"""v991 -- SEAM.DETLINE.UNIFICATION.01 [O update: finite bulk-edge determinant-line shadow [E]; continuum Bismut-Freed stays Open]. THE POINT (promote round 2026-08-28). Re-derives the executable content of detline_bulk_edge_probe (not imported) on the v367 QWZ collar: [E] BULK: occupied-frame FHS Chern of the determinant line over the twist torus is +1 at M = 1 (L = 4 and L = 6); curvature integral = 2 pi * (+1). M = 3 control is 0 both sizes. [E] EDGE: bottom-weighted seam holonomy winding = +1 (phase 2 pi); occupied-only winding = +1; signed spectral flow of the in-gap bottom branch = +1, top = -1. [E] CORRESPONDENCE: 2 pi * C_bulk == holonomy winding (diff machine zero; probe 6e-16). M = 3 control 0 = 0 both sides. [E] CONJUGATION MUTANT (SY -> -SY): both signs flip coherently (C -> -1 and W -> -1). HONEST SCOPE (firewall): 2+1D bulk / 1D edge finite lattice shadow. Continuum Bismut-Freed and the 4D chiral-measure unification stay [O]. No marker move. Python-only / Wolfram mirror deferred. Provenance: experiments/tfpt-discovery/detline_bulk_edge_probe.py (ALL PASS; not imported). """ import numpy as np from tfpt_constants import check, summary, reset SX = np.array([[0, 1], [1, 0]], complex) SY = np.array([[0, -1j], [1j, 0]], complex) SZ = np.array([[1, 0], [0, -1]], complex) TX = (SX / (2j) - SZ / 2) L_SMALL, L_LARGE = 4, 6 N_GRID = 8 NY_STRIP, NK_STRIP = 16, 64 EPS_DET, XI_BOT = 0.02, 1.2 GAP_WINDOW = 0.8 NX_CYL, NY_CYL, NT_CYL = 8, 10, 40 TOL_CHERN = 1e-8 TOL_WIND = 1e-6 def _TY(sy): return (sy * SY / (2j) - SZ / 2) def torus_H(L, M, thx, thy, sy=1.0): n = L * L H = np.zeros((2 * n, 2 * n), complex) TYm = _TY(sy) def blk(x, y): return 2 * ((x % L) + L * (y % L)) for x in range(L): for y in range(L): i = blk(x, y) H[i:i + 2, i:i + 2] += M * SZ for dx, dy, T, th in ((1, 0, TX, thx), (0, 1, TYm, thy)): j = blk(x + dx, y + dy) phase = (np.exp(1j * th) if (x + dx == L or y + dy == L) else 1.0) H[j:j + 2, i:i + 2] += phase * T H[i:i + 2, j:j + 2] += np.conj(phase) * T.conj().T return H def detline_chern(L, M, n_grid=N_GRID, sy=1.0): ths = np.linspace(0.0, 2.0 * np.pi, n_grid, endpoint=False) frames = {} min_gap, n_ref = np.inf, None for i, tx in enumerate(ths): for j, ty in enumerate(ths): w, v = np.linalg.eigh(torus_H(L, M, tx, ty, sy)) n_occ = int(np.sum(w < 0.0)) frames[(i, j)] = v[:, :n_occ] min_gap = min(min_gap, float(w[n_occ] - w[n_occ - 1])) if n_ref is None: n_ref = n_occ assert n_occ == n_ref def link(a, b): u = np.linalg.det(frames[a].conj().T @ frames[b]) return u / abs(u) F = 0.0 for i in range(n_grid): for j in range(n_grid): ip, jp = (i + 1) % n_grid, (j + 1) % n_grid F += np.angle(link((i, j), (ip, j)) * link((ip, j), (ip, jp)) * np.conj(link((i, jp), (ip, jp))) * np.conj(link((i, j), (i, jp)))) C = F / (2.0 * np.pi) return float(C), float(min_gap), float(F) def strip_H(k, M, Ny, sy=1.0): ons = np.sin(k) * SX + (M - np.cos(k)) * SZ hop = -0.5 * SZ + sy * (1.0 / (2j)) * SY H = np.zeros((2 * Ny, 2 * Ny), complex) for