#!/usr/bin/env python3 """v645 -- QGEO.KLEINRP.01: parafermionic Klein-twist reflection positivity of the Z3 twist sector -- the precise RP question left open by v639_twist_ope.py S4 (the naive antiunitary mirror pairing is indefinite, N-stable min/max ~ -4.5e-3; the Z2 control is exactly PSD). The named open item of QGEO.TWISTOPE.01 is RESOLVED at the lattice OS level: the positivity-carrying pairing is mirror COMPOSED WITH charge conjugation, with omega^{k q} zero-mode phases and eta = (1, 1) unique in the scanned family. THE CONSTRUCTION (three sentences): 1. On the seam-double chain (complex free fermion, NS sector, half filling -- the lattice of orbifold_twist_ope_probe.py), the twist field sigma_k(x) is the zero-mode-dressed axis string sigma_k(x) = exp(i lam_k (N_[b, b+x) - x/2)), lam_k = 2 pi k/3, supported in the right half of the bond-centered diameter reflection r(j) = 2b - 1 - j (v622 cut geometry, axis bond b; all coordinates even). 2. The bond reflection reverses the string orientation and therefore flips the endpoint winding, and the antilinear conjugation flips it AGAIN -- so the naive antiunitary pairing pairs sigma_k with sigma_k (SAME Z3 charge) and pins a spurious e^{2 i lam} defect on the axis bond; for Z2 this is invisible (e^{-i pi N} = e^{+i pi N} identically), which is exactly why the predecessor's Z2 control was PSD while the Z3 sector failed. 3. The parafermionic Klein-twist pairing composes the mirror with charge conjugation (particle-hole): Theta_K(sigma_k(x)) = eta_k sigma_{3-k}(r(x)) = reflected support with the SAME string angle lam_k plus the zero-mode phase omega^{-k (x/2 mod 3)} (the "omega^{k q}" phases, q = half-position mod 3) -- the glued configuration is then the single smooth flux line through the axis, and the exact particle-hole identity = e^{i u |S|} (I - C = D C D, D = diag((-1)^j), at half filling) makes every dressed Gram entry EXACTLY real. Checks: (K0) MACHINERY / STRUCTURE [exact]: (i) the dressed-string reality identity at machine precision for random multi-arc, mixed-angle configurations; (ii) for lam = pi the naive and Klein pairings COINCIDE identically (the Z2 control of the predecessor could not distinguish them); (iii) axis-defect bookkeeping: the naive glued configuration carries the spurious axis jump 2 lam != 0 mod 2 pi, the Klein glue is defect-free. (K1) KLEIN PSD OF THE CHARGED BLOCKS [numeric]: the sector-diagonal Gram G^(k)_ij = (charge conservation: only the -k+l = 0 mod 3 blocks are nontrivial) is exactly real symmetric and PSD (eigenvalue floor -1e-10 relative) for k = 1, 2, on BOTH diameter axes (through-mark b = 0 and between-mark b = N/8), at N = 48/96/192/384; the two charge blocks coincide exactly (charge conjugation) and the two axes coincide exactly (translation invariance of the seam double -- the mark distinction lives in the covering bookkeeping, not in the free chain). (K2) THE PHASE SCAN [numeric, uniqueness]: over the finite family (eta_1, eta_2) in {+-1, +-omega, +-omega^2}^2, EXACTLY ONE pair satisfies {Klein consistency eta_k eta_{3-k}* = 1, Hermiticity, PSD}: eta = (1, 1) on top of the omega^{k q} zero-mode dressing. Sharpening: (a) NO phase in the family rescues