"""v994 -- SEAM.EQUIV.MMST.01 / SEAM.SIMPLECURRENT.GENERATOR.01 [O updates: 5 of 7 MMST identification criteria EXECUTED [E-measured]; C6 = the quasi-free convergence theorem now WRITTEN OUT (cited [C]); C7 modular invariance stays cited/open]. Provenance: experiments/tfpt-discovery/mmst_identification_criteria_probe.py (ALL PASS) + articles/2026-08-28/psi_lambda_convergence_theorem_en.tex. THE POINT (completeness wave, 2026-08-28). Turn the remaining MMST identification QWZ free-fermion edge scaling limit = the (E8)_1 sector theory into an executable sufficient-criterion battery. Cited operator-algebraic inputs: Osborne--Stottmeister, CMP 398 (2023) 219 (arXiv:2107.13834), in the MMST framework (arXiv:2010.11121). OS23 covers free-fermion/Virasoro and bilinear current limits, not the mu4/GSO spin-field extension. [E-measured] C1 one QWZ boundary has c=1/2 per Majorana from the interval entropy coefficient; 16 copies give c=8 (measured c_carrier = 8.005 at L=128). [E-measured] C2 the QWZ edge branch is linear with v~1 (lattice units) and its first four NS levels approach r=1/2,3/2,5/2,7/2 (measured [0.506, 1.516, 2.523, 3.524]); even single-Majorana Fock degeneracies begin 1,0,1,1,2. [E] C3 finite-mode NS/R character sums and the mu4/GSO selector O+S give [1,0,248,0,4124,0,34752] on the q^(1/2) grid three orders deep (flagship). [E] C4 the edge two-point amplitude gives the level-one current normalisation by Wick contraction: k = 1 exactly at 1e-10. [E] C5 four SO(16)_1 sectors recombine to one projected E8 sector; det D8 = 4 -> det E8 = 1 exactly. [C cited] C6 the quasi-free convergence theorem is now WRITTEN OUT in articles/2026-08-28/psi_lambda_convergence_theorem_en.tex (analytic log coefficient 2/pi^2 = 0.202642... vs measured v988 slope 0.203, 0.18%). Not re-proved here. [O cited] C7 modular covariance/invariance of the completed extension stays cited/open. MUST-FAIL: dropping GSO (O alone) misses the E8 vacuum character; N_MAJORANA=8 cannot produce the 248 current coefficient. HONEST SCOPE (firewall): finite QWZ strips and finite character truncations; the written C6 memorandum is a quasi-free / finite-window theorem, not a conformal-net isomorphism. C7 is not executed. Display markers of both contracts stay [O]/[C] as booked -- no silent upgrade. 1+1D in and out. Python-only / Wolfram mirror deferred. """ from __future__ import annotations from math import comb, pi import numpy as np from tfpt_constants import check, summary, reset SX = np.array([[0, 1], [1, 0]], dtype=complex) SY = np.array([[0, -1j], [1j, 0]], dtype=complex) SZ = np.array([[1, 0], [0, -1]], dtype=complex) M_TOP = 1.0 LY = 20 L_ENTROPY = (64, 96, 128) L_SPECTRUM = 128 MAX_T_POWER = 10 N_MAJORANA = 16 TOL = 1e-10 # v988 measured global HS^2 slope on the QWZ collar (cited, not re-fit). V988_HS2_SLOPE = 0.203 def strip_hamiltonian(k: float, mass: float = M_TOP, ly: int = LY) -> np.ndarray: """The v367/v444 QWZ/p+ip strip, periodic momentum k and open y.""" onsite = np.sin(k) * SX + (mass - np.cos(k)) * SZ hop = -0.5 * SZ + SY / (2j) h = np.zeros((2 * ly, 2 * ly), dtype=complex) for y in range(ly): sl = slice(2 * y, 2 * y + 2) h[sl, sl] = onsite for y in range(ly - 1): a = slice(2 * y, 2 * y + 2) b = slice(2 * y + 2, 2 * y + 4) h[a, b] = hop h[b, a] = hop.conj().T return h def top_edge_branch(length: int) -> tuple[np.ndarray, np.ndarray, np.ndarray]: """Extract the most top-localised eigenmode at NS momenta.""" ks = 2 * pi * (np.arange(length) + 0.5) / length - pi energies = np.empty(length) localisations = np.empty(length) for i, k in enumerate(ks): eigenvalues, eigenvectors = np.linalg.eigh(strip_hamiltonian(float(k))) weights = np.abs(eigenvectors.reshape(LY, 2, -1)) ** 2 top_weight = weights.sum(axis=1)[:4].sum(axis=0) selected = int(np.argmax(top_weight)) energies[i] = float(eigenvalues[selected]) localisations[i] = float(top_weight[selected]) return ks, energies, localisations def edge_correlation(ks: np.ndarray, energies: np.ndarray) -> np.ndarray: """Scalar correlation matrix of the occupied extracted edge branch.""" length = len(ks) occupied = energies < 0 x = np.arange(length) fourier = np.exp(1j * np.outer(x, ks)) / np.sqrt(length) frame = fourier[:, occupied] return frame @ frame.conj().T def interval_entropy(correlation: np.ndarray, ell: int) -> float: eigenvalues = np.linalg.eigvalsh(correlation[:ell, :ell]) eigenvalues = np.clip(eigenvalues, 1e-14, 1 - 1e-14) return float(-np.sum( eigenvalues * np.log(eigenvalues) + (1 - eigenvalues) * np.log(1 - eigenvalues) )) def entropy_c_fit(length: int) -> tuple[float, float]: ks, energies, _ = top_edge_branch(length) correlation = edge_correlation(ks, energies) ells = np.arange(6, length // 3) log_chord = np.log((length / pi) * np.sin(pi * ells / length)) # Majorana/BdG factor 1/2 and one-cut factor 1/2. entropies = np.array( [interval_entropy(correlation, int(ell)) / 4 for ell in ells] ) slope, intercept = np.polyfit(log_chord, entropies, 1) predicted = slope * log_chord + intercept residual = entropies - predicted r2 = 1 - np.sum(residual ** 2) / np.sum( (entropies - entropies.mean()) ** 2 ) return float(6 * slope), float(r2) def polynomial_multiply(a: list[int], b: list[int]) -> list[int]: out = [0] * (MAX_T_POWER + 1) for i, ai in enumerate(a): for j, bj in enumerate(b): if i + j <= MAX_T_POWER: out[i + j] += ai * bj return out def fermion_factor(power: int, sign: int, flavors: int) -> list[int]: out = [0] * (MAX_T_POWER + 1) for number in range(flavors + 1): exponent = number * power if exponent <= MAX_T_POWER: out[exponent] = comb(flavors, number) * sign ** number return out def so16_characters(n_majorana: int = N_MAJORANA) -> dict[str, list[int]]: """Finite products on the t=q^(1/2) grid, enough for q^5.""" plus = [1] + [0] * MAX_T_POWER minus = plus.copy() for power in range(1, MAX_T_POWER + 1, 2): plus = polynomial_multiply( plus, fermion_factor(power, +1, n_majorana) ) minus = polynomial_multiply( minus, fermion_factor(power, -1, n_majorana) ) vacuum = [(a + b) // 2 for a, b in zip(plus, minus)] vector = [(a - b) // 2 for a, b in zip(plus, minus)] spinor = [0] * (MAX_T_POWER + 1) spinor[2] = 2 ** (n_majorana // 2 - 1) for power in range(2, MAX_T_POWER + 1, 2): spinor = polynomial_multiply( spinor, fermion_factor(power, +1, n_majorana) ) return { "O": vacuum, "V": vector, "S": spinor, "C": spinor.copy(), } def single_majorana_vacuum_degeneracies(max_level: int = 4) -> list[int]: max_t = 2 * max_level plus = [1] + [0] * max_t minus = plus.copy() def multiply_local(a: list[int], power: int, sign: int) -> list[int]: out = a.copy() for i in range(len(a) - power): out[i + power] += sign * a[i] return out for power in range(1, max_t + 1, 2): plus = multiply_local(plus, power, +1) minus = multiply_local(minus, power, -1) even = [(a + b) // 2 for a, b in zip(plus, minus)] return [even[2 * level] for level in range(max_level + 1)] def bareiss_determinant(matrix: list[list[int]]) -> int: """Fraction-free exact determinant for the small Cartan matrices.""" a = [row.copy() for row in matrix] n = len(a) previous = 1 sign = 1 for k in range(n - 1): if a[k][k] == 0: swap = next(i for i in range(k + 1, n) if a[i][k] != 0) a[k], a[swap] = a[swap], a[k] sign *= -1 pivot = a[k][k] for i in range(k + 1, n): for j in range(k + 1, n): a[i][j] = ( a[i][j] * pivot - a[i][k] * a[k][j] ) // previous previous = pivot return sign * a[-1][-1] def cartan(rank: int, edges: tuple[tuple[int, int], ...]) -> list[list[int]]: matrix = [[2 if i == j else 0 for j in range(rank)] for i in range(rank)] for i, j in edges: matrix[i][j] = matrix[j][i] = -1 return matrix def run(): reset() print("v994 SEAM.EQUIV.MMST.01 / SEAM.SIMPLECURRENT.GENERATOR.01: " "5-of-7 MMST identification criteria (completeness wave)") c_fits = [] r2_fits = [] for length in L_ENTROPY: c_fit, entropy_r2 = entropy_c_fit(length) c_fits.append(c_fit) r2_fits.append(entropy_r2) print(" entropy L=%3d: c_Majorana=%.6f R^2=%.8f" % (length, c_fit, entropy_r2)) c_single = c_fits[-1] c_carrier = N_MAJORANA * c_single check("C1 EXECUTED [E-measured]: one-cut Majorana c~1/2 and 16 copies " "give c~8 (convention S_M,1cut = S_scalar/4)", abs(c_single - 0.5) < 0.015 and abs(c_carrier - 8.0) < 0.25 and max(c_fits) - min(c_fits) < 0.01 and min(r2_fits) > 0.99) print(" measured c_single=%.6f; 16*c=%.6f" % (c_single, c_carrier)) ks, energies, localisations = top_edge_branch(L_SPECTRUM) near = np.abs(ks) < 0.35 velocity, intercept = np.polyfit(ks[near], energies[near], 1) predicted = velocity * ks[near] + intercept velocity_r2 = 1 - np.sum((energies[near] - predicted) ** 2) / np.sum( (energies[near] - energies[near].mean()) ** 2 ) positive = np.where((energies > 0) & (ks > 0) & (ks < 0.4))[0] positive = positive[np.argsort(energies[positive])][:4] scaled_levels = ( energies[positive] * L_SPECTRUM / (2 * pi * abs(velocity)) ) expected_levels = np.arange(4) + 0.5 level_degeneracies = single_majorana_vacuum_degeneracies() check("C2 EXECUTED [E-measured]: velocity ~1, NS levels ~ r+1/2, " "single-Majorana vacuum degeneracies [1,0,1,1,2]", abs(abs(velocity) - 1.0) < 0.03 and velocity_r2 > 0.999 and np.max(np.abs(scaled_levels - expected_levels)) < 0.04 and level_degeneracies == [1, 0, 1, 1, 2] and float(np.min(localisations[near])) > 0.99) print(" v=%.6f R2=%.7f; scaled=%s; deg=%s" % (velocity, velocity_r2, [round(float(value), 6) for value in scaled_levels], level_degeneracies)) characters = so16_characters() e8_projected = [ characters["O"][i] + characters["S"][i] for i in range(MAX_T_POWER + 1) ] e8_flagship = [1, 0, 248, 0, 4124, 0, 34752] e8_expected = e8_flagship + [0, 213126] check("C3 EXECUTED [E]: mu4/GSO-projected lattice character sums " "reproduce the (E8)_1 vacuum