"""v444 -- SEAM.EQUIV.CONTINUUM.EDGE.01: the CHIRAL EDGE scaling-limit MOTOR for the seam collar -- the edge half of the one SEAM.EQUIV.01 continuum residual. v439 (SEAM.EQUIV.CONTINUUM.LR.01) grounded the BULK half of the cited MMST existence (v336): the gapped collar has a uniform gap + exponential clustering + finite light cone, so its thermodynamic limit exists. That module explicitly narrowed the open analytic step to the CHIRAL massless EDGE scaling limit (the c=8 (E8)_1 CFT). This module attacks THAT residual head-on on the SAME explicit v367/v368 p+ip model: it shows the lattice edge already carries the kinematic AND conformal data of a chiral c=1/2 Majorana CFT, and that those data CONVERGE (scaling-collapse) as the lattice is refined -- the numerical hallmark that the edge scaling limit EXISTS. It does NOT supply the rigorous uniform-convergence theorem (MMST, v336) and does NOT close SEAM.EQUIV.01; it grounds the edge half the way v439 grounded the bulk half, so the keystone's one continuum input is now numerically grounded on BOTH halves and pinned to a single cited existence theorem. Edge data of a chiral free-fermion (Majorana) CFT: * a RELATIVISTIC (linear) edge dispersion E(k)=v k with a DEFINITE chirality (opposite group velocities on the two boundaries: one chiral Majorana/edge); * an ALGEBRAIC equal-time edge two-point function ~ r^{-eta} with eta=1 (the free-fermion exponent), in contrast to the EXPONENTIAL bulk decay (gapped); * those numbers must be STABLE under refinement (scaling-collapse) -- the signature of an existing continuum limit. [E] 1. CHIRAL LINEAR EDGE DISPERSION. the in-gap branch is linear near its zero-crossing (R^2>0.999, velocity v~0.98); the two boundaries carry OPPOSITE velocities -- one chiral Majorana per edge (relativistic, chiral kinematics). [E] 2. ALGEBRAIC EDGE TWO-POINT. the equal-time edge correlator decays as a power law r^{-eta} with eta~1 (the chiral free-fermion CFT exponent) -- a CRITICAL edge. [E] 3. EXPONENTIAL BULK (neg control). the bulk-row correlator decays EXPONENTIALLY (finite xi) -- only the edge is critical, confirming the power law is the edge CFT, not a geometric artefact. [E] 4. TRIVIAL-PHASE NEG CONTROL. M=3 (no edge, v368) has an EXPONENTIAL edge-row correlator -- the algebraic decay is a CONSEQUENCE of the topological edge, not of the boundary geometry. [E] 5. SCALING-COLLAPSE / CONVERGENCE. eta is N-stable and the edge correlator profile COLLAPSES across refinements (Nx=36..60) -- the lattice edge correlators converge to a fixed conformal profile (existing scaling limit, the edge analogue of v439's bulk thermodynamic limit). [C] 6. EDGE CENTRAL CHARGE. eta=1 <=> one chiral Majorana (c=1/2) per edge, so the N_Maj=2^{g_car-1}=16-copy carrier has a chiral c_-=8 edge CFT (=(E8)_1, consistent with v367/v368/v376): linear chiral dispersion + algebraic 1/r correlator + c_-=8 ARE the conformal data of the edge. [C]/[O] 7. VERDICT. the edge half of the cited MMST scaling limit (v336) is now numerically GROUNDED on our explicit collar (kinematics + conformal correlator + scaling-collapse); SEAM.EQUIV.01 stays [O] because the rigorous uniform convergence of all edge n-point functions (the MMST existence theorem) is not supplied here. With v439 (bulk) + v444 (edge) both halves of the one continuum residual are grounded. Python-only (numpy; free-fermion BdG, no new exact identity to Wolfram-mirror). """ import numpy as np from tfpt_constants import check, summary, reset, g_car, N_fam SX = np.array([[0, 1], [1, 0]], complex) SY = np.array([[0, -1j], [1j, 0]], complex) SZ = np.array([[1, 0], [0, -1]], complex) def _strip_H(kx, M, Ly): """v367/v368 p+ip strip: Ly sites in y (open), kx periodic. 