the NAIVE pairing (both Hermitian choices stay indefinite -- the fix is structural charge conjugation, not a phase); (b) on the 6Z sub-grid the bare sigma-dagger pairing is already PSD (the dressing is trivial there), while off the 6Z grid the bare pairing is NOT Hermitian and the omega^{-k (x/2 mod 3)} dressing is exactly what restores reality. (K3) N-STABILITY [numeric]: the K1 minimal eigenvalues at N = 48/96/192/384 stay above the -1e-10 relative floor and do not drift towards 0^-. (K4) MUST-FAIL CONTROLS [numeric]: (a) the naive pairing reproduces the predecessor violation (charged axis strings min/max ~ -4.5e-3 at N = 96/192/384, N-stable; neutral pairs ~ -2.5e-2); (b) the deliberately wrong phases fail: eta = -1 flips the spectrum (PSD lost), eta = omega breaks Hermiticity. (K5) THE MIXED SECTOR / LATTICE OS STATEMENT [numeric]: the FULL Gram over {1, sigma_1(x_i), sigma_2(x_i)} (+ neutral sigma-dagger-sigma dipoles at N = 48/96), with NO sector projection (all cross blocks kept as lattice numbers), is real symmetric and PSD under the Klein pairing; Z2 mixed control likewise. (K6) TYPING [C]: what stands and what is still open for the full CFT-level statement (continuum OS reconstruction, R / fermion-parity sectors, modularity, non-abelian content). Verdict enums (frozen): KLEIN-RP-POSITIVE (K1-K5 pass incl. the mixed OS matrix), KLEIN-RP-SECTOR-ONLY (K1-K4 pass, mixed K5 fails), KLEIN-RP-FAILS (the Klein pairing itself is not PSD), MIXED (controls break). FIREWALL: GATE.QGEO does not move; no marker changes; the continuum OS reconstruction / modularity / non-abelian content stay typed open (K6). Python-only, counted per GATE.WOLFRAM.02. Machinery (cov, string_det) copied verbatim from the v639 probe orbifold_twist_ope_probe.py (conventions h_sigma = 1/36, Delta = 1/9, lam = 2 pi/3 unchanged). PROVENANCE: discovery probe parafermion_klein_rp_probe.py (2026-08-02, 17/17, verdict KLEIN-RP-POSITIVE). """ import numpy as np CHECKS = [] def check(name, ok, detail=""): CHECKS.append((name, bool(ok))) print("[%s] %s%s" % ("PASS" if ok else "FAIL", name, (" -- " + detail) if detail else "")) # ------------------------------------------------------------------ constants LAM = 2.0 * np.pi / 3.0 # Z3 twist string angle LAMK = {1: LAM, 2: 2.0 * LAM, "z2": np.pi} # sector -> string angle OMEGA = np.exp(2j * np.pi / 3.0) ROOTS6 = [np.exp(1j * np.pi * j / 3.0) for j in range(6)] ROOT_LBL = ["+1", "-w2", "+w", "-1", "+w2", "-w"] TOL_MACH = 1e-10 # machine-precision identities TOL_RP = 1e-10 # RP eigenvalue floor (relative) TOL_EQ = 1e-9 # exact-symmetry equalities N_LIST = (48, 96, 192, 384) RNG = np.random.default_rng(61) # ------------------------------------------------------------------ machinery def cov(N): """NS (antiperiodic) half-filled covariance on N sites (verbatim orbifold_twist_ope_probe.py).""" assert N % 4 == 0 m = np.arange(-(N // 4), N // 4) k = 2.0 * np.pi * (m + 0.5) / N j = np.arange(N) V = np.exp(1j * np.outer(k, j)) / np.sqrt(N) return V.conj().T @ V def string_det(C, sites, u): """ = det(I + D C_S), D = diag(e^{iu}-1).""" S = np.asarray(sites, dtype=int) Cs = C[np.ix_(S, S)] M = np.eye(len(S), dtype=complex) \ + (np.exp(1j * np.asarray(u, dtype=float)) - 1.0)[:, None] * Cs return np.linalg.det(M) COVS = {Nv: cov(Nv) for Nv in N_LIST} def xgrid(N): """11 even right-half insertion points, the predecessor grid scaled.""" return [4 * i * N // 96 for i in range(1, 12)] def dip_pairs(N): """Neutral sigma^dag-sigma dipole windows, predecessor PAIRS scaled.""" return [(s * N // 96, (s + l) * N // 96) for s in (0, 4, 8, 16, 24) for l in (4, 8, 16) if s + l <= 48] def ket_ops(el, b, N): """Right-half string realization of a basis element at axis bond b.""" if el[0] == "id": return [] if el[0] == "sig": _, k, x = el return [(np.arange(b, b + x) % N, LAMK[k])] _, k, a1, a2 = el return [(np.arange(b + a1, b + a2) % N, LAMK[k])] def bra_ops(el, b, N, klein): """Reflected (left-half) string: Klein keeps the angle (+lam_k, the charge-conjugated anti-twist), naive flips it to -lam_k.""" if el[0] == "id": return [] if el[0] == "sig": _, k, x = el ang = LAMK[k] if klein else -LAMK[k] return [(np.arange(b - x, b) % N, ang)] _, k, a1, a2 = el ang = LAMK[k] if klein else -LAMK[k] return [(np.arange(b - a2, b - a1) % N, ang)] def gram(N, b, els, klein=True, dress=True): """Mirror Gram M_AB = ; dress = zero-mode phase e^{-i sum(u)/2} (the parafermionic omega^{k q} Klein phases).""" C = COVS[N] n = len(els) M = np.zeros((n, n), dtype=complex) for i, A in enumerate(els): ba = bra_ops(A, b, N, klein) for j, B in enumerate(els): ops = ba + ket_ops(B, b, N) if not ops: M[i, j] = 1.0 continue sites = np.concatenate([s for s, _ in ops]) u = np.concatenate([a * np.ones(len(s)) for s, a in ops]) val = string_det(C, sites, u) M[i, j] = np.exp(-0.5j * u.sum()) * val if dress else val return M def herm_dev(M): return np.abs(M - M.conj().T).max() / np.abs(M).max() def imag_dev(M): return np.abs(M.imag).max() / np.abs(M).max() def eig_ratio(M): """(eigenvalues, lambda_min / max|lambda|). The floor ratio uses max|lambda| (NOT lambda_max) so that negative-definite matrices report -1, not a spurious positive min/max quotient.""" ev = np.linalg.eigvalsh(0.5 * (M + M.conj().T)) return ev, ev.min() / np.abs(ev).max() # ================================================================== K0 print("=" * 72) print("K0: machinery and the structural bookkeeping") print("=" * 72) # K0.1 the dressed-string reality identity (exact PH identity # = e^{iu|S|} at half filling) C96 = COVS[96] dev_re = 0.0 for _ in range(40): cuts = np.sort(RNG.choice(np.arange(0, 96, 2), size=6, replace=False)) ops = [] for (a1, a2) in ((cuts[0], cuts[1]), (cuts[2], cuts[3]), (cuts[4], cuts[5])): if a2 > a1: ops.append((np.arange(a1, a2), RNG.choice([LAM, 2 * LAM, np.pi]))) sites = np.concatenate([s for s, _ in ops]) u = np.concatenate([a * np.ones(len(s)) for s, a in ops]) w = np.exp(-0.5j * u.sum()) * string_det(C96, sites, u) dev_re = max(dev_re, abs(w.imag) / max(1.0, abs(w))) check("K0.1 the zero-mode-dressed string e^{-i sum(u)/2} " "is EXACTLY real for random multi-arc mixed-angle configurations " "(the particle-hole identity I - C = DCD at half filling): reality " "IS the lattice fingerprint of the charge-conjugating Klein pairing", dev_re < TOL_MACH, "max |Im|/|w| = %.3e (40 configs)" % dev_re) # K0.2 Z2 degeneracy of the pairings: e^{-i