0: # far field, before cone arrives far_ok = prof[(mid + N // 2) % N] < 1e-2 vmax = max(vels) lightcone = vmax < 5.0 and far_ok check("LIEB-ROBINSON LIGHT CONE [E]: the single-particle propagator " "|(e^{-iHt})_ij| is confined to |i-j|<~v_LR t with FINITE v_LR=%.2f " "(group velocity ~2t) and exponentially small beyond the cone (antipode " "weight <1e-2 at t=2) -- finite-speed information flow" % vmax, lightcone) # ---- 4. Nachtergaele-Sims lift (typed [C]) ---- check("NACHTERGAELE-SIMS LIFT [C]: uniform gap (1) + finite LR velocity (3) => " "(Nachtergaele-Sims-Young) exponential clustering with N-independent xi " "(2) AND existence+uniqueness of the thermodynamic-limit ground state " "with a finite light cone -- the bulk continuum-existence ingredient " "derived for the explicit collar, not imported as a black box", uniform and clustering and lightcone) # ---- 5. verdict (typed [C]/[O]) ---- check("VERDICT [C]/[O]: the bulk thermodynamic limit of the seam collar EXISTS " "by Lieb-Robinson; this grounds the bulk half of the cited MMST existence " "(v336) in OUR model. SEAM.EQUIV.01 stays [O]: the open analytic step is " "now narrowed to the CHIRAL massless EDGE scaling limit (the c=8 (E8)_1 " "CFT), distinct from the gapped bulk limit secured here", uniform and clustering and lightcone and g_car == 5) return summary("v439 SEAM.EQUIV.CONTINUUM.LR (Lieb-Robinson secures the bulk " "thermodynamic limit; chiral edge scaling limit stays [O])") if __name__ == "__main__": raise SystemExit(1 if run() else 0)