#!/usr/bin/env python3 """v650 -- QGEO.ORBMOD.01: MODULAR DATA of the Z3 orbifold of the seam on the lattice -- the nine twisted-sector partition functions Z[g,h] of the seam double (free complex fermion) on the N x L cylinder, their modular S-covariance (MEASURED, with a convergence rate in N), the twisted character exponents against h_sigma = 1/36, and the deck^3 = -1 spin-structure bookkeeping (v623) as exact Z6 phase arithmetic. CONVENTIONS (declared up front; everything below hangs on these): * SEAM DOUBLE: one complex free fermion = the doubled Majorana seam (v622 doubled bookkeeping; a single Majorana cannot be omega-twisted -- orbifold_twist_ope_probe.py T0). Lattice: N-site hopping chain, dispersion eps(k) = -cos k, half filling, Fermi velocity v = 1. NS BASIS: the untwisted spatial sector is ANTIPERIODIC, momenta k = 2 pi (m + 1/2)/N. * TWIST SECTORS: the spatial Z3 twist g is the phase boundary condition psi(x+N) = -omega^g psi(x) (NS times omega^g), i.e. momenta k = 2 pi (m + 1/2 + g/3)/N. The temporal insertion h is the CHARGE-SYMMETRIC Z3 phase U_h = exp(i lam_h (Nhat - N/2)), lam_h = 2 pi h/3 (the zero-mode dressing of parafermion_klein_rp_probe.py; N even makes U_{h+3} consistent). Euclidean time is CONTINUOUS (Hamiltonian transfer matrix, H = sum_k eps_k (n_k - 1/2)), time length L, tau = i L/N (v = 1). Exact free-fermion product: Z[g,h] = prod_k 2 cosh((L eps_k - i lam_h)/2), plain trace = ANTIPERIODIC time; the (-1)^F insertion is lam -> lam + pi. Z[g,h] is exactly REAL AND POSITIVE (k -> -k reflection + particle-hole at half filling; machine-checked). * EXACT LATTICE DEGENERACIES (checked): Z[g,h] = Z[-g,h] = Z[g,-h] (charge conjugation of the double): only the four sectors (g,h) in {0,1}^2 are independent -- exactly the continuum conjugate-sector degeneracy of the Z3 orbifold. * CONTINUUM TARGET: Z_cft[g,h](tau) = |theta[g/3, h/3](0|tau) / eta(tau)|^2 (theta with characteristics; theta[0,0] = theta_3 = the known NS-NS Dirac answer), S: Z[g,h](-1/tau) = Z[h,-g](tau). Lattice Casimir coefficient in pi/N units: C_g = 8 (h_nu - c/24)|_{c=1/2} = 4 x the v628 per-class datum: C = (-1/6, 1/18, 1/18) for g = (0, 1, 2), i.e. C_g/4 = (-1/24, 1/72, 1/72) -- the seam double carries FOUR real chiral deck families (complex doubling x two chiralities), each with the v628 coefficient (6 nu^2 - 6 nu + 1)/12 = 2(h_nu - c/24). * MODULAR MAPS ON THE LATTICE: S is realized by exchanging the N <-> L roles (rotated lattice) with (g,h) -> (h,-g); it is NOT an exact lattice symmetry (space is a discrete cos-band, time is continuous) -- the S-deviation vs N IS the measurement (M2). T is realized EXACTLY: the discrete Dehn twist = translation by N sites, which acts on the g-sector as (-omega^g)^{Nhat}, i.e. (g,h) -> (g, g+h) PLUS a fermion-parity flip -- the Z6 spin-structure shift, an exact operator identity (M2.4); the continuum T-PHASES per character are NOT resolvable in the real-positive trace convention and stay OPEN (M5), so only S is MEASURED. Checks: (C0) CONTINUUM REFERENCE INTEGRITY [machine]: the product form of Z_cft equals the theta_3-series/eta form at (0,0) (the known NS partition function) and |theta[g/3,h/3]/eta|^2 for all nine sectors, and the reference itself is exactly S-covariant (theta-transform, numerically < 1e-9). (M1) THE NINE Z[g,h] [E-float]: exact reality/positivity, half filling in every twist sector, the exact lattice degeneracies Z[g,h] = Z[-g,h] = Z[g,-h], Z3 periodicity of the insertion (N even); the 9 x 3 table (tau = i/2, i, 2i; N = 48) matches Z_cft at < 1 % on ln Z after removing the exact bulk N L/pi; the L -> infinity Casimir slope fit reproduces C_g/4 = (-1/24, 1/72, 1/72) per g-sector at < 1 % (v628). (M2) MODULAR S-COVARIANCE, MEASURED [central]: Z[g,h] at tau vs the rotated lattice Z[h,-g] (N <-> L), relative deviation as a function of N (N = 24/48/96): tau = i (the self-dual point, one dynamical relation Z[1,0] vs Z[0,1]) and tau = i/2 <-> 2i (all nine, four independent). PASS = every dynamical deviation decreases monotonically in N with a stable rate; the rate is REPORTED. MEASURED RESULT: the S-deviation falls ~ N^{-4}, FASTER than the naive N^{-2} dispersion artifact -- the single-lattice deviation from the continuum reference falls ~ N^{-2} (measured alongside), so the leading N^{-2} lattice artifact is itself S-COVARIANT and cancels in the S-relation; only the N^{-4} tail survives. Both rates are measured. T [E]: the exact Dehn identity Z_T[g,h] = Z^F[g, g+h] at machine precision (the Z6 shift; phases beyond it: open). (M3) q-EXPANSION / CHARACTER EXPONENTS [E-float, N = 48]: the leading exponent of Z[1,0]/Z[0,0] gives the twisted ground state Delta_sigma = 1/9 = 4 h_sigma, i.e. h_sigma = 1/36 per deck class (v639 dictionary) at < 1 %; the next gap in the twisted sector is the parafermionic moding 1/6 (= 1/2 - 1/3, the {1/6, 5/6} twisted pair) with multiplicity 2 at < 1 %/2 %; the omega^{k q} charge bookkeeping in TIME: the level-1/6 weight in Z[1,1]/Z[1,0] is 2 cos(2 pi/3) - 2 = -3 (< 2 %). (M4) SPIN-STRUCTURE BOOKKEEPING [E, machine]: on the 48-site cover (deck = translation by 16, v623) the deck insertion phases are the Z6 values e^{i pi (2r+1)/3} per deck class (r = m mod 3), deck^3 = (-1)^F and deck^6 = 1 as EXACT partition-function identities; the cover trace FACTORIZES exactly into the three base-sector (N = 16) traces with per-class insertion (-1)^h omega^{g~ h} -- the Z3 twist lifts to a Z6 structure on the double, exactly as deck^3 = -1 demands; parity projections Z^{+-} = (Z +- Z^F)/2 are nonnegative state counters in the h = 0 column, every sector ground state is parity-EVEN (charge-0), the odd towers open at gap (1/2, 1/6, 1/6) for g = (0, 1, 2) (< 1 %), and the POSITIVE bookkeeping of the full Z6 insertion family is the Z6 CHARGE PROJECTION P_c[g] = (1/6) sum_j e^{-i pi j c/3} Tr[e^{i pi j (Nhat-N/2)/3} e^{-LH}] (projector on (Nhat - N/2) mod 6): all six P_c >= 0 per g-sector, summing to Z[g,0], ground charge class c = 0. HONEST NOTE: for h != 0 the naive ratio Z^F[g,h]/Z[g,h] EXCEEDS 1 (measured up to 1.93) -- Z^{+-}[g, h!=0] are omega-weighted parity traces, not counters; the Z6 projection is the counter. (M5) TYPING [C]: what stands on the lattice (S measured with rate, characters consistent, Z6/spin bookkeeping exact) and what is missing (exact continuum eta/theta identification of chiral PHASES, complete T-phases per character, the non-abelian sector assembly / orbifold projection sum, continuum OS) -- named, not claimed. Verdict enums (frozen): ORB-MODULAR-COHERENT (all pass: S measured with a clean rate, characters and Casimir consistent, Z6 bookkeeping exact), ORB-MODULAR-S-OPEN (M1/M3/M4 pass but the S-deviation has no clean monotone rate), ORB-MODULAR-FAILS (reference or M1 control level fails), MIXED (anything else). FIREWALL: GATE.QGEO does not move; no marker changes; the chiral phases / continuum T-phases / non-abelian assembly / continuum OS stay typed open (M5). Python-only, counted per GATE.WOLFRAM.02. Conventions inherited read-only from v639_twist_ope.py, v645_klein_rp.py, v622/v623/v628. PROVENANCE: discovery probe orbifold_modular_probe.py (2026-08-02, 17/17, verdict ORB-MODULAR-COHERENT). """ 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 OMEGA = np.exp(2j * np.pi / 3.0) N_MAIN = 48 # the covered-seam scale (v623) N_BASE = 16 # the base seam (v622) NS_M2 = (24, 48, 96) # S-covariance N series TAUS = ((0.5, 24), (1.0, 48), (2.0, 