pi N} = e^{+i pi N} els_z2 = [("sig", "z2", x) for x in xgrid(96)] Mz2_naive = gram(96, 0, els_z2, klein=False, dress=False) Mz2_klein = gram(96, 0, els_z2, klein=True, dress=False) dev_z2 = np.abs(Mz2_naive - Mz2_klein).max() ev_z2, r_z2 = eig_ratio(gram(96, 0, els_z2, klein=True, dress=True)) check("K0.2 for lam = pi the naive and Klein pairings COINCIDE " "identically (e^{-i pi N} = e^{+i pi N}): the predecessor's Z2 " "control validated the machinery but could not distinguish the " "pairings; the dressed Z2 Gram is PSD", dev_z2 < 1e-14 and r_z2 >= -TOL_RP, "max |naive - klein| = %.3e, Z2 min/max = %.3e" % (dev_z2, r_z2)) # K0.3 axis-defect bookkeeping (angle arithmetic of the glued profiles) naive_jump = (LAM - (-LAM)) % (2 * np.pi) # bra -lam -> ket +lam klein_jump = (LAM - LAM) % (2 * np.pi) # bra +lam -> ket +lam z2_jump = (np.pi - (-np.pi)) % (2 * np.pi) # Z2: 2 pi = 0 check("K0.3 axis-defect bookkeeping: the naive glue jumps by 2 lam = " "4 pi/3 != 0 mod 2 pi at the axis bond (spurious charge-2k twist " "pinned ON the mirror -- it pairs sigma_k with sigma_k), the " "Klein glue is defect-free (sigma_k with sigma_{3-k}), and for Z2 " "the naive jump 2 pi is invisible", abs(naive_jump - 4 * np.pi / 3) < 1e-12 and klein_jump < 1e-12 and z2_jump < 1e-12, "naive jump = %.6f, klein jump = %.6f" % (naive_jump, klein_jump)) # ================================================================== K1 print("=" * 72) print("K1: Klein PSD of the charged blocks (both axes, N = 48..384)") print("=" * 72) GK = {} # (N, axis_label, k) -> dressed Klein block K1_ROWS = [] k1_real, k1_psd = True, True for Nv in N_LIST: axes = (("through-mark", 0), ("between-mark", Nv // 8)) for lbl, b in axes: for k in (1, 2): els = [("sig", k, x) for x in xgrid(Nv)] M = gram(Nv, b, els, klein=True, dress=True) GK[(Nv, lbl, k)] = M ev, r = eig_ratio(M) k1_real &= (imag_dev(M) < TOL_MACH and herm_dev(M) < TOL_MACH) k1_psd &= (r >= -TOL_RP) K1_ROWS.append((Nv, lbl, k, ev, r)) print(" N=%4d %-13s k=%d min=%.6e max=%.6e min/max=%.3e" % (Nv, lbl, k, ev.min(), ev.max(), r)) print(" eigenvalues (N=96, through-mark, k=1):") print(" %s" % np.array2string( [row for row in K1_ROWS if row[:3] == (96, "through-mark", 1)][0][3], precision=4, max_line_width=70)) check("K1.1 every Klein block G^(k)_ij = is EXACTLY real symmetric (machine precision) -- " "the omega^{k q} zero-mode dressing removes every phase", k1_real, "max Im/herm dev over %d blocks" % len(K1_ROWS)) check("K1.2 every Klein block is PSD at the -1e-10 relative floor " "(k = 1, 2; both axes; N = 48/96/192/384): the charge-conjugated " "sigma-dagger pairing carries the positivity the naive pairing " "lost", k1_psd, "worst min/max = %.3e" % min(r for *_, r in K1_ROWS)) dev_sector = max(np.abs(GK[(Nv, lbl, 1)] - GK[(Nv, lbl, 2)]).max() / np.abs(GK[(Nv, lbl, 1)]).max() for Nv in N_LIST for lbl in ("through-mark", "between-mark")) check("K1.3 the two charge blocks coincide exactly, G^(1) = G^(2) " "(charge conjugation of the seam double)", dev_sector < TOL_EQ, "max rel dev = %.3e" % dev_sector) dev_axis = max(np.abs(GK[(Nv, "through-mark", k)] - GK[(Nv, "between-mark", k)]).max() / np.abs(GK[(Nv, "through-mark", k)]).max() for Nv