"""v504 -- CELEST.WP5DB.01: WP5d-beta of the research contract CELEST.SEAM.01 -- "split + strong additivity from the lattice": the two remaining Kawahigashi-Longo-Mueger (KLM) legs of complete rationality, witnessed for the orbifold/quotient prescription on the seam edge. Python (numpy + scipy Gaussian machinery + exact GF2/integer algebra). Question: complete rationality (KLM) = split property + strong additivity + finite two-interval mu-index. WP5d-alpha (v501) witnessed the mu-index (entropic offset ln 2 => [F : F_even] = 2, mu_gauged = 4, chain 16 -> 4 -> 1); this module witnesses the OTHER TWO legs for the same orbifold prescription on the critical Majorana ring (v367/v450 reduction, Peschel-exact), with the U(1)/Dirac number-conserving subnet as the preregistered negative control -- the current-net analogue where strong additivity famously FAILS in the continuum (Buchholz-Schulz-Mirbach). With v501, all THREE KLM ingredients of complete rationality now have lattice witnesses. [E-machine] S0. MACHINERY VALIDATED: closed-form NS covariance == eigh route (4.3e-15); the touching (adjacent-interval) parity-deficit Gaussian formula == exact diagonalisation (4.4e-16); the U(1) Fock-built rho (covariance round trip 5.6e-16) with charge-dephased Tr sigma^2 == exact finite quadrature (5.6e-17 -- R(alpha) is a trig polynomial of degree <= n_A, so the (2 n_A + 1)-node grid is EXACT); the second-compound identity (cross-Gram of even bilinears = Lambda^2 C, singular values {sigma_a sigma_b}, 2.8e-14); parity flip C -> -C leaves sigma_k invariant (8.3e-17); sigma_X = rho_X for every single region (Gaussian parity superselection, exact). [E] S1. STRONG ADDITIVITY, EXACT ALGEBRA: "touching" = shared boundary Majorana => Even(A) v Even(B) = Even(A u B) EXACTLY (GF2 spans full 64/64, 256/256, 512/512, 1024/1024 on four sizes; literal matrix rank 32/32); DISJOINT intervals give exactly HALF (rank 16/32, index 2 -- the missing sector is odd(x)odd, the single ln-2 bit of WP5d-alpha, localised at the split point); U(1)/neutral algebras do NOT generate the union even WITH the shared site (gaps 2/10/52, GROWING -- the charge-transfer tower is missing): the exact lattice shadow of the current-net strong-additivity failure (infinite index). [N] S2. STRONG ADDITIVITY, ENTROPIC SCALING: the fermionic touching MI diverges with exact (c/3) ln 2 increments per doubling (0.115525, dev 1.2e-6 -- pure UV point-gap, present for EVERY net, hence NOT the discriminator); the honest discriminator is BOUNDED vs DIVERGENT sector-sum deficit: the Z2/orbifold deficit rises 0.4045 -> 0.5899 (N = 64 -> 4096), stays < ln 2, with (ln 2 - Delta_2) ~ N^(-p), p = 0.2444 vs 2 Delta_mu = 1/4 (the Ising disorder-operator dimension); split-point scan exactly reflection symmetric (3.9e-14); ED vN deficits same direction; gap continuity to the v501 x -> 1 endpoint. HONEST NOTE: the naive "defect -> 0" expectation is FALSE on the sharp lattice split -- bounded <=> finite index (Longo-Xu) is the correct statement. U(1) CONTRAST: D_U1(touch) BURSTS the finite-group ceiling ln 2 from L = 128 on and keeps growing as (1/2) ln Var Q_A with Var Q_A = (1/pi^2) ln L + const (Klich-Levitov