96)) # (beta, L) at N = 48 C_TARGET = {0: -1.0 / 6.0, 1: 1.0 / 18.0, 2: 1.0 / 18.0} V628 = {0: -1.0 / 24.0, 1: 1.0 / 72.0, 2: 1.0 / 72.0} H_SIGMA = 1.0 / 36.0 # v639 per-deck-class twist weight DELTA_SIGMA = 4.0 * H_SIGMA # lattice twist dimension 1/9 GAP2 = 1.0 / 6.0 # parafermionic next gap (1/2 - 1/3) TOL_MACH = 1e-9 # machine-precision identities (on lnZ) TOL_FIT = 0.01 # 1 % on exponents / Casimir / table TOL_MULT = 0.02 # 2 % on multiplicities / weights MAX_IMAG = [0.0] # running reality-deviation tracker # ------------------------------------------------------------------ machinery def modes(N, g): """g-twisted NS momenta k = 2 pi (m + 1/2 + g/3)/N.""" m = np.arange(N) return 2.0 * np.pi * (m + 0.5 + (g % 3) / 3.0) / N def eps_of(N, g): return -np.cos(modes(N, g)) def ln2cosh(z): """Stable elementwise log(2 cosh z) for complex z.""" zs = np.where(z.real >= 0, z, -z) return zs + np.log(1.0 + np.exp(-2.0 * zs)) def lnZ_lam(N, L, g, lam): """Complex ln Z for insertion exp(i lam (Nhat - N/2)): sum_k log 2cosh((L eps_k - i lam)/2).""" z = 0.5 * (L * eps_of(N, g) - 1j * lam) return ln2cosh(z).sum() def lnZ(N, L, g, h, parity=False): """Real ln Z[g,h] (optionally with (-1)^F); returns (lnZ, sign) and tracks the reality deviation |e^{i Im} -+ 1|.""" lam = 2.0 * np.pi * (h % 3) / 3.0 + (np.pi if parity else 0.0) val = lnZ_lam(N, L, g, lam) ph = np.exp(1j * val.imag) dev_p, dev_m = abs(ph - 1.0), abs(ph + 1.0) MAX_IMAG[0] = max(MAX_IMAG[0], min(dev_p, dev_m)) return val.real, (1.0 if dev_p <= dev_m else -1.0) def lnZr(N, L, g, h): """Plain real ln Z[g,h] (must be positive; sign asserted).""" v, s = lnZ(N, L, g, h) assert s > 0 return v def lnZraw(N, L, g, theta): """Complex ln of Tr[prod_k e^{i theta_k n_k} e^{-L H}] with RAW per-mode phases theta_k (no zero-mode dressing): prod_k (e^{L eps_k/2} + e^{i theta_k} e^{-L eps_k/2}).""" a = 0.5 * L * eps_of(N, g) th = np.broadcast_to(np.asarray(theta, dtype=float), (N,)) out = np.empty(N, dtype=complex) pos = a >= 0 out[pos] = a[pos] + np.log(1.0 + np.exp(1j * th[pos] - 2.0 * a[pos])) out[~pos] = (-a[~pos] + 1j * th[~pos] + np.log(1.0 + np.exp(-1j * th[~pos] + 2.0 * a[~pos]))) return out.sum() def E0_exact(N, g): """Exact twisted-sector ground-state energy (mode sum).""" eps = eps_of(N, g) assert (eps < 0).sum() == N // 2 # half filling in every sector return eps[eps < 0].sum() def lnZcft(g, h, beta, M=2000): """Continuum ln Z_cft[g,h](tau = i beta): q^{C_g/2} times the four real chiral deck families, q = e^{-2 pi beta}.""" lnq = -2.0 * np.pi * beta lam = 2.0 * np.pi * (h % 3) / 3.0 a0 = (0.5 + (g % 3) / 3.0) % 1.0 m = np.arange(M) tot = 0.5 * C_TARGET[g % 3] * lnq for a in (m + a0, m + 1.0 - a0): qa = np.exp(lnq * a) tot += np.sum(np.log(1.0 + 2.0 * np.cos(lam) * qa + qa * qa)) return tot def theta_ab(a, b, beta, M=80): """theta[a,b](0 | i beta) = sum_n e^{-pi beta (n+a)^2 + 2 pi i (n+a) b}.""" n = np.arange(-M, M + 1) return np.sum(np.exp(-np.pi * beta * (n + a) ** 2 + 2j * np.pi * (n + a) * b)) def ln_eta(beta, M=4000): n = np.arange(1, M) return -np.pi * beta / 12.0 + np.sum(np.log1p(-np.exp(-2.0 * np.pi * beta * n))) # ================================================================== C0 print("=" * 72) print("C0: continuum reference integrity (theta/eta bookkeeping)") print("=" * 72) # C0.1 the reference is exactly S-covariant: Z[g,h](i/beta) = Z[h,-g](i beta) dev_s = 0.0 for beta in (0.5, 1.0, 2.0): for g in range(3): for h in range(3): dev_s = max(dev_s, abs(lnZcft(g, h, 1.0 / beta) - lnZcft(h, (-g) % 3, beta))) check("C0.1 the continuum reference Z_cft[g,h] = |theta[g/3,h/3]/eta|^2 " "is EXACTLY S-covariant, Z[g,h](-1/tau) = Z[h,-g](tau) (all nine " "sectors, beta = 1/2, 1, 2)", dev_s < TOL_MACH, "max |dev lnZ| = %.3e" % dev_s) # C0.2 (0,0) = the known NS-NS Dirac partition function |theta_3/eta|^2 dev_ns = max(abs(lnZcft(0, 0, beta) - 2.0 * (np.log(theta_ab(0.0, 0.0, beta).real) - ln_eta(beta))) for beta in (0.5, 1.0, 2.0)) check("C0.2 Z_cft[0,0] equals the KNOWN NS partition function " "|theta_3/eta|^2 (theta_3 as its own lattice-independent series)", dev_ns < TOL_MACH, "max dev = %.3e" % dev_ns) # C0.3 all nine sectors are |theta[g/3, h/3]/eta|^2 dev_th = 0.0 for beta in (0.5, 1.0): for g in range(3): for h in range(3): t = theta_ab(g / 3.0, h / 3.0, beta) dev_th = max(dev_th, abs(lnZcft(g, h, beta) - 2.0 * (np.log(abs(t)) - ln_eta(beta)))) check("C0.3 all nine reference sectors are theta quotients: " "Z_cft[g,h] = |theta[g/3, h/3](0|tau)/eta(tau)|^2 exactly", dev_th < TOL_MACH, "max dev = %.3e" % dev_th) # ================================================================== M1 print("=" * 72) print("M1: the nine Z[g,h] on the lattice (N = 48; tau = i/2, i, 2i)") print("=" * 72) # M1.1 exact structure: reality, degeneracies, Z3 periodicity dev_deg = 0.0 for (beta, L) in TAUS: for g in range(3): for h in range(3): z = lnZr(N_MAIN, L, g, h) dev_deg = max(dev_deg, abs(z - lnZr(N_MAIN, L, (-g) % 3, h)), abs(z - lnZr(N_MAIN, L, g, (-h) % 3))) # Z3 periodicity of the RAW (unreduced) insertion lam -> lam + 2 pi (N even) per = lnZ_lam(N_MAIN, 48, 1, 2.0 * np.pi / 3.0) per3 = lnZ_lam(N_MAIN, 48, 1, 2.0 * np.pi / 3.0 + 2.0 * np.pi) dev_per = (abs(per.real - per3.real) + abs(np.exp(1j * (per.imag - per3.imag)) - 1.0)) check("M1.1 exact lattice structure: Z[g,h] is REAL POSITIVE (max " "reality dev %.1e), every twist sector is half filled, the " "charge-conjugation degeneracies Z[g,h] = Z[-g,h] = Z[g,-h] hold " "at machine precision, and the insertion is Z3-periodic " "(U_{h+3} = U_h, N even)" % MAX_IMAG[0], MAX_IMAG[0] < TOL_MACH and dev_deg < TOL_MACH and dev_per < TOL_MACH, "max degeneracy dev = %.3e, periodicity dev = %.3e" % (dev_deg, dev_per)) # M1.2 the 9 x 3 table vs the continuum reference print(" lnZhat = lnZ - N L/pi vs continuum (N = 48):") print(" %-6s %-6s %14s %14s %11s" % ("tau", "(g,h)", "lattice", "continuum", "|dev|")) dev_tab = 0.0 for (beta, L) in TAUS: for g in range(3): for h in range(3): zl = lnZr(N_MAIN, L, g, h) - N_MAIN * L / np.pi zc = lnZcft(g, h, beta) d = abs(zl - zc) dev_tab = max(dev_tab, d) if (g, h) in ((0, 0), (0, 1), (1, 0), (1, 1)): print(" %-6s (%d,%d) %14.8f %14.8f %11.3e" % ("i" if beta == 1 else ("i/2" if beta == 0.5 else "2i"), g, h, zl, zc, d)) check("M1.2 the full 9 x 3 lattice table (tau = i/2, i, 2i at N = 48) " "matches the continuum |theta[g/3,h/3]/eta|^2 at < 1 %% on ln Z " "after removing the exact bulk N L/pi (four independent sectors " "shown; the other five are exact degenerates)" % (), dev_tab < TOL_FIT, "max |dev lnZ| = %.3e" % dev_tab) # M1.3 Casimir limit L -> infinity: slope fit vs v628 per-class data # (L = 768/960: the twisted-sector e^{-delta L} tail is < 1e-9) print(" Casimir coefficients (pi/N units), N = 48:") cas_ok, fit_vs_exact = True, 0.0 for g in range(3): lz1, lz2 = lnZr(N_MAIN, 768, g, 0), lnZr(N_MAIN, 960, g, 0) E0_fit = -(lz2 - lz1) / 192.0 E0_ex = E0_exact(N_MAIN, g) fit_vs_exact = max(fit_vs_exact, abs(E0_fit - E0_ex)) C_fit = (E0_fit + N_MAIN / np.pi) * N_MAIN / np.pi rel = abs(C_fit / 4.0 - V628[g]) / abs(V628[g]) cas_ok &= rel < TOL_FIT print(" g=%d E0_fit=%.9f E0_exact=%.9f C=%.6f C/4=%.6f " "vs v628 %.6f (rel dev %.2e)" % (g, E0_fit, E0_ex, C_fit, C_fit / 4.0, V628[g], rel)) check("M1.3 the L -> infinity Casimir slope of ln Z[g,0] reproduces the " "v628 coefficients per deck class: C_g/4 = (-1/24, 1/72, 1/72) " "for g = (0,1,2) at < 1 %% (dictionary C_g = 4 x v628: complex " "doubling x two chiralities; slope fit vs