in N_LIST for k in (1, 2)) check("K1.4 through-mark and between-mark diameter axes give the SAME " "Gram (translation invariance of the free seam double: the mark " "distinction lives in the covering bookkeeping, not in the " "chain) -- both axes tested, both PSD", dev_axis < TOL_EQ, "max rel dev = %.3e" % dev_axis) # ================================================================== K2 print("=" * 72) print("K2: the phase scan (uniqueness of the Klein phase)") print("=" * 72) G1, G2 = GK[(96, "through-mark", 1)], GK[(96, "through-mark", 2)] winners, cons_rows = [], [] for i1, e1 in enumerate(ROOTS6): for i2, e2 in enumerate(ROOTS6): cons = abs(e1 * np.conj(e2) - 1.0) < 1e-9 h = max(herm_dev(e1 * G1), herm_dev(e2 * G2)) herm = h < TOL_MACH psd = False if herm: r = min(eig_ratio(e1 * G1)[1], eig_ratio(e2 * G2)[1]) psd = r >= -TOL_RP if cons: cons_rows.append((ROOT_LBL[i1], ROOT_LBL[i2], herm, psd)) if cons and herm and psd: winners.append((ROOT_LBL[i1], ROOT_LBL[i2])) print(" consistent pairs (eta_1 eta_2* = 1):") for lbl1, lbl2, herm, psd in cons_rows: print(" (%s, %s): hermitian=%s psd=%s" % (lbl1, lbl2, herm, psd)) check("K2.1 UNIQUENESS: over the 36 pairs (eta_1, eta_2) in " "{+-1, +-w, +-w^2}^2, exactly ONE satisfies {consistency " "eta_k eta_{3-k}* = 1, Hermiticity, PSD}: eta = (+1, +1) on top " "of the omega^{k q} zero-mode dressing -- the Klein phase is " "unique in the scanned family", winners == [("+1", "+1")], "winners = %s" % winners) els_n96 = [("sig", 1, x) for x in xgrid(96)] Mn96 = gram(96, 0, els_n96, klein=False, dress=False) rescue = [] for i1, e1 in enumerate(ROOTS6): if herm_dev(e1 * Mn96) < TOL_MACH: if eig_ratio(e1 * Mn96)[1] >= -TOL_RP: rescue.append(ROOT_LBL[i1]) check("K2.2 [must-fail] NO phase in the family rescues the NAIVE " "pairing (the Hermitian choices +-1 both leave it indefinite): " "the fix is the structural charge conjugation, not a phase", rescue == [], "rescuing phases = %s, naive min/max = %.3e" % (rescue, eig_ratio(Mn96)[1])) x6 = [6 * i for i in range(1, 8)] M6_bare = gram(96, 0, [("sig", 1, x) for x in x6], klein=True, dress=False) ev6, r6 = eig_ratio(M6_bare) Mg_bare = gram(96, 0, els_n96, klein=True, dress=False) hd_bare = herm_dev(Mg_bare) dev_dress = 0.0 for k in (1, 2): for xa in xgrid(96): for xb in xgrid(96): d1 = np.exp(-0.5j * LAMK[k] * (xa + xb)) d2 = OMEGA ** (-k * ((xa // 2 + xb // 2) % 3)) dev_dress = max(dev_dress, abs(d1 - d2)) check("K2.3 the omega^{k q} structure: on the 6Z sub-grid the BARE " "sigma-dagger pairing is already PSD (the dressing is trivial " "there, herm dev %.1e); on the general even grid the bare " "pairing is NOT Hermitian (dev %.3f) and the zero-mode dressing " "e^{-i lam_k (x_a+x_b)/2} = omega^{-k (x_a/2 + x_b/2 mod 3)} " "(parafermionic sector-charge phases) is exactly what restores " "reality (identity dev %.1e)" % (herm_dev(M6_bare), hd_bare, dev_dress), herm_dev(M6_bare) < TOL_MACH and r6 >= -TOL_RP and hd_bare > 0.1 and dev_dress < 1e-12, "6Z min/max = %.3e" % r6) # ================================================================== K3 print("=" * 72) print("K3: N-stability of the Klein positivity") print("=" * 72) ratios_k1 = {Nv: [r for (n, lbl, k, ev, r) in