slope 0.10134 vs 1/pi^2 = 0.10132): infinite index, no saturation. [N] S3. SPLIT PROPERTY (nuclearity witness): the singular values of the off-diagonal covariance block for DISJOINT intervals fall exponentially in k with rate = the elliptic nome pi K(1-x)/K(x) to 1.3-2.0% (d = 16..512 at N = 4096: rates 2.29/3.24/4.82/5.81); sigma_1 ~ x^0.5044 vs 1/2 (the dimension-1/2 Majorana two-point law); trace norm summable and falling (Buchholz-Wichmann nuclearity proxy); the dimerised control decays exponentially per site (a gapped seam splits trivially -- the CFT power laws are properties of the seam). ORBIFOLD INHERITANCE IS EXACT AT MACHINE LEVEL: P_A maps C -> -C (sigma_k identical, 8.3e-17) and the even-bilinear coupling Gram is the second compound Lambda^2 C with singular values {sigma_a sigma_b} (2.8e-14) -- the SAME ladder at doubled rate; the orbifold Renyi-2 MI is < 3% of the fermionic one and falls with x (Longo heredity, [C]). [E] S4. PIMSNER-POPA / LONGO-REHREN, EXACT: E(a) - a/2 = PaP/2 holds IDENTICALLY (the PP inequality with lambda = 1/2 = 1/[F : F_even], one line, exact); ATTAINMENT in exact integers: all 16384 (even, odd) monomial pairs at N = 8 Majoranas -- 8192 extremal with M^2 = 4, Tr M = 0, PMP = -M exactly, 0 violations; all 2048 hermitianised odd monomials at N = 12 (2048/2048); random pencil sweep min lambda = 0.5000000000; the two-interval group {1, P_A, P_B, P_AB} gives lambda = 1/4 = 1/mu exactly; INDEX CONSISTENCY, TWO ROUTES: exp(Delta_inf) = 2.000000000 = 1/lambda_PP (the v501 entropic route) and exp(2 Delta_inf) = 4 = 1/lambda_E4 = mu -- the Longo-Rehren index witnessed entropically AND operator-algebraically, independently; U(1) control: lambda = 1/(m+1) -> 0 (infinite index). [E/N] S5. NEGATIVE CONTROLS + FINITE-SIZE HONESTY: dimerised/trivial phase -- every indicator trivial (touching deficit N-independent to 3.6e-15, MI bounded, sigma_1 exponential); U(1) small-L honesty (non-monotone below L ~ 16, filling-parity artefacts -- documented); Z2 finite-size honesty: at N = 4096 the touching deficit still sits 14.9% below ln 2 (the N^{-1/4} Ising approach is SLOW -- the verdict rests on the bound + fitted exponent, stated as witness, not proof); wrong prescription {1, P_AB} on the touching pair: deficit EXACTLY 0 (machine identity) -- the bookkeeping is falsifiable. [C] The lattice -> continuum KLM dictionary: bounded deficit = finite index = complete rationality (Longo-Xu dichotomy); split heredity under finite index (Longo); the entropic order parameters (CHMP); the nome ladder = the free-fermion two-interval modular package. [O] The continuum proof: the theorem "finite-group orbifolds of completely rational nets are completely rational (hence split + strongly additive)" is XU'S THEOREM (algebraic orbifold CFTs) -- CITED, NOT CLAIMED; the continuum proof for THE seam quotient net itself and for the interacting condensed (E8)_1 net stays open (WP5e / Costello-Li territory); SEAM.EQUIV.01 stays [O], untouched; no marker moves. Verdict: with v501 (mu-index), all THREE KLM ingredients of complete rationality -- split, strong additivity, finite mu -- are witnessed on the lattice for the orbifold prescription; the