exact mode sum " "agrees to %.1e)" % fit_vs_exact, cas_ok and fit_vs_exact < 1e-6) # ================================================================== M2 print("=" * 72) print("M2: modular S-covariance, MEASURED (N = 24/48/96)") print("=" * 72) # tau = i, the self-dual point: the single dynamical relation dev_i = {N: abs(lnZr(N, N, 1, 0) - lnZr(N, N, 0, 2)) for N in NS_M2} rates_i = [np.log(dev_i[24] / dev_i[48]) / np.log(2.0), np.log(dev_i[48] / dev_i[96]) / np.log(2.0)] print(" tau = i (self-dual): |lnZ[1,0] - lnZ[0,-1]| per N:") print(" " + " ".join("N=%d: %.3e" % (N, dev_i[N]) for N in NS_M2) + " rates: %.3f, %.3f" % tuple(rates_i)) check("M2.1 S at the self-dual point tau = i: the dynamical relation " "Z[1,0] = Z[0,-1] (space twist <-> time twist) converges " "monotonically in N with a stable rate (the other seven " "relations at tau = i are EXACT lattice degeneracies)", dev_i[24] > dev_i[48] > dev_i[96] and abs(rates_i[0] - rates_i[1]) < 0.5, "rates %.3f / %.3f" % tuple(rates_i)) # tau = i/2 <-> 2i: all nine (four independent) INDEP = ((0, 0), (0, 1), (1, 0), (1, 1)) print(" tau = i/2 <-> 2i: |lnZ^{(N,N/2)}[g,h] - lnZ^{(N/2,N)}[h,-g]|:") print(" %-6s %12s %12s %12s %9s %9s" % ("(g,h)", "N=24", "N=48", "N=96", "rate1", "rate2")) dev_half, rates_half = {}, {} for g in range(3): for h in range(3): ds = [abs(lnZr(N, N // 2, g, h) - lnZr(N // 2, N, h, (-g) % 3)) for N in NS_M2] dev_half[(g, h)] = ds r = [np.log(ds[0] / ds[1]) / np.log(2.0), np.log(ds[1] / ds[2]) / np.log(2.0)] rates_half[(g, h)] = r tag = " *" if (g, h) in INDEP else " " print(" (%d,%d)%s %12.4e %12.4e %12.4e %9.3f %9.3f" % (g, h, tag, ds[0], ds[1], ds[2], r[0], r[1])) mono = all(dev_half[k][0] > dev_half[k][1] > dev_half[k][2] for k in INDEP) check("M2.2 S at tau = i/2 <-> 2i (rotated lattice, N <-> L): all nine " "(g,h) deviations DECREASE monotonically over N = 24/48/96 " "(four independent relations, marked *; the rest are exact " "degenerates of these)", mono, "worst N=96 deviation = %.3e" % max(dev_half[k][2] for k in INDEP)) all_rates = [r for k in INDEP for r in rates_half[k]] + rates_i fine_rates = ([rates_half[k][1] for k in INDEP] + [rates_i[1]]) rate_mean = float(np.mean(fine_rates)) rate_spread = float(np.max(fine_rates) - np.min(fine_rates)) # the single-lattice deviation from the continuum reference (tau = i): # measures the RAW lattice artifact that S could have seen cft_dev = {} for N in NS_M2: cft_dev[N] = max(abs((lnZr(N, N, g, h) - N * N / np.pi) - lnZcft(g, h, 1.0)) for (g, h) in INDEP) cft_rates = [np.log(cft_dev[24] / cft_dev[48]) / np.log(2.0), np.log(cft_dev[48] / cft_dev[96]) / np.log(2.0)] print(" single-lattice vs continuum (tau = i, worst of the four " "independent sectors):") print(" " + " ".join("N=%d: %.3e" % (N, cft_dev[N]) for N in NS_M2) + " rates: %.3f, %.3f" % tuple(cft_rates)) check("M2.3 the S-convergence RATE is clean and common across sectors " "AND is N^{-4}, not N^{-2}: fine-pair rate %.3f (spread %.3f " "over the five dynamical relations), while the SINGLE-LATTICE " "deviation from the continuum falls with rate ~ %.2f (= the " "naive N^{-2} dispersion artifact) -- the leading N^{-2} " "lattice artifact is itself S-covariant and CANCELS in the " "S-relation; the measured S-answer is N^{-4}" % (rate_mean, rate_spread, float(np.mean(cft_rates))), 3.5 < rate_mean < 4.5 and rate_spread < 0.5 and 1.5 < float(np.mean(cft_rates)) < 2.5, "S rates = %s; cft rates = %s" % (np.array2string(np.array(all_rates), precision=3), np.array2string(np.array(cft_rates), precision=3))) # M2.4 T as the EXACT discrete Dehn twist (Z6 shift bookkeeping) dev_T = 0.0 for g in range(3): for h in range(3): lam_h = 2.0 * np.pi * h / 3.0 theta = lam_h + N_MAIN * modes(N_MAIN, g) # U_h