2Ly x 2Ly BdG.""" ons = np.sin(kx) * SX + (M - np.cos(kx)) * SZ hop = -0.5 * SZ + (1 / (2j)) * SY H = np.zeros((2 * Ly, 2 * Ly), complex) for y in range(Ly): H[2 * y:2 * y + 2, 2 * y:2 * y + 2] = ons for y in range(Ly - 1): H[2 * y:2 * y + 2, 2 * y + 2:2 * y + 4] = hop H[2 * y + 2:2 * y + 4, 2 * y:2 * y + 2] = hop.conj().T return H def _real2d(Nx, Ny, M): """The SAME p+ip model in 2D real space: periodic x, open y (the v368 strip on a torus-in-x), so the edge correlator along x can be read off.""" onx = -0.5 * SZ + (1 / (2j)) * SX # x -> x+1 ony = -0.5 * SZ + (1 / (2j)) * SY # y -> y+1 ons = M * SZ N = Nx * Ny def idx(x, y): return (x % Nx) * Ny + y H = np.zeros((2 * N, 2 * N), complex) for x in range(Nx): for y in range(Ny): a = idx(x, y) H[2 * a:2 * a + 2, 2 * a:2 * a + 2] += ons b = idx(x + 1, y) H[2 * a:2 * a + 2, 2 * b:2 * b + 2] += onx H[2 * b:2 * b + 2, 2 * a:2 * a + 2] += onx.conj().T if y < Ny - 1: c = idx(x, y + 1) H[2 * a:2 * a + 2, 2 * c:2 * c + 2] += ony H[2 * c:2 * c + 2, 2 * a:2 * a + 2] += ony.conj().T return H, idx def _edge_branch_slope(M, Ly, which, kxs): """slope of the in-gap branch localised on the chosen boundary, fit near 0.""" E = [] for kx in kxs: w, v = np.linalg.eigh(_strip_H(kx, M, Ly)) cand = [] for i in range(len(w)): site = (np.abs(v[:, i]) ** 2).reshape(Ly, 2).sum(1) loc = site[:5].sum() if which == "top" else site[-5:].sum() if loc > 0.6 and abs(w[i]) < 0.8: cand.append(w[i]) E.append(min(cand, key=abs) if cand else np.nan) E = np.array(E) near = (~np.isnan(E)) & (np.abs(E) < 0.4) p = np.polyfit(kxs[near], E[near], 1) resid = E[near] - np.polyval(p, kxs[near]) r2 = 1 - np.var(resid) / np.var(E[near]) return p[0], r2, int(near.sum()) def _row_corr(Nx, Ny, M, y0): """|| equal-time correlator along row y0 vs x-separation r=1..Nx/2-1.""" H, idx = _real2d(Nx, Ny, M) w, v = np.linalg.eigh(H) occ = v[:, w < -1e-9] G = occ @ occ.conj().T a0 = idx(0, y0) return np.array([np.linalg.norm(G[2 * a0:2 * a0 + 2, 2 * idx(dx, y0):2 * idx(dx, y0) + 2]) for dx in range(1, Nx // 2)]) def run(): reset() print("v444 SEAM.EQUIV.CONTINUUM.EDGE: chiral edge scaling-limit motor " "(the edge half of the continuum residual)") M = 1.0 kxs = np.linspace(-np.pi, np.pi, 161) # ---- 1. chiral linear edge dispersion ---- vt, r2t, nt = _edge_branch_slope(M, 40, "top", kxs) vb, r2b, _ = _edge_branch_slope(M, 40, "bot", kxs) chiral = (vt * vb) < 0 linear = r2t > 0.999 and r2b > 0.999 check("CHIRAL LINEAR EDGE DISPERSION [E]: the in-gap branch is linear near its " "zero-crossing (top v=%.3f R^2=%.4f over %d pts) and the two boundaries " "carry OPPOSITE velocities (top %.3f, bot %.3f, product<0) -- one chiral " "Majorana per edge (relativistic, chiral kinematics)" % (vt, r2t, nt, vt, vb), linear and chiral) # ---- 2/3/5. edge algebraic, bulk exponential, scaling-collapse ---- sizes = [(36, 14), (48, 16), (60, 18)] etas, edge_profiles = [], [] bulk_slope = None for Nx, Ny in sizes: edge = _row_corr(Nx, Ny, M, 0) bulk = _row_corr(Nx, Ny, M, Ny // 2) rs = np.arange(1, Nx // 2) fit = (rs >= 2) & (rs <= Nx // 4) etas.append(-np.polyfit(np.log(rs[fit]), np.log(np.maximum(edge[fit], 1e-15)), 1)[0]) edge_profiles.append(edge[1:7]) # r=2..7, common to all sizes if Nx == 48: bulk_slope = np.polyfit(rs[fit], np.log(np.maximum(bulk[fit], 1e-15)), 1)[0] etas = np.array(etas) eta_mean = etas.mean() algebraic = abs(eta_mean - 1.0) < 0.15 and etas.std() < 0.05 check("ALGEBRAIC EDGE TWO-POINT [E]: the equal-time edge correlator decays as " "r^{-eta} with eta=%.3f ~ 1 (the chiral free-fermion CFT exponent) -- a " "CRITICAL edge" % eta_mean, algebraic) xi_bulk = -1.0 / bulk_slope bulk_exp = bulk_slope < -0.3 and xi_bulk < 3.0 check("EXPONENTIAL BULK [E] (neg control): the bulk-row correlator decays " "EXPONENTIALLY (xi=%.2f sites, slope %.3f) -- only the edge is critical, " "so the power law is the edge CFT, not a geometric artefact" % (xi_bulk, bulk_slope), bulk_exp) # ---- 4. trivial-phase neg control (no edge => exponential edge row) ---- triv = _row_corr(48, 16, 3.0, 0) rs = np.arange(1, 24) fit = (rs >= 2) & (rs <= 12) triv_slope = np.polyfit(rs[fit], np.log(np.maximum(triv[fit], 1e-15)), 1)[0] triv_exp = triv_slope < -0.3 check("TRIVIAL-PHASE NEG CONTROL [E]: M=3 (no edge, v368) has an EXPONENTIAL " "edge-row correlator (slope %.3f, xi=%.2f) -- the algebraic decay is a " "CONSEQUENCE of the topological edge, not the boundary geometry" % (triv_slope, -1.0 / triv_slope), triv_exp) # ---- 5. scaling-collapse of the edge profile across refinements ---- prof = np.array(edge_profiles) # (n_sizes, 6); r=2..7 # successive-refinement spreads must SHRINK (Cauchy convergence to a limit), # not just be small: the coarse pair (36,48) vs the fine pair (48,60). coarse = float(np.max(np.abs(prof[1] - prof[0]) / prof[1])) fine = float(np.max(np.abs(prof[2] - prof[1]) / prof[2])) collapse = fine < 0.02 and fine < coarse and etas.std() < 0.01 check("SCALING-COLLAPSE / CONVERGENCE [E]: eta is N-stable (%.3f+-%.3f) and the " "edge correlator profile is CAUCHY-CONVERGENT under refinement (successive " "spread shrinks: Nx 36->48 %.4f then 48->60 %.4f at r=2..7) -- the lattice " "edge correlators converge to a fixed conformal profile (existing scaling " "limit, the edge analogue of v439's bulk thermodynamic limit)" % (eta_mean, etas.std(), coarse, fine), collapse) # ---- 6. edge central charge (typed [C]) ---- N_Maj = 2 ** (g_car - 1) c_minus = N_Maj * 0.5 check("EDGE CENTRAL CHARGE [C]: eta=1 <=> one chiral Majorana (c=1/2) per edge, " "so the N_Maj=%d-copy carrier has a chiral c_-=%g edge CFT (=(E8)_1, " "consistent with v367/v368/v376; c_-=g_car+N_fam=%d) -- linear chiral " "dispersion + algebraic 1/r correlator + c_-=8 are the conformal data of " "the edge" % (N_Maj, c_minus, g_car + N_fam), N_Maj == 16 and c_minus == 8 and c_minus == g_car + N_fam and algebraic and chiral) # ---- 7. verdict (typed [C]/[O]) ---- grounded = linear and chiral and algebraic and bulk_exp and triv_exp and collapse check("VERDICT [C]/[O]: the EDGE half of the cited MMST scaling limit (v336) is " "now numerically GROUNDED on our explicit collar (chiral linear kinematics " "+ algebraic conformal correlator + scaling-collapse). SEAM.EQUIV.01 stays " "[O]: the rigorous uniform convergence of all edge n-point functions is not " "supplied here. With v439 (bulk) + v444 (edge) BOTH halves of the one " "continuum residual are grounded, pinned to a single cited theorem", grounded and g_car == 5) return summary("v444 SEAM.EQUIV.CONTINUUM.EDGE: the v367/v368 lattice edge carries " "the conformal data of a chiral c=1/2 Majorana CFT (linear chiral " "dispersion v~0.98 with opposite per-edge velocities; algebraic edge " "correlator eta~1 vs exponential bulk; trivial-phase neg control; " "N-stable scaling-collapse), so the 16-copy edge is a chiral c_-=8 " "(E8)_1 CFT. Grounds the EDGE half of the cited MMST scaling limit " "(v336) -- the edge analogue of v439's bulk; SEAM.EQUIV.01 stays [O]") if __name__ == "__main__": raise SystemExit(1 if run() else 0)