K1_ROWS if n == Nv] for Nv in N_LIST} mins_k1 = {Nv: min(v) for Nv, v in ratios_k1.items()} print(" worst min/max per N: %s" % {Nv: "%.3e" % v for Nv, v in mins_k1.items()}) check("K3.1 the Klein minimal eigenvalues stay above the -1e-10 " "relative floor at every N = 48/96/192/384 (no drift towards " "0^-): the positivity is N-stable, in contrast to the N-stable " "VIOLATION of the naive pairing", min(mins_k1.values()) >= -TOL_RP, "floor across N = %.3e" % min(mins_k1.values())) # ================================================================== K4 print("=" * 72) print("K4: must-fail controls") print("=" * 72) # (a) the naive pairing reproduces the predecessor violation naive_c = {} for Nv in (96, 192, 384): els = [("sig", 1, x) for x in xgrid(Nv)] naive_c[Nv] = eig_ratio(gram(Nv, 0, els, klein=False, dress=False))[1] naive_n = {} for Nv in (96, 192): els = [("dip", 1, a1, a2) for a1, a2 in dip_pairs(Nv)] naive_n[Nv] = eig_ratio(gram(Nv, 0, els, klein=False, dress=False))[1] rc = np.array(sorted(naive_c.values())) stable_naive = (rc.max() < -1e-3 and rc.min() > -1e-2 and (rc.max() - rc.min()) / abs(rc.mean()) < 0.5 and abs(naive_c[96] - (-4.5e-3)) < 2e-3 and min(naive_n.values()) < -1.5e-2) check("K4.1 [must-fail control] the NAIVE pairing still violates, " "reproducing the predecessor: charged axis strings in the " "-4.5e-3 range at N = 96/192/384 (N-stable), neutral pairs " "~ -2.5e-2", stable_naive, "charged min/max: %s; neutral min/max: %s" % ({k: "%.3e" % v for k, v in naive_c.items()}, {k: "%.3e" % v for k, v in naive_n.items()})) # (b) deliberately wrong phases r_neg = eig_ratio(-1.0 * G1)[1] hd_om = herm_dev(OMEGA * G1) check("K4.2 [must-fail control] the deliberately wrong phases fail: " "eta = -1 makes the block negative definite (floor ratio = " "%.3e, PSD lost), eta = omega breaks Hermiticity (dev = %.3f)" % (r_neg, hd_om), r_neg < -0.5 and hd_om > 0.1) # ================================================================== K5 print("=" * 72) print("K5: the mixed sector -- the lattice OS statement") print("=" * 72) k5_rows = [] k5_real, k5_psd = True, True for Nv in (48, 96): els = ([("id",)] + [("sig", 1, x) for x in xgrid(Nv)] + [("sig", 2, x) for x in xgrid(Nv)] + [("dip", 1, a1, a2) for a1, a2 in dip_pairs(Nv)] + [("dip", 2, a1, a2) for a1, a2 in dip_pairs(Nv)]) M = gram(Nv, 0, els, klein=True, dress=True) ev, r = eig_ratio(M) k5_real &= (imag_dev(M) < TOL_MACH and herm_dev(M) < TOL_MACH) k5_psd &= (r >= -TOL_RP) k5_rows.append((Nv, len(els), ev, r)) print(" N=%3d full basis (dim %d: 1 + 2x11 sigma + 2x15 dipoles):" % (Nv, len(els))) print(" min=%.6e max=%.6e min/max=%.3e" % (ev.min(), ev.max(), r)) print(" lowest eigenvalues: %s" % np.array2string(ev[:6], precision=4)) els_192 = ([("id",)] + [("sig", 1, x) for x in xgrid(192)] + [("sig", 2, x) for x in xgrid(192)]) M192 = gram(192, 0, els_192, klein=True, dress=True) ev192, r192 = eig_ratio(M192) k5_real &= (imag_dev(M192) < TOL_MACH and herm_dev(M192) < TOL_MACH) print(" N=192 {1, sigma_1, sigma_2} (dim %d): min=%.6e max=%.6e " "min/max=%.3e" % (len(els_192), ev192.min(), ev192.max(), r192)) els_z2m = ([("id",)] + [("sig", "z2", x) for x in xgrid(96)] + [("dip", "z2", a1, a2) for