U(1) control fails both new legs exactly as the continuum current net does. Status: [E] the exact algebra spans/indices, the PP identities and integer attainments, the quadrature/compound/machine identities; [N] all scaling witnesses (deficit tables, nome ladder, power laws, KL slope; fine type Lattice/Numerical in the ledger); [C]/[O] as fenced above. Python; Wolfram-mirrored (the exact parts only: GF2 span counts and index 2, the PP identity E(a) - a/2 = PaP/2 with lambda = 1/2 and lambda_E4 = 1/4 = 1/mu, the index consistency exp(Delta) = 1/lambda_PP and exp(2 Delta) = mu, the Lambda^2-compound identity -- the numerical lattice curves stay Python-only), counted per GATE.WOLFRAM.02 convention. Deterministic (fixed seeds); full sizes (N up to 4096) run in ~12 s.""" import numpy as np import scipy.sparse as sp from scipy.linalg import schur, toeplitz from scipy.special import ellipk from itertools import combinations from math import comb from tfpt_constants import check, summary, reset, g_car, N_fam G_CAR = g_car # 5, the carrier rank (P2) N_FAM = N_fam # 3 SEAM_LAYERS = 2 * (G_CAR + N_FAM) # 16 layers = 2 c_-, c_- = 8 (v489/v501) LN2 = np.log(2.0) # --------------------------------------------------------------------------- # Gaussian machinery (identical conventions to v501 / WP5d-alpha) # --------------------------------------------------------------------------- def ring_A(N, t_weak=None, antiperiodic=True): A = np.zeros((N, N)) for j in range(N): t = 1.0 if (t_weak is None or j % 2 == 0) else t_weak if j == N - 1 and antiperiodic: t = -t A[j, (j + 1) % N] += t A[(j + 1) % N, j] -= t return A def ground_gamma(A): w, U = np.linalg.eigh(1j * A) G = 1j * (U * np.sign(w)) @ U.conj().T assert np.abs(G.imag).max() < 1e-11 return 0.5 * (G.real - G.real.T) def gamma_analytic(N): """NS ground-state covariance of the uniform critical ring, closed form: Gamma_jk = -2 / (N sin(pi (k-j)/N)) for odd k-j, 0 for even.""" j = np.arange(N) D = j[None, :] - j[:, None] with np.errstate(divide="ignore"): M = -2.0 / (N * np.sin(np.pi * D / N)) M[D % 2 == 0] = 0.0 np.fill_diagonal(M, 0.0) return M def sub(G, idx): return G[np.ix_(idx, idx)] def S_vN(G): lam = np.linalg.eigvalsh(1j * G) p = np.clip((1 + lam) / 2, 1e-15, 1 - 1e-15) return float(-0.5 * np.sum(p * np.log(p) + (1 - p) * np.log(1 - p))) def S_R2(G): lam = np.linalg.eigvalsh(1j * G) return float(-0.5 * np.sum(np.log((1 + lam ** 2) / 2))) def ln_tr_prod(G1, G2): n = G1.shape[0] sign, ld = np.linalg.slogdet(np.eye(n) - G1 @ G2) assert sign > 0 return 0.5 * ld - (n // 2) * LN2 def deficit2_flip(GAB, la, layers=1): """Renyi-2 deficit of the P_A sector sum; la = # flipped Majoranas.""" n = GAB.shape[0] s = np.ones(n) s[:la] = -1.0 GS = GAB * np.outer(s, s) R = np.exp(ln_tr_prod(GAB, GS) - ln_tr_prod(GAB, GAB)) return float(-np.log((1 + R ** layers) / 2)), float(R) def chord(z, N): return abs(np.sin(np.pi * z / N)) def cross_ratio(cuts, N): u1, v1, u2, v2 = cuts den = chord(u2 - u1, N) * chord(v2 - v1, N) x = chord(v1 - u1, N) * chord(v2 - u2, N) / den xb = chord(u2 - v1, N) * chord(v2 - u1, N) / den return x, xb # --------------------------------------------------------------------------- # ED