character [1,0,248,0,4124,0,34752] " "three orders deep", e8_projected[:7] == e8_flagship and e8_projected[:9] == e8_expected) print(" O(q even): %s; S(q even): %s; O+S: %s" % (characters["O"][::2][:5], characters["S"][::2][:5], e8_projected[:9])) correlation = edge_correlation(ks, energies) odd_separations = np.arange(1, L_SPECTRUM // 8, 2) amplitudes = [] for separation in odd_separations: chord = L_SPECTRUM / pi * np.sin(pi * separation / L_SPECTRUM) amplitudes.append(0.5 * abs(correlation[0, separation]) * chord) majorana_amplitude = float(np.mean(amplitudes)) amplitude_spread = float(np.std(amplitudes)) current_level = (2 * pi * majorana_amplitude) ** 2 check("C4 EXECUTED [E]: current-algebra two-point normalisation " "k=(2 pi A_psi)^2 = 1 (Wick)", abs(majorana_amplitude - 1 / (2 * pi)) < 1e-10 and abs(current_level - 1.0) < 1e-10 and amplitude_spread < 1e-10) print(" A_psi=%.12f; k=%.12f" % (majorana_amplitude, current_level)) d8_edges = tuple((i, i + 1) for i in range(6)) + ((5, 7),) e8_edges = tuple((i, i + 1) for i in range(6)) + ((2, 7),) det_d8 = bareiss_determinant(cartan(8, d8_edges)) det_e8 = bareiss_determinant(cartan(8, e8_edges)) nonzero_sector_heads = { name: next(i for i, value in enumerate(series) if value) for name, series in characters.items() } check("C5 EXECUTED [E]: four SO(16)_1 sectors -> mu-index-one lattice " "shadow (det D8=4 -> det E8=1; selector O+S)", det_d8 == 4 and det_e8 == 1 and set(characters) == {"O", "V", "S", "C"} and nonzero_sector_heads == {"O": 0, "V": 1, "S": 2, "C": 2}) print(" heads(t)=%s; det(D8)=%d -> det(E8)=%d" % (nonzero_sector_heads, det_d8, det_e8)) analytic_slope = 2.0 / (pi ** 2) relative_vs_measured = abs(V988_HS2_SLOPE / analytic_slope - 1.0) check("C6 CITED [C]: quasi-free log coefficient 2/pi^2 matches the " "written theorem and the v988 measured slope 0.203 at 0.18% " "(articles/2026-08-28/psi_lambda_convergence_theorem_en.tex; " "NOT re-proved here)", abs(analytic_slope - 0.202642367) < 1e-9 and relative_vs_measured < 0.002) print(" 2/pi^2=%.12f vs measured %.3f (rel %.4f%%)" % (analytic_slope, V988_HS2_SLOPE, 100 * relative_vs_measured)) check("C7 CITED [O]: modular invariance/covariance of the completed " "mu4/GSO extension remains cited; finite q-coefficients are not " "that theorem", True) check("MUST-FAIL [X]: dropping GSO (O alone) misses the (E8)_1 vacuum " "character -- the 248 lives in O+S", characters["O"][:7] != e8_flagship and characters["O"][2] != 248) eight_copy = so16_characters(8) eight_projected = [ eight_copy["O"][i] + eight_copy["S"][i] for i in range(7) ] check("MUST-FAIL [X]: N_MAJORANA=8 cannot produce the 248 current " "coefficient of (E8)_1", eight_projected != e8_flagship and eight_projected[2] != 248) executed = 5 check("CRITERIA TALLY: 5 of 7 EXECUTED (C1-C5); C6 written-out [C]; " "C7 cited [O] -- verdict MMST_CRITERIA_5_OF_7_EXECUTED", executed == 5) check("FIREWALL (scope): finite QWZ strips + finite character " "truncations; no conformal-net isomorphism; no marker move; " "1+1D in and out; Python-only", True) return summary("v994 MMST criteria battery: 5/7 executed; " "O+S character [1,0,248,0,4124,0,34752]; C6 written-out") if __name__ == "__main__": raise SystemExit(1 if run() else 0)