"""v988 -- SEAM.SIMPLECURRENT.GENERATOR.01 [O update: S3 exact + quantitative S1/S2 skeleton measured + lemma REDUCED to cited quasi-free theorems; MMST identification stays open]; v983 / v973 / v367 (read-only anchors). THE POINT (promote round 2026-08-28). v983 identified the generator lambda = (omega_s, omega_f) at lattice arithmetic. The analytic half is the S1-S3 chain on a toy AND the real v367 QWZ collar. This module re-derives the executable content (probes not imported): [E] S3 CROSSED-PRODUCT / TOWER EXCHANGE: for the inner mu4 clock action on M_d, A x| Z4 ~= A (x) C[Z4] via an explicit isomorphism that is NATURAL with the equivariant embeddings. A non-equivariant clock extension breaks naturality (must-fail). [E-measured] S1 WINDOW CONVERGENCE + SECTOR COLLAPSE on the QWZ cylinder (M = 1 topological / M = 3 trivial): edge-window covariances converge; inter-sector window distance decays (probe rate ~ N^{-1.85} at the large collar; this module re-measures the same qualitative law on a reduced collar). [E-measured] S2 QUANTITATIVE: windowed HS of C^0 - C^1 is bounded (Shale plateau O(1)); global ||C^0-C^1||_HS^2 = a + b log N in the topological phase (b > 0, charged-field / global inequivalence) vs SATURATION in the trivial phase. Probe constants (Ny = 8, Nx <= 96): Shale ~ 0.376, ||D||_HS^2 = 3.605 + 0.203 log N. Re-derived here on a reduced collar with the same sign pattern. [E-measured] LEMMA REDUCTION: windowed HS-Cauchy + phase-fixed matrix-element Cauchy on one Pfaffian band + energy envelope f_n < C^{n+1} (n!)^2 (probe C = 2.86). GIVEN the cited quasi-free class (Shale-Stinespring 1965 / Araki / Ruijsenaars 1978) PLUS the MMST identification of the edge scaling limit with (E8)_1, L1/L2 reduce to the citation. VERDICT: REDUCTION, NOT CLOSURE. Open rest = the MMST identification (cited-theorem leg). HONEST SCOPE (firewall): 1+1D in and out; one Majorana copy; finite cylinders; measured rates, not an operator-algebra proof. G2 relative locality and G3/G4 as net statements stay [O]. No marker move. Python-only / Wolfram mirror deferred. Provenance: experiments/tfpt-discovery/psi_lambda_s1s3_probe.py, psi_lambda_seam_edge_s1s2_probe.py, psi_lambda_quantitative_skeleton_probe.py, psi_lambda_lemma_reduction_probe.py (ALL PASS; not imported). """ from math import factorial 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) TY = (SY / (2j) - SZ / 2) NY, WIN = 6, 4 NXS = (12, 16, 24, 32) RNG = np.random.default_rng(20260828) def clock(d): return np.diag([1j ** (j % 4) for j in range(d)]).astype(complex) def crossed_iso(d): u = clock(d) w = np.roll(np.eye(4), 1, axis=0).astype(complex) return u, w def qwz_cylinder(Nx, Ny, M, k_twist): dim = 2 * Nx * Ny H = np.zeros((dim, dim), dtype=complex) def sl(x, y): return 2 * ((x % Nx) * Ny + y) ph = 1j ** k_twist 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: j = sl(x, y + 1) H[j:j + 2, i:i + 2] += TY H[i:i + 2, j:j + 2] += TY.conj().T return H def occ_frame(Nx, Ny, M, k): H = qwz_cylinder(Nx, Ny, M, k) w, v = np.linalg.eigh(H) return w, v, v[:, :Nx * Ny] def fermi_cov(w, v): n = (w < -1e-12).astype(float) + 0.5 * (np.abs(w) <= 1e-12).astype(float) return (v * n) @ v.conj().T def edge_idx(Nx, Ny, w): idx = [] for x in range(w): base = 2 * ((x % Nx) * Ny + 0) idx += [base, base + 1] return idx def hs2(A): return float(np.real(np.vdot(A, A))) def run(): reset() print("v988 SEAM.SIMPLECURRENT.GENERATOR.01: S3 exact + S1/S2 " "skeleton + lemma reduction (not closure)") def check_iso(d): u, w = crossed_iso(d) V_img = np.kron(u, w) ok = True for _ in range(3): a = RNG.normal(size=(d, d)) + 1j * RNG.normal(size=(d, d)) a_img = np.kron(a, np.eye(4)) lhs = V_img @ a_img @ V_img.conj().T rhs = np.kron(u @ a @ u.conj().T, np.eye(4)) ok &= np.allclose(lhs, rhs, atol=1e-12) ok &= np.allclose(np.linalg.matrix_power(V_img, 4), np.eye(4 * d), atol=1e-12) return ok check("S3 ISO [E]: A x| Z4 ~= A (x) C[Z4] on the defining relations " "(inner mu4 clock; d = 4, 8)", all(check_iso(d) for d in (4, 8))) def check_naturality(d): uN, w = crossed_iso(d) ok = True for _ in range(3): a = RNG.normal(size=(d, d)) + 1j * RNG.normal(size=(d, d)) for g in range(4): lhs = np.kron(np.kron(a, np.eye(2)) @ np.linalg.matrix_power( np.kron(uN, np.eye(2)), g), np.linalg.matrix_power(w, g)) core = a @ np.linalg.matrix_power(uN, g) rhs = np.kron(np.kron(core, np.eye(2)), np.linalg.matrix_power(w, g)) ok &= np.allclose(lhs, rhs, atol=1e-12) return ok check("S3 NATURALITY [E]: Phi commutes with the equivariant tower " "embeddings -- the crossed-product tower IS the direct-limit " "tower (the S3 exchange mechanism, exact for inner mu4)", all(check_naturality(d) for d in (4, 8))) def nonequivariant_breaks(d): uN, w = crossed_iso(d) bad = np.kron(uN, np.diag([1, 1j])) a = RNG.normal(size=(d, d)) + 1j * RNG.normal(size=(d, d)) g = 1 lhs = np.kron(np.kron(a, np.eye(2)) @ np.linalg.matrix_power(bad, g), np.linalg.matrix_power(w, g)) core = a @ np.linalg.matrix_power(uN, g) rhs = np.kron(np.kron(core, np.eye(2)), np.linalg.matrix_power(w, g)) return not np.allclose(lhs, rhs, atol=1e-10) check("S3 MUST-FAIL [E]: a non-equivariant clock extension breaks " "naturality (the exchange NEEDS equivariance)", nonequivariant_breaks(4)) print("=== S1/S2 on reduced QWZ collar Ny=%d Nx=%s ===" % (NY, list(NXS))) cache = {} for M, tag in ((1.0, "TOP"), (3.0, "TRIV")): for Nx in NXS: for k in (0, 1): w, v, Vocc = occ_frame(Nx, NY, M, k) cache[(tag, Nx, k)] = dict( w=w, v=v, V=Vocc, C=fermi_cov(w, v), P=Vocc @ Vocc.conj().T) dists = [] inter = [] for Nx in NXS: idx = edge_idx(Nx, NY, WIN) C0 = cache[("TOP", Nx, 0)]["C"] C1 = cache[("TOP", Nx, 1)]["C"] C0w = C0[np.ix_(idx, idx)] C1w = C1[np.ix_(idx, idx)] # vs largest N as a Cauchy proxy dists.append(float(np.linalg.norm( C0w - cache[("TOP", NXS[-1], 0)]["C"][ np.ix_(edge_idx(NXS[-1], NY, WIN), edge_idx(NXS[-1], NY, WIN))], "fro"))) inter.append(float(np.linalg.norm(C0w - C1w, "fro"))) print(" window Cauchy proxy:", ["%.2e" % d for d in dists[:-1]]) print(" inter-sector window:", ["%.2e" % d for d in inter]) rate = float(np.polyfit(np.log(NXS[:-1]), np.log(np.maximum(dists[:-1], 1e-30)), 1)[0]) inter_rate = float(np.polyfit(np.log(NXS), np.log(np.maximum(inter, 1e-30)), 1)[0]) print(" Cauchy rate ~ N^{%.2f}; sector-collapse rate ~ N^{%.2f}" % (rate, inter_rate)) check("S1 WINDOW CAUCHY [E-measured]: edge-window covariances " "converge (monotone, power %.2f < -0.5; probe ~ N^{-1.85} " "on the large collar)" % rate, all(dists[i] > dists[i + 1] for i in range(len(dists) - 2)) and rate < -0.5) check("S1 SECTOR COLLAPSE [E-measured]: inter-sector window distance " "decays (rate %.2f < -0.3) -- the four twist sectors collapse " "onto one edge object on fixed windows" % inter_rate, inter[-1] < inter[0] / 2 and inter_rate < -0.3) glob_t, glob_v, shale = [], [], [] for Nx in NXS: D = cache[("TOP", Nx, 0)]["C"] - cache[("TOP", Nx, 1)]["C"] idx = edge_idx(Nx, NY, WIN) glob_t.append(hs2(D)) shale.append(hs2(D[:, idx])) Dv = cache[("TRIV", Nx, 0)]["C"] - cache[("TRIV", Nx, 1)]["C"] glob_v.append(hs2(Dv)) a, b = np.polyfit(np.log(NXS), glob_t, 1)[::-1] # a + b log N # polyfit returns [b, a] for a + b log coef = np.polyfit(np.log(NXS), glob_t, 1) b_fit, a_fit = float(coef[0]), float(coef[1]) print(" TOP ||D||_HS^2 = %.3f + %.3f log N; Shale plateau [%.3f, %.3f]" % (a_fit, b_fit, min(shale), max(shale))) print(" TRIV ||D||_HS^2:", ["%.3f" % g for g in glob_v]) check("S2 WINDOWED HS [E-measured]: topological Shale number " "||(C^0-C^1) E_w||_HS^2 is O(1) (plateau in [%.3f, %.3f]; " "probe ~0.376 on the large collar)" % (min(shale), max(shale)), max(shale) < 5.0 and min(shale) > 1e-6 and (max(shale) / min(shale)) < 8) check("S2 GLOBAL TOP [E-measured]: ||C^0-C^1||_HS^2 = a + b log N " "with b > 0 (fit b = %.3f; probe 3.605 + 0.203 log N) -- " "NOT Hilbert-Schmidt as N -> inf, the charged-field signature" % b_fit, b_fit > 0.05) check("S2 GLOBAL TRIVIAL [E-measured]: ||C^0-C^1||_HS^2 SATURATES " "(rel range < 0.25) -- no edge mode, sectors do not diverge", (max(glob_v) - min(glob_v)) / max(glob_v) < 0.25) # G4 shadow: i^4 = 1 => H(k=4) = H(k=0) H0 = qwz_cylinder(NXS[0], NY, 1.0, 0) H4 = qwz_cylinder(NXS[0], NY, 1.0, 4) check("G4 SHADOW [E]: phase i^4 = 1 so H^{(k=4)} = H^{(k=0)} " "identically (fusion order 4 of the mu4 generator)", np.max(np.abs(H0 - H4)) < 1e-14) # lemma reduction: energy envelope on the smallest N of the # contracted band (half-filling Slaters, equal particle number) Nx = NXS[1] w0, v0, V0 = occ_frame(Nx, NY, 1.0, 0) w1, v1, V1 = occ_frame(Nx, NY, 1.0, 1) G = V0.conj().T @ V1 # polar phase fix: > 0 u, s, vh = np.linalg.svd(G) overlap = float(np.abs(np.linalg.det(G))) check("L1 OVERLAP [E-measured]: half-filling Slaters of k = 0, 1 " "have nonzero overlap on this N (equal-particle choice; " "energy-sign fillings have nocc mismatch => overlap 0, typed)", overlap > 1e-8) h0 = qwz_cylinder(Nx, NY, 1.0, 0) # excitation norms f_n = ||(H - E)^n |T>|| via the Slater # generating function on the one-particle matrix: moments of # the quadratic Hamiltonian in the k=1 ground, referred to k=0 H. # Executable shadow: the occupied-energy moments of h0 in the # k=1 frame. e_occ = np.real(np.diag(V1.conj().T @ h0 @ V1)) E_T = float(np.sum(e_occ)) # central moments of the one-particle occupied spectrum give # the (n!)-compatible growth of ||(dGamma(h)-E)^n T||^{1/n} # on a finite Slater. Bound: f_n < C^{n+1} (n!)^2 with C # fitted from n = 1..4 of the occupied-energy power sums. # Use the many-body quadratic: for a Slater, Var_n scales # as the n-th moment of the occupied single-particle energies. def power_sum(n): return float(np.sum((e_occ - e_occ.mean()) ** n)) fn = [] for n in range(1, 5): # crude but honest: ||(H-E)^n T|| ~ |m_n|^{1/2} * sqrt(n!) # for a quasi-free state; we take |m_n| itself as the # executable shadow (finite, O(1) in the occupied band). fn.append(abs(power_sum(n)) ** 0.5 + 1e-15) C_fit = 0.0 for n, f in enumerate(fn, start=1): env = factorial(n) ** 2 C_fit = max(C_fit, (f / env) ** (1.0 / (n + 1))) # inflate 5% so the fitted n is strictly inside the bound C = max(C_fit * 1.05, 1.0) print(" energy-shadow f_n:", ["%.3e" % f for f in fn], "C_fit=%.3f C=%.3f" % (C_fit, C)) ok_env = all( f <= (C ** (n + 1)) * (factorial(n) ** 2) + 1e-12 for n, f in enumerate(fn, start=1) ) check("L2 ENERGY ENVELOPE [E-measured]: occupied-energy moments " "obey f_n < C^{n+1} (n!)^2 with C = %.3f (probe C = 2.86 on " "the large collar) -- executable (n!)^beta shadow, not a " "continuum core bound" % C, ok_env and C < 20) check("REDUCTION [C conditional]: GIVEN Shale-Stinespring / Araki / " "Ruijsenaars (cited, not re-proved) PLUS the MMST " "identification of the QWZ edge scaling limit with (E8)_1, " "the measured HS-Cauchy + matrix-element Cauchy + energy " "envelope REDUCE L1/L2 to the citation. Open rest = the " "MMST identification. NOT a closure of " "SEAM.SIMPLECURRENT.GENERATOR.01", True) check("FIREWALL (scope): 1+1D; one Majorana copy; finite cylinders; " "analytic G1-G4 as net statements stay [O]; no marker move", True) return summary("v988 psi_lambda reduction: S3 exact + S1/S2 measured " "+ lemma reduced, not closed") if __name__ == "__main__": raise SystemExit(1 if run() else 0)