y in range(Ny): H[2 * y:2 * y + 2, 2 * y:2 * y + 2] = ons for y in range(Ny - 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 _wind(logdets): im = np.unwrap([z.imag for z in logdets] + [logdets[0].imag]) return float((im[-1] - im[0]) / (2.0 * np.pi)) def seam_detline(M, Ny=NY_STRIP, Nk=NK_STRIP, sy=1.0, eps=EPS_DET, xi=XI_BOT, gap_win=GAP_WINDOW): ks = np.linspace(0.0, 2.0 * np.pi, Nk, endpoint=False) y_site = np.repeat(np.arange(Ny), 2).astype(float) ld_bot, ld_occ, ld_top = [], [], [] E_bot, E_top = [], [] for k in ks: w, v = np.linalg.eigh(strip_H(k, M, Ny, sy)) acc_b = acc_o = acc_t = 0.0j bot_in, top_in = [], [] for n, E in enumerate(w): yexp = float(np.abs(v[:, n]) ** 2 @ y_site) wb = np.exp(-yexp / xi) wt = np.exp(-(Ny - 1.0 - yexp) / xi) z = np.log(E + 1j * eps) acc_b += wb * z acc_t += wt * z if E < 0.0: acc_o += wb * z if abs(E) < gap_win: if yexp < Ny / 2.0: bot_in.append(E) else: top_in.append(E) ld_bot.append(acc_b) ld_occ.append(acc_o) ld_top.append(acc_t) E_bot.append(bot_in[int(np.argmin(np.abs(bot_in)))] if bot_in else np.nan) E_top.append(top_in[int(np.argmin(np.abs(top_in)))] if top_in else np.nan) def signed_flow(energies): sf = 0 seq = list(energies) + [energies[0]] for a, b in zip(seq, seq[1:]): if np.isnan(a) or np.isnan(b): continue if a < 0.0 <= b: sf += 1 if a > 0.0 >= b: sf -= 1 return sf W_seam = _wind(ld_bot) return dict(W_seam=W_seam, W_occ=_wind(ld_occ), W_top=_wind(ld_top), phase=2.0 * np.pi * W_seam, sf_bot=signed_flow(E_bot), sf_top=signed_flow(E_top)) def qwz_cylinder(Nx, Ny, M, theta, sy=1.0): TYm = _TY(sy) dim = 2 * Nx * Ny H = np.zeros((dim, dim), complex) def sl(x, y): return 2 * ((x % Nx) * Ny + y) ph = np.exp(1j * theta) for x in range(Nx): for y in range(Ny): i = sl(x, y) H[i:i + 2, i:i + 2] += M * SZ j = sl(x + 1, y) amp = ph if x == Nx - 1 else 1.0 H[j:j + 2, i:i + 2] += amp * TX H[i:i + 2, j:j + 2] += np.conj(amp) * TX.conj().T if y + 1 < Ny: jy = sl(x, y + 1) H[jy:jy + 2, i:i + 2] += TYm H[i:i + 2, jy:jy + 2] += TYm.conj().T return H def cylinder_edge_flow(M, Nx=NX_CYL, Ny=NY_CYL, Nt=NT_CYL, sy=1.0, gap_win=GAP_WINDOW): ths = 2.0 * np.pi * (np.arange(Nt) + 0.5) / Nt y_site = np.array([y for _x in range(Nx) for y in range(Ny) for _or in (0, 1)], float) flow = {"bottom": 0, "top": 0} prev = {} for th in ths: w, v = np.linalg.eigh(qwz_cylinder(Nx, Ny, M, th, sy)) ingap = {} for i in np.where(np.abs(w) < gap_win)[0]: yexp = float(np.abs(v[:, i]) ** 2 @ y_site) edge = "bottom" if yexp < Ny / 2.0 else "top" if edge not in ingap or abs(w[i]) < abs(ingap[edge]): ingap[edge] = float(w[i]) for edge, E_new in ingap.items(): E_old = prev.get(edge) if E_old is not None and E_old * E_new < 0.0: flow[edge] += 1 if E_new > E_old else -1 prev = ingap return flow def near_int(x, n, tol): return abs(x - n) < tol def run(): reset() print("v991 SEAM.DETLINE.UNIFICATION.01: finite bulk-edge " "determinant-line correspondence") bulk = {} for (M, sy, tag) in ((1.0, 1.0, "TOP"), (3.0, 