"""v480 -- QGEO.MULTILOCAL.01: the four mu4 marks make the seam circle a FOUR- INTERVAL free-fermion geometry -- the class whose multi-interval modular theory is EXACTLY KNOWN (Casini-Huerta 2009 mixing term; Longo-Xu multi-interval subfactors; Rehren-Tedesco multilocal fermions; Kawahigashi-Longo-Mueger: holomorphic <=> trivial 2-interval inclusion, mu = 1 -- the SAME one bit as det K = 1). The bedrock invariance omega o rho = omega (QGEO.SYM.01) is exhibited MANIFEST at the 4-interval state level on the exactly solvable ring, with the mu4 clock appearing in its BINARY (double-cover) order-8 form. Mechanism + exhibit; it does NOT close the raw-seam premise (SEAM.EQUIV.01 / QGEO.SYM.01 stay [O]). Why this geometry: the seam circle minus the four Gauss-Bonnet marks (v216) IS the n = 4 disjoint-interval region of a chiral free fermion; the carrier clock rho is the quarter rotation permuting the intervals. Until now the repo used only the single-ball CHM boost (v358); the ESTABLISHED multi-interval modular apparatus is the natural continuum frame for BOTH the bedrock (omega o rho = omega) and the AP2 continuum leg (v476/v478). On the antiperiodic (RP-admissible, no zero mode) critical ring, mu4-symmetrically placed intervals (N = 256, Q = 64, ell = 48): [E-num] 1. BINARY CLOCK: the quarter-turn clock is a signed permutation of the region with rho^4 = -1 EXACTLY -- the fermionic mu4 clock is the order-8 double-cover lift (the same Z2-lift pattern as 2I over I, v346/v465). [E-num] 2. EXACT SECTOR DECOUPLING: the primitive-8th-root character projectors P_k of rho resolve the identity, are idempotent, and commute with the reduced covariance at 1e-15 -- the Rehren-Tedesco multilocal ("diagonal 4-fold cover") isomorphism at covariance level. [E-num] 3. COVER/TWIST IDENTITY: each sector kernel C_k(u,u') = sum_m lambda_k^{-m} c(u'-u+mQ) is the SINGLE-interval kernel on the quarter ring with boundary twist lambda_k; the union of the four sector spectra equals the full 4-interval spectrum to 1e-14. [E-num] 4. CLOCK INVARIANCE MANIFEST: rho C rho^{-1} = C at 1e-14, hence [rho, K] = 0 for the Peschel modular Hamiltonian K = ln((1-C)/C) by functional calculus -- omega o rho = omega HOLDS at the 4-interval state level (the log-amplified numerical residual is 3e-5 relative). [E-num] 5. NEGATIVE CONTROL: displacing ONE interval by one site breaks the clock map and re-couples the character sectors at 1e-2 (vs 1e-15 symmetric) -- the invariance is the mu4-symmetric configuration. [N] 6. CONJUGATE-POINT MIXING: every row of the cross-interval K block peaks on the conjugate diagonal |s'-s| <= 2 at both sizes (ell = 24, 48) -- the lattice shadow of the Casini-Huerta multi-interval mixing term (band ENERGY spreads with size: long-range conformal tails, honestly noted). TYPING. [E-num] finite exact identities (machine precision); [N] the mixing structure; [C] the continuum multi-interval modular theory is CITED (Casini-Huerta J. Stat. Mech. (2009) P12012-class; Longo-Xu CMP 251 (2004); Rehren-Tedesco Lett. Math. Phys. 104 (2014); KLM CMP 219 (2001) mu-index); [O] the raw-seam premise is NOT closed -- this exhibits the mechanism in the exactly solvable class the seam is