x shift by N lhs = lnZraw(N_MAIN, 48, g, theta) - 1j * lam_h * (N_MAIN / 2.0) lam_hg = 2.0 * np.pi * ((h + g) % 3) / 3.0 + np.pi rhs = lnZ_lam(N_MAIN, 48, g, lam_hg) dev_T = max(dev_T, abs(lhs.real - rhs.real) + abs(np.exp(1j * (lhs.imag - rhs.imag)) - 1.0)) check("M2.4 [E] T on the lattice is the EXACT discrete Dehn twist " "(translation by N sites): per occupied mode e^{i k N} = " "-omega^g, hence Z_T[g,h] = Z^F[g, g+h] -- the sector map " "(g,h) -> (g, g+h) PLUS a fermion-parity flip (the Z6 " "spin-structure shift), machine-exact for all nine sectors; the " "continuum T-PHASES per character are not resolvable in the " "real-positive trace convention: only S is MEASURED (honest " "scope, see M5)", dev_T < TOL_MACH, "max dev = %.3e" % dev_T) # ================================================================== M3 print("=" * 72) print("M3: q-expansion -- twisted character exponents (N = 48)") print("=" * 72) # M3.1 the leading twisted exponent: Delta_sigma = 1/9 = 4 h_sigma # (L = 768/960: the e^{-delta L} tail of the fit is < 1e-9) lr1 = lnZr(N_MAIN, 768, 1, 0) - lnZr(N_MAIN, 768, 0, 0) lr2 = lnZr(N_MAIN, 960, 1, 0) - lnZr(N_MAIN, 960, 0, 0) dE_fit = -(lr2 - lr1) / 192.0 dE_ex = E0_exact(N_MAIN, 1) - E0_exact(N_MAIN, 0) Delta_lat = dE_fit * N_MAIN / (2.0 * np.pi) rel_D = abs(Delta_lat - DELTA_SIGMA) / DELTA_SIGMA h_lat = Delta_lat / 4.0 check("M3.1 the lowest twisted state: the leading exponent of " "Z[1,0]/Z[0,0] gives Delta_sigma = %.8f vs 1/9 = %.8f (rel dev " "%.2e < 1 %%), i.e. h_sigma = Delta/4 = %.8f vs 1/36 = %.8f -- " "the v639 twist dimension carried by the lowest twisted state" % (Delta_lat, DELTA_SIGMA, rel_D, h_lat, H_SIGMA), rel_D < TOL_FIT and abs(dE_fit - dE_ex) < 1e-7, "slope fit vs exact gap: %.2e" % abs(dE_fit - dE_ex)) # M3.2 the next gap: F(L) = ln(Z[1,0]/Z[0,0]) + L dE_exact # = d ln(1 + e^{-delta L}) with d = 2, delta = (2 pi/N)/6 Ls3 = (240, 288, 336) Fs = [lnZr(N_MAIN, L, 1, 0) - lnZr(N_MAIN, L, 0, 0) + dE_ex * L for L in Ls3] L1, L2 = Ls3[0], Ls3[-1] F1, F2 = Fs[0], Fs[-1] lo, hi = 1e-4, 1.0 for _ in range(200): # bisection: ratio monotone mid = 0.5 * (lo + hi) ratio = (np.log1p(np.exp(-mid * L1)) / np.log1p(np.exp(-mid * L2))) if ratio < F1 / F2: lo = mid else: hi = mid delta_fit = 0.5 * (lo + hi) d_fit = F1 / np.log1p(np.exp(-delta_fit * L1)) resid = abs(Fs[1] - d_fit * np.log1p(np.exp(-delta_fit * Ls3[1]))) gap_lat = delta_fit * N_MAIN / (2.0 * np.pi) rel_gap = abs(gap_lat - GAP2) / GAP2 check("M3.2 the next twisted gap: F(L) = d ln(1 + e^{-delta L}) fit " "gives gap = %.7f vs 1/6 = %.7f (rel dev %.2e < 1 %%) with " "multiplicity d = %.5f vs 2 (< 2 %%) -- the parafermionic " "twisted-pair moding {1/6, 5/6} (one R-hole + one L-particle), " "mid-point residual %.1e" % (gap_lat, GAP2, rel_gap, d_fit, resid), rel_gap < TOL_FIT and abs(d_fit - 2.0) / 2.0 < TOL_MULT and resid < 1e-4) # M3.3 omega^{k q} charge bookkeeping in TIME: the level-1/6 weight in # Z[1,1]/Z[1,0] is (omega + omega^-1) - 2 = -3 LG = 240 G = lnZr(N_MAIN, LG, 1, 1) - lnZr(N_MAIN, LG, 1, 0) xG = np.exp(-delta_fit * LG) c_fit = G / xG rel_c = abs(c_fit - (-3.0)) / 3.0 check("M3.3 the omega^{k q} charge bookkeeping in time: the level-1/6 " "weight of Z[1,1]/Z[1,0] is c = %.5f vs 2 cos(2 pi/3) - 2 = -3 " "(rel dev %.2e < 2 %%): the h-insertion resolves the +-1 charges " "of the twisted pair with exactly the omega phases of the Klein " "pairing" % (c_fit, rel_c), rel_c < TOL_MULT) # ================================================================== M4 print("=" * 72) print("M4: spin-structure bookkeeping -- deck^3 = -1 as Z6 arithmetic") print("=" * 72) # M4.1 per-mode deck phases are Z6; deck^3 = (-1)^F, deck^6 = 1 exactly kc = modes(N_MAIN, 0) # 48-site cover, NS ph_deck = np.exp(1j * N_BASE * kc) # deck = translation by 16 m_idx = np.arange(N_MAIN) ph_expect = np.exp(1j * np.pi * (2 * (m_idx % 3) + 1) / 3.0) dev_ph = np.abs(ph_deck - ph_expect).max() dev_cube = np.abs(ph_deck ** 3 + 1.0).max() dev_z1, dev_z2 = 0.0, 0.0 for L in (32, 72): z3 = lnZraw(N_MAIN, L, 0, 3.0 * N_BASE * kc) zF = lnZ_lam(N_MAIN, L, 0, np.pi) dev_z1 = max(dev_z1, abs(z3.real - zF.real) + abs(np.exp(1j * (z3.imag - zF.imag)) - 1.0)) z6 = lnZraw(N_MAIN, L, 0, 6.0 * N_BASE * kc) z0 = lnZ_lam(N_MAIN, L, 0, 0.0) dev_z2 = max(dev_z2, abs(z6.real - z0.real) + abs(np.exp(1j * (z6.imag - z0.imag)) - 1.0)) check("M4.1 the fermionic deck lift on the 48-site cover: per-mode " "phases e^{i 16 k} take EXACTLY the Z6 values e^{i pi (2r+1)/3} " "on the three deck classes (max dev %.1e), each cubes to -1 " "(max dev %.1e), and as partition-function identities " "deck^3 = (-1)^F and deck^6 = 1 at machine precision -- " "deck^3 = -1 (v623) closes exactly in the trace" % (dev_ph, dev_cube), dev_ph < 1e-12 and dev_cube < 1e-12 and dev_z1 < TOL_MACH and dev_z2 < TOL_MACH, "Z-identity devs: deck^3 vs (-1)^F %.2e, deck^6 vs 1 %.2e" % (dev_z1, dev_z2)) # M4.2 exact factorization: cover deck^h trace = product of the three # base twist sectors with per-class insertion (-1)^h omega^{g~ h} GTIL = {0: 2, 1: 0, 2: 1} # class r -> base twist g~ dev_fac, dev_z6 = 0.0, 0.0 for h in range(6): for L in (32, 72): lhs = lnZraw(N_MAIN, L, 0, N_BASE * kc * h) rhs = 0.0 + 0.0j for r in range(3): nu = (2 * r + 1) / 6.0 rhs += lnZraw(N_BASE, L, GTIL[r], 2.0 * np.pi * nu * h) dev_fac = max(dev_fac, abs(lhs.real - rhs.real) + abs(np.exp(1j * (lhs.imag - rhs.imag)) - 1.0)) for r in range(3): nu = (2 * r + 1) / 6.0 dev_z6 = max(dev_z6, abs(np.exp(2j * np.pi * nu * h) - (-1.0) ** h * OMEGA ** (GTIL[r] * h))) check("M4.2 EXACT factorization of the cover trace: " "Tr_48[deck^h e^{-LH}] = prod_{g~=0,1,2} Tr_16-base[g~-twist, " "insertion ((-1)^h omega^{g~ h})^{Nhat} e^{-LH}] for all h in " "Z6 -- on the seam double the Z3 deck insertion IS the Z3 phase " "times a fermion-parity flip PER STEP: the Z6 spin structure " "demanded by deck^3 = -1, exact", dev_fac < TOL_MACH and dev_z6 < 1e-12, "max factorization dev = %.3e, Z6 phase identity dev = %.3e" % (dev_fac, dev_z6)) # M4.3 parity / Z6-charge projections of the sectors print(" parity data per g-sector (h = 0), N = 48:") par_pos, par_even, gap_ok = True, True, True z6_pos, z6_sum, z6_ground = True, 0.0, True rmax_h = 0.0 GAP_ODD = {0: 0.5, 1: GAP2, 2: GAP2} D_ODD = {0: 4.0, 1: 2.0, 2: 2.0} for g in range(3): # h = 0 column: Z^{+-} are state counters, must be nonnegative z0, _ = lnZ(N_MAIN, 48, g, 0) zF0, sF0 = lnZ(N_MAIN, 48, g, 0, parity=True) R0 = sF0 * np.exp(zF0 - z0) par_pos &= (1.0 + R0 >= -1e-12) and (1.0 - R0 >= -1e-12) # h != 0: the naive ratio exceeds 1 (omega-weighted, NOT a counter) for h in (1, 2): z, _ = lnZ(N_MAIN, 48, g, h) zF, sF = lnZ(N_MAIN, 48, g, h, parity=True) rmax_h = max(rmax_h, sF * np.exp(zF - z)) # the POSITIVE bookkeeping: Z6 charge projection on (Nhat-N/2) mod 6 zj = np.array([lnZ_lam(N_MAIN, 48, g, np.pi * j / 3.0) for j in range(6)]) dev_im = max(abs(np.exp(1j * v.imag) - np.sign(np.cos(v.imag))) for v in zj) zvals = np.array([np.sign(np.cos(v.imag)) * np.exp(v.real - z0) for v in zj]) P = np.array([np.mean([np.exp(-1j * np.pi * j * c / 3.0) * zvals[j] for j in range(6)]).real for c in range(6)]) z6_pos &= (P.min() >= -1e-12) and dev_im < TOL_MACH z6_sum = max(z6_sum, abs(P.sum() - 1.0)) z6_ground &= (P[0] == P.max()) if g < 2: print(" g=%d Z6 charge classes P_c/Z[g,0], c=0..5: %s" % (g, np.array2string(P, precision=5))) # ground-state parity and the odd tower. R = Z^F/Z -> +1 with the # measured odd gap: demand R > 0 growing