1.0, "TRIV"), (1.0, -1.0, "CONJ")): bulk[tag] = {} for L in (L_SMALL, L_LARGE): C, gap, F = detline_chern(L, M, sy=sy) bulk[tag][L] = dict(C=C, gap=gap, F=F) print(" %s L=%d: C=%+.10f int F=%+.10f gap=%.4f" % (tag, L, C, F, gap)) C_top4 = bulk["TOP"][L_SMALL]["C"] C_top6 = bulk["TOP"][L_LARGE]["C"] F_top6 = bulk["TOP"][L_LARGE]["F"] check("BULK TOP [E]: det-line Chern = +1 at L = 4 and L = 6 " "(C4=%+.8f C6=%+.8f)" % (C_top4, C_top6), near_int(C_top4, 1, TOL_CHERN) and near_int(C_top6, 1, TOL_CHERN)) check("BULK CURVATURE [E]: integral F = 2 pi * (+1) at L = 6 " "(diff %.1e)" % abs(F_top6 - 2.0 * np.pi), abs(F_top6 - 2.0 * np.pi) < 1e-7) check("BULK CONTROL [E]: M = 3 Chern = 0 on both sizes", all(near_int(bulk["TRIV"][L]["C"], 0, TOL_CHERN) for L in (L_SMALL, L_LARGE))) edge = {} for (M, sy, tag) in ((1.0, 1.0, "TOP"), (3.0, 1.0, "TRIV"), (1.0, -1.0, "CONJ")): r = seam_detline(M, sy=sy) edge[tag] = r print(" EDGE %s: W_seam=%+.8f W_occ=%+.8f W_top=%+.8f " "sf_bot=%+d sf_top=%+d" % (tag, r["W_seam"], r["W_occ"], r["W_top"], r["sf_bot"], r["sf_top"])) et = edge["TOP"] check("EDGE SEAM [E]: bottom-weighted holonomy winding = +1", near_int(et["W_seam"], 1, TOL_WIND)) check("EDGE SLATER [E]: occupied-only winding = +1 and " "sf_bot = +1, sf_top = -1", near_int(et["W_occ"], 1, TOL_WIND) and et["sf_bot"] == 1 and et["sf_top"] == -1) ev = edge["TRIV"] check("EDGE CONTROL [E]: M = 3 W_seam = W_occ = 0 and " "sf_bot = sf_top = 0", near_int(ev["W_seam"], 0, TOL_WIND) and near_int(ev["W_occ"], 0, TOL_WIND) and ev["sf_bot"] == 0 and ev["sf_top"] == 0) cyl = {} for (M, sy, tag) in ((1.0, 1.0, "TOP"), (3.0, 1.0, "TRIV"), (1.0, -1.0, "CONJ")): cyl[tag] = cylinder_edge_flow(M, sy=sy) check("CYLINDER [E]: M = 1 bottom flow +1, top -1; M = 3 none", cyl["TOP"]["bottom"] == 1 and cyl["TOP"]["top"] == -1 and cyl["TRIV"]["bottom"] == 0 and cyl["TRIV"]["top"] == 0) C = C_top6 W = et["W_seam"] diff = C - W print(" C - W = %+.3e" % diff) check("CORRESPONDENCE [E]: C_bulk = W_seam = +1 " "(diff %.1e; probe ~6e-16)" % abs(diff), near_int(C, 1, TOL_CHERN) and near_int(W, 1, TOL_WIND) and et["sf_bot"] == 1 and abs(diff) < 1e-6) check("CORRESPONDENCE CONTROL [E]: M = 3 gives 0 = 0 both sides", near_int(bulk["TRIV"][L_LARGE]["C"], 0, TOL_CHERN) and near_int(ev["W_seam"], 0, TOL_WIND) and ev["sf_bot"] == 0) Cc = bulk["CONJ"][L_LARGE]["C"] Wc = edge["CONJ"]["W_seam"] check("MUTANT [E]: conjugation flips C and W together " "(C=%+.8f W=%+.8f)" % (Cc, Wc), near_int(Cc, -1, TOL_CHERN) and near_int(Wc, -1, TOL_WIND) and edge["CONJ"]["sf_bot"] == -1 and cyl["CONJ"]["bottom"] == -1 and cyl["CONJ"]["top"] == 1 and near_int(Cc + C, 0, 1e-7) and near_int(Wc + W, 0, 1e-6)) check("FIREWALL (scope): 2+1D/1D finite lattice shadow; " "continuum Bismut-Freed and 4D unification stay [O]; " "SEAM.DETLINE.UNIFICATION.01 not closed", True) return summary("v991 detline bulk-edge: finite shadow [E], " "continuum stays [O]") if __name__ == "__main__": raise SystemExit(1 if run() else 0)