layer (Jordan-Wigner; validates the Gaussian formulas) # --------------------------------------------------------------------------- def jw_majoranas(nq): I2 = sp.identity(2, format="csr", dtype=complex) X = sp.csr_matrix(np.array([[0, 1], [1, 0]], complex)) Y = sp.csr_matrix(np.array([[0, -1j], [1j, 0]], complex)) Z = sp.csr_matrix(np.array([[1, 0], [0, -1]], complex)) cs = [] for k in range(nq): for P in (X, Y): op = None for j in range(nq): f = Z if j < k else (P if j == k else I2) op = f if op is None else sp.kron(op, f, format="csr") cs.append(op) return cs def gauss_rho(G, cs): n = G.shape[0] T, Q = schur(G, output="real") assert max(abs(T[i, i]) for i in range(n)) < 1e-8 dim = cs[0].shape[0] rho = np.eye(dim, dtype=complex) for k in range(n // 2): b = T[2 * k, 2 * k + 1] d1 = sum(Q[i, 2 * k] * cs[i] for i in range(n)) d2 = sum(Q[i, 2 * k + 1] * cs[i] for i in range(n)) fac = 0.5 * (sp.identity(dim, dtype=complex, format="csr") + 1j * b * (d1 @ d2)) rho = rho @ fac.toarray() return rho def parity_first(nq_a, nq): idx = np.arange(2 ** nq) signs = np.ones(2 ** nq) for j in range(nq_a): bit = (idx >> (nq - 1 - j)) & 1 signs *= np.where(bit == 1, -1.0, 1.0) return signs def ed_S(rho): lam = np.clip(np.linalg.eigvalsh(rho).real, 0.0, 1.0) lam = lam[lam > 1e-14] return float(-np.sum(lam * np.log(lam))) # --------------------------------------------------------------------------- # Dirac (number-conserving) machinery: C_jl = , half filling, NS # --------------------------------------------------------------------------- def dirac_C(L): ks = 2 * np.pi * (np.arange(L) + 0.5) / L filled = ks[np.cos(ks) < 0] j = np.arange(L) f = np.zeros(L, complex) for k in filled: f += np.exp(1j * k * j) / L return toeplitz(f.conj(), f) def R_alpha(Csub, nA, alpha): """Tr(rho U rho U+)/Tr rho^2 for the U(1) phase on the first nA modes.""" n = Csub.shape[0] v = np.ones(n, complex) v[:nA] = np.exp(1j * alpha) Ca = (v[:, None] * Csub) * v.conj()[None, :] I = np.eye(n) s1, l1 = np.linalg.slogdet(Csub @ Ca + (I - Csub) @ (I - Ca)) s0, l0 = np.linalg.slogdet(Csub @ Csub + (I - Csub) @ (I - Csub)) return float((np.exp(l1 - l0) * s1 / s0).real) def u1_touch_deficit(L, nodes=None): """full-U(1) sector-sum Renyi-2 deficit for touching halves of [0, L/2); exact quadrature: R(alpha) is a trig polynomial of degree <= nA.""" C = dirac_C(L) nAB, nA = L // 2, L // 4 Csub = C[:nAB, :nAB] m = nodes if nodes else 2 * nA + 1 Rs = [R_alpha(Csub, nA, 2 * np.pi * t / m) for t in range(m)] Rbar = float(np.mean(Rs)) CA = C[:nA, :nA] varQ = float(np.trace(CA @ (np.eye(nA) - CA)).real) return -np.log(Rbar), varQ # --------------------------------------------------------------------------- # exact algebra spans (GF2 monomial labels + literal matrix ranks) # --------------------------------------------------------------------------- def even_labels(sites, n_all): out = [] for r in range(0, len(sites) + 1, 2): for c in combinations(sites, r): m = 0 for s in c: m |= 1 << s out.append(m) return out def even_span_count(nA, nB, shared): """|{a XOR b : a in Even(A), b in Even(B)}| vs dim Even(union). Majorana monomials c_S are linearly independent iff labels differ and c_a