argued to reach. Numerical (numpy), Python-only by nature. """ import numpy as np from tfpt_constants import check, summary, reset N = 256 Q = N // 4 P0 = 8 ELL = 48 EPS = 1e-12 def make_cf(n): ks = 2 * np.pi * (np.arange(n) + 0.5) / n ks = np.where(ks > np.pi, ks - 2 * np.pi, ks) occ = ks[np.abs(ks) < np.pi / 2] def cf(d): d = np.asarray(d, dtype=float) return (np.exp(1j * occ[:, None] * d.ravel()[None, :]).sum(axis=0) / n ).reshape(d.shape) return cf def region_sites(p0=P0, ell=ELL): return np.concatenate([p0 + j * Q + np.arange(ell) for j in range(4)]) def clock_matrix(sites): n = len(sites) idx = {s: i for i, s in enumerate(sites)} R = np.zeros((n, n)) broken = 0 for i, s in enumerate(sites): t, sign = s + Q, 1.0 if t >= N: t, sign = t - N, -1.0 if t in idx: R[idx[t], i] = sign else: broken += 1 return R, broken def peschel_K(C): w, U = np.linalg.eigh((C + C.conj().T) / 2) w = np.clip(w.real, EPS, 1 - EPS) return (U * np.log((1 - w) / w)) @ U.conj().T def run(): reset() print("v480 QGEO.MULTILOCAL.01: the mu4 marks as the exactly solvable " "4-interval modular geometry") cf = make_cf(N) sites = region_sites() C = cf(sites[None, :] - sites[:, None]) R, broken = clock_matrix(sites) n = 4 * ELL # 1. binary clock rho^4 = -1 R4 = np.linalg.matrix_power(R, 4) check("BINARY CLOCK [E-num]: the quarter-turn clock on the antiperiodic " "(RP-admissible) ring satisfies rho^4 = -1 EXACTLY (residual %.1e) -- the " "fermionic mu4 clock is the BINARY order-8 double-cover lift, the same " "Z2-lift pattern as 2I over the icosahedral group" % np.abs(R4 + np.eye(n)).max(), broken == 0 and np.allclose(R4, -np.eye(n), atol=1e-14)) # character projectors at the primitive 8th roots lams = [np.exp(1j * np.pi * (2 * k + 1) / 4) for k in range(4)] Ps = [sum(lam ** (-m) * np.linalg.matrix_power(R, m) for m in range(4)) / 4 for lam in lams] # 2. exact sector decoupling resolution = np.abs(sum(Ps) - np.eye(n)).max() idem = max(np.abs(P @ P - P).max() for P in Ps) commC = max(np.abs(P @ C - C @ P).max() for P in Ps) check("EXACT SECTOR DECOUPLING [E-num]: the primitive-8th-root character " "projectors resolve the identity (%.1e), are idempotent (%.1e) and commute " "with the reduced covariance (max %.1e) -- the Rehren-Tedesco multilocal / " "diagonal 4-fold-cover isomorphism at covariance level" % (resolution, idem, commC), resolution < 1e-12 and idem < 1e-12 and commC < 1e-12) # 3. cover/twist identity u = np.arange(ELL) d0 = u[None, :] - u[:, None] spec_full = np.sort(np.linalg.eigvalsh((C + C.conj().T) / 2)) herm, spec_sect = 0.0, [] for lam in lams: Ck = sum(lam ** (-m) * cf(d0 + m * Q) for m in range(4)) herm = max(herm, np.abs(Ck - Ck.conj().T).max()) spec_sect.append(np.linalg.eigvalsh((Ck + Ck.conj().T) / 2)) spec_sect = np.sort(np.concatenate(spec_sect)) dspec = np.abs(spec_full - spec_sect).max() check("COVER/TWIST IDENTITY [E-num]: each sector kernel is the SINGLE-interval " "quarter-ring kernel with boundary twist lambda_k; union of sector spectra " "= full 4-interval spectrum (mismatch %.1e; hermitian to %.1e)" % (dspec, herm), dspec < 1e-12 and herm < 1e-12) # 4. clock invariance => modular invariance dC = np.abs(R @ C @ R.T - C).max() K = peschel_K(C) dK_rel = np.abs(R @ K @ R.T - K).max() / np.abs(K).max() check("CLOCK INVARIANCE MANIFEST [E-num]: rho C rho^-1 = C (%.1e) => [rho, K] = 0 " "by functional calculus -- omega o rho = omega HOLDS at the 4-interval " "state level (log-amplified numerical residual %.1e relative)" % (dC, dK_rel), dC < 1e-12 and dK_rel < 1e-3) # 5. negative control: displace one interval by one site sites_bad = np.concatenate([P0 + np.arange(ELL) + 1] + [P0 + j * Q + np.arange(ELL) for j in range(1, 4)]) Cb = cf(sites_bad[None, :] - sites_bad[:, None]) _, broken_b = clock_matrix(sites_bad) def max_cross_coupling(Cmat): worst = 0.0 for k, lam in enumerate(lams): W = np.zeros((ELL, 4 * ELL), dtype=complex) for j in range(4): W[:, j * ELL:(j + 1) * ELL] = np.eye(ELL) * lam ** (-j) / 2 for kk, lam2 in enumerate(lams): if kk == k: continue W2 = np.zeros((ELL, 4 * ELL), dtype=complex) for j in range(4): W2[:, j * ELL:(j + 1) * ELL] = np.eye(ELL) * lam2 ** (-j) / 2 worst = max(worst, np.abs(W @ Cmat @ W2.conj().T).max()) return worst couple_bad = max_cross_coupling(Cb) couple_sym = max_cross_coupling(C) check("NEGATIVE CONTROL [E-num]: displacing ONE interval by one site breaks the " "clock map (%d sites) and re-couples the character sectors at %.1e (vs " "%.1e symmetric) -- the invariance is the mu4-symmetric configuration, not " "an artifact" % (broken_b, couple_bad, couple_sym), broken_b > 0 and couple_bad > 1e-3 and couple_sym < 1e-12) # 6. conjugate-point mixing structure def band_stats(ell_val, band=2): sts = region_sites(P0, ell_val) Kv = peschel_K(cf(sts[None, :] - sts[:, None])) B = Kv[0:ell_val, ell_val:2 * ell_val] peaked = float(np.mean([abs(int(np.argmax(np.abs(B[s]))) - s) <= band for s in range(ell_val)])) tot = float(np.sum(np.abs(B) ** 2)) near = float(sum(np.abs(B[s, t]) ** 2 for s in range(ell_val) for t in range(max(0, s - band), min(ell_val, s + band + 1)))) return peaked, near / tot p24, f24 = band_stats(24) p48, f48 = band_stats(48) check("CONJUGATE-POINT MIXING [N]: every row of the cross-interval K block peaks " "on the conjugate diagonal |s'-s| <= 2 at both sizes (peaked %.2f / %.2f " "for ell = 24 / 48) -- the lattice shadow of the Casini-Huerta mixing " "term; band energy %.2f -> %.2f (long-range conformal tails, honest)" % (p24, p48, f24, f48), p24 >= 0.9 and p48 >= 0.9) check("SCOPE [C]/[O]: the continuum multi-interval modular theory is CITED " "(Casini-Huerta mixing term; Longo-Xu multi-interval subfactors; " "Rehren-Tedesco multilocal fermions; KLM mu-index -- holomorphic <=> " "trivial multi-interval inclusion, the SAME bit as det K = 1); the " "raw-seam premise (QGEO.SYM.01 / SEAM.EQUIV.01) stays OPEN -- mechanism " "exhibited, not closed", True) return summary("v480 QGEO.MULTILOCAL.01: the four mu4 marks = the exactly " "solvable 4-interval free-fermion modular geometry -- binary clock " "rho^4 = -1 [E], exact 8th-root sector decoupling 1e-15 [E], " "cover/twist spectra 1e-14 [E], omega o rho = omega manifest [E] + " "configuration-forced (negative control 1e-2) [E], Casini-Huerta " "conjugate-point mixing shadow [N]; continuum theory cited [C]; " "raw-seam premise stays [O]") if __name__ == "__main__": raise SystemExit(1 if run() else 0)