a1, a2 in dip_pairs(96)]) Mz2m = gram(96, 0, els_z2m, klein=True, dress=True) evz2m, rz2m = eig_ratio(Mz2m) check("K5.1 Z2 mixed control: the full {1, disorder, dipole} Gram " "(dim %d) is PSD" % len(els_z2m), rz2m >= -TOL_RP, "min/max = %.3e" % rz2m) check("K5.2 the FULL mixed Z3 Gram over {1, sigma_1(x), sigma_2(x), " "dipoles} with NO sector projection (all cross blocks kept as " "lattice numbers) is exactly real symmetric AND PSD at N = 48 " "and 96 -- the lattice Osterwalder-Schrader statement of the " "orbifold twist sector under the Klein pairing", k5_real and k5_psd, "min/max: %s" % {n: "%.3e" % r for n, d, e, r in k5_rows}) check("K5.3 the mixed {1, sigma_1, sigma_2} Gram stays PSD at N = 192 " "(stability of the mixed statement)", r192 >= -TOL_RP, "min/max = %.3e" % r192) # ================================================================== K6 print("=" * 72) print("K6: the typing") print("=" * 72) check("K6.1 [C, honest typing] WHAT STANDS: reflection positivity of " "the Z3 twist sector on the seam-double lattice with the " "parafermionic Klein-twist pairing -- mirror composed with " "charge conjugation, sigma_k paired with sigma_{3-k} at the " "reflected point, omega^{k q} zero-mode phases, eta = (1, 1) " "unique in the scanned family -- N-stable 48..384, both diameter " "axes, including the mixed identity+twist+dipole Gram (lattice " "OS level). STILL OPEN for the full CFT statement: the " "continuum OS reconstruction (operator convergence, not just " "Gram entries), the R / fermion-parity sectors (deck^3 = -1 spin " "structure), modular invariance of the full orbifold partition " "function, chiral factorization, and any non-abelian content -- " "named, not claimed. GATE.QGEO does not move", True) # ================================================================== summary print("=" * 72) n_pass = sum(1 for _, ok in CHECKS if ok) print("%d/%d checks passed" % (n_pass, len(CHECKS))) k1_ok = all(ok for n, ok in CHECKS if n.startswith("K1")) k5_ok = all(ok for n, ok in CHECKS if n.startswith("K5")) ctrl_ok = all(ok for n, ok in CHECKS if n.startswith(("K0", "K4"))) if n_pass == len(CHECKS): print("ALL CHECKS PASSED") print("VERDICT: KLEIN-RP-POSITIVE -- the Z3 twist sector is reflection") print("positive on the seam-double lattice under the parafermionic") print("Klein-twist pairing (mirror x charge conjugation, omega^{k q}") print("zero-mode phases, eta = (1,1) unique), N-stable 48..384, both") print("axes, including the mixed identity+twist+dipole OS Gram; the") print("naive pairing keeps its N-stable violation (must-fail).") elif k1_ok and ctrl_ok and not k5_ok: print("SOME CHECKS FAILED") print("VERDICT: KLEIN-RP-SECTOR-ONLY -- charged Klein blocks PSD but") print("the mixed OS Gram fails: the sector projection is load-bearing.") elif not k1_ok: print("SOME CHECKS FAILED") print("VERDICT: KLEIN-RP-FAILS -- the Klein pairing does not carry") print("positivity on the lattice: honest negative.") else: print("SOME CHECKS FAILED") print("VERDICT: MIXED") def run(): """run_all entry point: the checks above already ran at import.""" return len([1 for _, ok in CHECKS if not ok]) if __name__ == "__main__": raise SystemExit(1 if run() else 0)