c_b = +- c_{a XOR b}, so span dim = # distinct labels (exact).""" n = nA + nB - (1 if shared else 0) A = list(range(nA)) B = list(range(nA - 1, n)) if shared else list(range(nA, n)) prod = set() for a in even_labels(A, n): for b in even_labels(B, n): prod.add(a ^ b) return len(prod), 2 ** (n - 1) def monomial(cs, S, dim): M = np.eye(dim, dtype=complex) for s_ in S: M = M @ cs[s_].toarray() return M def matrix_span_rank(cs, sitesA, sitesB, dim, even_only=True): def monos(sites): step = 2 if even_only else 1 return [monomial(cs, S, dim) for r in range(0, len(sites) + 1, step) for S in combinations(sites, r)] vecs = [(a @ b).ravel() for a in monos(sitesA) for b in monos(sitesB)] return int(np.linalg.matrix_rank(np.array(vecs), tol=1e-9)) def neutral_span_rank(nA, nB, shared): """number-conserving (U(1)-invariant) span: products of a+_S a_T (|S| = |T|) supported in A times the same in B, vs dim of the neutral algebra on the union = sum_q C(n, q)^2. Exact Fock matrices.""" n = nA + nB - (1 if shared else 0) A = list(range(nA)) B = list(range(nA - 1, n)) if shared else list(range(nA, n)) dim = 2 ** n def neutral_ops(sites): ops = [] for r in range(0, len(sites) + 1): for S in combinations(sites, r): for T in combinations(sites, r): M = np.zeros((dim, dim)) for st in range(dim): state, amp, ok = st, 1.0, True for tt in sorted(T, reverse=True): if not (state >> tt) & 1: ok = False break amp *= (-1) ** bin(state & ((1 << tt) - 1)).count("1") state &= ~(1 << tt) if not ok: continue for ss in sorted(S): if (state >> ss) & 1: ok = False break amp *= (-1) ** bin(state & ((1 << ss) - 1)).count("1") state |= 1 << ss if ok: M[state, st] = amp ops.append(M) return ops vecs = [(a @ b).ravel() for a in neutral_ops(A) for b in neutral_ops(B)] rank = int(np.linalg.matrix_rank(np.array(vecs), tol=1e-8)) target = sum(comb(n, q) ** 2 for q in range(n + 1)) return rank, target # --------------------------------------------------------------------------- # Pimsner-Popa pencil # --------------------------------------------------------------------------- def pp_lambda(a, Emap, tol=1e-11): """largest lambda with E(a) >= lambda a, via the pencil on supp E(a).""" Ea = Emap(a) w, V = np.linalg.eigh(Ea) keep = w > tol * max(w.max(), 1e-30) W = V[:, keep] * (w[keep] ** -0.5) return float(1.0 / np.linalg.eigvalsh(W.conj().T @ a @ W).max()) # =========================================================================== # S0 -- machinery validation # =========================================================================== def section0(): print(" -- S0: machinery validated (analytic covariance, ED, quadrature," " compound)") dev = np.abs(gamma_analytic(256) - ground_gamma(ring_A(256))).max() check("S0.1: ANALYTIC COVARIANCE [E-machine] -- closed-form NS Gamma " "equals the eigh route at N = 256 to %.1e (used for N up to 4096)" % dev, dev < 1e-12) # touching deficit: Gaussian == ED (N = 16 ring, region 12, split 6) N = 16 G = ground_gamma(ring_A(N)) GAB = sub(G, list(range(12))) d2g, Rg = deficit2_flip(GAB, 6) cs = jw_majoranas(6) rho = gauss_rho(GAB, cs) P = parity_first(3, 6) rhoP = (P[:, None] * rho) * P[None, :] R_ed = float(np.trace(rho @ rhoP).real / np.trace(rho @ rho).real) dvn = ed_S(0.5 * (rho + rhoP)) - ed_S(rho) check("S0.2: TOUCHING DEFICIT vs ED [E-machine] -- adjacent intervals " "(region 12 Maj, split at 6): Gaussian R = %.10f == ED R = %.10f " "(dev %.1e); Delta_2 = %.6f, Delta_vN = %.6f (same direction)" % (Rg, R_ed, abs(Rg - R_ed), d2g, dvn), abs(Rg - R_ed) < 1e-12 and dvn > d2g > 0) # U(1): Fock-built rho from C; dephasing == exact quadrature L = 8 C = dirac_C(L) Csub = C[:4, :4] nvals, V = np.linalg.eigh(Csub) cs4 = jw_majoranas(4) a_ops = [0.5 * (cs4[2 * q] + 1j * cs4[2 * q + 1]).toarray() for q in range(4)] b_ops = [sum(V[jj, kk] * a_ops[jj] for jj in range(4)) for kk in range(4)] rho = np.eye(16, dtype=complex) for kk in range(4): nk = float(np.clip(nvals[kk], 0, 1)) num = b_ops[kk].conj().T @ b_ops[kk] rho = rho @ (nk * num + (1 - nk) * (np.eye(16) - num)) devC = max(abs(np.trace(rho @ (a_ops[p].conj().T @ a_ops[q])) - Csub[p, q]) for p in range(4) for q in range(4)) QA = sum(a_ops[p].conj().T @ a_ops[p] for p in range(2)) wq, Vq = np.linalg.eigh(QA) sig = np.zeros_like(rho) for q in (0, 1, 2): Pq = Vq[:, np.abs(wq - q) < 1e-9] Pq = Pq @ Pq.conj().T sig += Pq @ rho @ Pq tr_s2 = float(np.trace(sig @ sig).real) m = 5 # 2*nA + 1 with nA = 2 Rq = np.mean([R_alpha(Csub, 2, 2 * np.pi * t / m) for t in range(m)]) tr_pred = float(np.trace(rho @ rho).real * Rq) check("S0.3: U(1) MACHINERY vs ED [E-machine] -- Fock rho from C " "(Tr = 1 dev %.1e, covariance round trip %.1e); charge-dephased " "Tr sigma^2 = %.10f == Tr rho^2 _quad = %.10f (dev %.1e)" % (abs(np.trace(rho).real - 1), abs(devC), tr_s2, tr_pred, abs(tr_s2 - tr_pred)), abs(devC) < 1e-12 and abs(tr_s2 - tr_pred) < 1e-12) d_a, _ = u1_touch_deficit(64) d_b, _ = u1_touch_deficit(64, nodes=2 * 16 + 17) check("S0.4: EXACT QUADRATURE [E-machine] -- R(alpha) is a trig " "polynomial of degree <= n_A (bounded charge transfer), so the " "(2 n_A + 1)-node average is exact: 33 vs 49 nodes agree to %.1e" % abs(d_a - d_b), abs(d_a - d_b) < 1e-12) # second compound: even-bilinear cross-Gram has s.v. {sigma_a sigma_b} rng = np.random.default_rng(4990) lsz = 10 Cb = rng.normal(size=(lsz, lsz)) sv = np.linalg.svd(Cb, compute_uv=False) pairs = list(combinations(range(lsz), 2)) M2 = np.zeros((len(pairs), len(pairs))) for pi, (i, jj) in enumerate(pairs): for qi, (k, l) in enumerate(pairs): M2[pi, qi] = Cb[i, k] * Cb[jj, l] - Cb[i, l] * Cb[jj, k] svM = np.sort(np.linalg.svd(M2, compute_uv=False))[::-1] prods = np.sort([sv[a] * sv[b] for a, b in pairs])[::-1] dev2 = np.abs(svM - prods).max() check("S0.5: SECOND COMPOUND [E-machine] -- the Wick cross-Gram of even " "bilinears c_i c_j (A) x c_k c_l (B) is Lambda^2(C): singular " "values {sigma_a sigma_b} exactly (dev %.1e on a random 10x10): " "the orbifold coupling ladder IS the fermionic one, squared" % dev2, dev2 < 1e-10) # parity flip: C -> -C, sigma_k invariant; single region sigma_X = rho_X G = gamma_analytic(512) Cx = G[np.ix_(range(64), range(128, 192))] devs = np.abs(np.linalg.svd(Cx, compute_uv=False) - np.linalg.svd(-Cx, compute_uv=False)).max() s = -np.ones(128) devG = np.abs(sub(G, list(range(128))) * np.outer(s, s) - sub(G, list(range(128)))).max() check("S0.6: PARITY STRUCTURE [E-machine] -- P_A maps the coupling block " "C -> -C: sigma_k identical (dev %.1e); flipping ALL Majoranas of " "a single region leaves Gamma invariant (dev %.1e): sigma_X = " "rho_X for every one-region state (Gaussian parity superselection)" % (devs, devG), devs < 1e-13 and devG < 1e-15) # =========================================================================== # S1 -- strong additivity: exact lattice algebra # =========================================================================== def section1(): print(" -- S1: strong additivity as exact lattice algebra (spans/ranks)") cs3 = jw_majoranas(3) r_car = matrix_span_rank(cs3, [0, 1, 2], [3, 4, 5], 8, even_only=False) check("S1.1: CAR REFERENCE [E-machine] -- CAR(A) . CAR(B) spans the FULL " "algebra (rank %d / %d) even for DISJOINT intervals: the graded " "net is additive, trivially -- the mu = 1 reference" % (r_car, 64), r_car == 64) rows = [] ok_sh = True for (nA, nB) in [(4, 4), (5, 5), (5, 6), (6, 6)]: c_sh, tot = even_span_count(nA, nB, shared=True) rows.append((nA, nB, c_sh, tot)) ok_sh = ok_sh and (c_sh == tot) r_sh = matrix_span_rank(cs3, [0, 1, 2, 3], [3, 4, 5], 8, even_only=True) print(" shared-site even spans: " + ", ".join( "%d+%d: %d/%d" % r for r in rows)) check("S1.2: TOUCHING = SHARED POINT [E] -- with the shared boundary " "Majorana, Even(A) v Even(B) = Even(A u B) EXACTLY: GF2 span full " "on all four sizes, literal matrix rank %d / 32: the lattice " "strong-additivity witness for the orbifold algebra" % r_sh, ok_sh and r_sh == 32) rows_d = [] ok_d = True for (nA, nB) in [(4, 4), (5, 5), (6, 6)]: c_d, tot = even_span_count(nA, nB, shared=False) rows_d.append((nA, nB, c_d, tot)) ok_d = ok_d and (2 * c_d == tot) r_d = matrix_span_rank(cs3, [0, 1, 2], [3, 4, 5], 8, even_only=True) print(" disjoint even spans: " + ", ".join( "%d+%d: %d/%d" % r for r in rows_d)) check("S1.3: DISJOINT INDEX 2 [E] -- without the shared site the even " "span is EXACTLY HALF (all sizes; matrix rank %d / 32): the " "missing sector is odd(x)odd -- the single ln 2 bit of WP5d-alpha " "(v501), localised at the split point" % r_d, ok_d and 2 * r_d == 32) rows_u = [] gaps = [] for (nA, nB) in [(2, 2), (3, 2), (3, 3)]: rk, tgt = neutral_span_rank(nA, nB, shared=True) rows_u.append((nA, nB, rk, tgt)) gaps.append(tgt - rk) rk_dd, tgt_dd = neutral_span_rank(2, 2, shared=False) print(" U(1) neutral shared-site spans: " + ", ".join( "%d+%d: %d/%d" % r for r in rows_u) + "; disjoint 2+2: %d/%d" % (rk_dd, tgt_dd)) check("S1.4: U(1) CONTRAST [E] -- the number-conserving (neutral) " "algebras do NOT generate the union even WITH the shared site: " "gaps %s GROW with size (the charge-transfer tower a+...a+ x a...a " "beyond one unit is missing): the exact lattice shadow of the " "current-net strong-additivity failure (infinite index)" % gaps, gaps == sorted(gaps) and gaps[0] > 0 and gaps[-1] > gaps[0] and rk_dd < tgt_dd) # =========================================================================== # S2 -- strong additivity: entropic scaling witnesses # =========================================================================== def section2(): print(" -- S2: strong additivity, entropic scaling (a -> 