towards 1 (at L = 336 the # residual odd weight is the fitted 2 e^{-delta L} ~ 1e-3 for g != 0) Rs, ys = [], [] for L in (240, 336): z, _ = lnZ(N_MAIN, L, g, 0) zF, sF = lnZ(N_MAIN, L, g, 0, parity=True) R = sF * np.exp(zF - z) Rs.append(R) ys.append((1.0 - R) / (1.0 + R)) par_even &= (Rs[0] > 0.9 and Rs[1] > 0.99 and Rs[1] > Rs[0]) gap_fit = np.log(ys[0] / ys[1]) / 96.0 * N_MAIN / (2.0 * np.pi) d_odd = ys[0] * np.exp(gap_fit * 2.0 * np.pi / N_MAIN * 240.0) rel_g = abs(gap_fit - GAP_ODD[g]) / GAP_ODD[g] rel_d = abs(d_odd - D_ODD[g]) / D_ODD[g] gap_ok &= (rel_g < TOL_FIT and rel_d < TOL_MULT) print(" g=%d ground parity EVEN, odd tower gap = %.7f vs %.7f " "(rel %.1e), multiplicity = %.4f vs %d" % (g, gap_fit, GAP_ODD[g], rel_g, d_odd, int(D_ODD[g]))) check("M4.3 parity / Z6-charge bookkeeping per sector: in the h = 0 " "column Z^{+-} = (Z +- Z^F)/2 are nonnegative state counters, " "every sector ground state is parity-EVEN (charge-0 twisted " "vacua), the parity-ODD towers open at gap (1/2, 1/6, 1/6) with " "multiplicities (4, 2, 2) for g = (0, 1, 2) at < 1 %%/2 %%; for " "h != 0 the naive Z^F/Z ratio EXCEEDS 1 (max %.3f measured: " "omega-weighted parity traces, not counters -- honest), and the " "POSITIVE bookkeeping of the full Z6 family is the Z6 charge " "projection: all six P_c >= 0, sum_c P_c = Z[g,0] (dev %.1e), " "ground class c = 0, in every g-sector -- the (-1)^F / Z6 " "transformation of the sectors, resolved" % (rmax_h, z6_sum), par_pos and par_even and gap_ok and z6_pos and z6_ground and z6_sum < TOL_MACH and rmax_h > 1.0) # ================================================================== M5 print("=" * 72) print("M5: the typing") print("=" * 72) check("M5.1 [C, honest typing] WHAT STANDS on the lattice: the nine " "Z[g,h] with exact conjugation degeneracies; the continuum " "match |theta[g/3,h/3]/eta|^2 at < 1 % (N = 48); the v628 " "Casimir table per g-sector; modular S-covariance MEASURED with " "a clean common N^{-4} rate (the N^{-2} lattice artifact is " "itself S-covariant and cancels); T realized exactly as the " "discrete Dehn twist = the Z6 shift (g,h) -> (g, g+h) x (-1)^F; " "the twisted characters (h_sigma = 1/36, parafermionic gap 1/6, " "omega charge weights); and the deck^3 = -1 spin structure as " "exact Z6 trace arithmetic including the cover factorization. " "STILL OPEN: the chiral PHASES of the eta/theta identification " "(only |.|^2 is matched), the full continuum T-phases per " "character (the lattice convention is real-positive), the " "non-abelian sector assembly / orbifold projection sum with its " "discrete-torsion choice, and the continuum OS statement -- " "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))) c_ok = all(ok for n, ok in CHECKS if n.startswith(("C0", "M1"))) s_ok = all(ok for n, ok in CHECKS if n.startswith("M2")) if n_pass == len(CHECKS): print("ALL CHECKS PASSED") print("VERDICT: ORB-MODULAR-COHERENT -- the modular data of the Z3") print("orbifold of the seam stands on the lattice: nine sectors match") print("|theta[g/3,h/3]/eta|^2, S-covariance is MEASURED with a clean") print("N^{-4} rate (the N^{-2} artifact is itself S-covariant and") print("cancels), T is the exact Z6 Dehn shift, the twisted") print("characters carry h_sigma = 1/36 with the parafermionic 1/6 gap,") print("and deck^3 = -1 closes exactly as Z6 trace arithmetic.") elif c_ok and not s_ok: print("SOME CHECKS FAILED") print("VERDICT: ORB-MODULAR-S-OPEN -- sectors and characters stand but") print("the S-relation does not converge with a clean rate: honest open.") elif not c_ok: print("SOME CHECKS FAILED") print("VERDICT: ORB-MODULAR-FAILS -- the reference or sector level") print("fails: 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)