0 tables)") # fermionic touching MI: pure point-gap UV, coefficient (c/3) ln Ns = [128, 256, 512, 1024, 2048, 4096] I_t = [] for N in Ns: G = gamma_analytic(N) la = N // 4 SA = S_vN(sub(G, list(range(la)))) SB = S_vN(sub(G, list(range(la, 2 * la)))) SAB = S_vN(sub(G, list(range(2 * la)))) I_t.append(SA + SB - SAB) incs = np.diff(I_t) dev_inc = np.abs(incs - LN2 / 6).max() print(" fermionic I(A:B) touching: " + " ".join( "N=%d: %.6f" % (n, v) for n, v in zip(Ns, I_t))) check("S2.1: POINT-GAP LAW [N] -- the fermionic touching MI diverges " "logarithmically with increments per doubling = %.6f..%.6f vs " "(c/3) ln 2 = ln2/6 = %.6f (max dev %.1e): the 'defect' at a " "touching point is pure UV, (c/3) ln(1/a), present for EVERY " "conformal net -- it cannot discriminate strong additivity" % (incs.min(), incs.max(), LN2 / 6, dev_inc), dev_inc < 1e-4) # Z2 sector-sum touching deficit: bounded by ln 2, Ising approach print(" Z2 orbifold touching deficit (A = B = N/4, adjacent):") Ns2 = [64, 128, 256, 512, 1024, 2048, 4096] dts = [] for N in Ns2: G = gamma_analytic(N) la = N // 4 d2, R = deficit2_flip(sub(G, list(range(2 * la))), la) dts.append((N, R, d2)) print(" N = %5d : R = %.6f, Delta_2 = %.6f, ln2 - Delta_2 = %.6f" % (N, R, d2, LN2 - d2)) mono = all(dts[i + 1][2] > dts[i][2] for i in range(len(dts) - 1)) bounded = all(d < LN2 for (_, _, d) in dts) check("S2.2: BOUNDED SECTOR-SUM DEFECT [N] -- Delta_2(touch) rises " "monotonically %.4f -> %.4f (N = 64 -> 4096) and stays < ln 2 = " "%.4f: the Z2 orbifold pays at most ONE bit at the split point -- " "bounded <=> finite index (the KLM-side discriminator; honest " "note: the naive 'defect -> 0' expectation is FALSE on the sharp " "lattice split -- bounded-vs-divergent is the real criterion)" % (dts[0][2], dts[-1][2], LN2), mono and bounded) gaps_ln = [np.log(LN2 - d) for (_, _, d) in dts[-4:]] lnN = [np.log(n) for (n, _, _) in dts[-4:]] p_fit = -np.polyfit(lnN, gaps_ln, 1)[0] check("S2.3: ISING APPROACH RATE [N] -- (ln 2 - Delta_2) ~ N^(-p) with " "fitted p = %.4f vs 2 Delta_mu = 1/4 (the Ising disorder-operator " "dimension h_mu = 1/16 per chirality; window 0.20..0.30): the " "missing bit decays with the CORRECT universal exponent" % p_fit, 0.20 < p_fit < 0.30) # split-point dependence N = 1024 G = gamma_analytic(N) GAB = sub(G, list(range(512))) fr_list = [1 / 8, 1 / 4, 3 / 8, 1 / 2, 5 / 8, 3 / 4, 7 / 8] dsp = {} for fr in fr_list: la = int(512 * fr) dsp[fr], _ = deficit2_flip(GAB, la) print(" split-point scan (N = 1024, region 512): " + " ".join( "%.3f: %.5f" % (f, dsp[f]) for f in fr_list)) sym_dev = max(abs(dsp[f] - dsp[1 - f]) for f in [1 / 8, 1 / 4, 3 / 8]) spread = (max(dsp.values()) - min(dsp.values())) / LN2 check("S2.4: SPLIT-POINT SCAN [N] -- exactly reflection symmetric " "(dev %.1e), maximal at the midpoint, spread only %.1f%% of ln 2: " "the witness does not depend on WHERE the interval is split" % (sym_dev, 100 * spread), sym_dev < 1e-8 and spread < 0.06 and max(dsp, key=lambda f: dsp[f]) == 1 / 2) # ED vN deficits (small N): same direction vn = [] for N in [12, 16, 20]: G = ground_gamma(ring_A(N)) la = N // 4 GAB = sub(G, list(range(2 * la))) nq = la cs = jw_majoranas(nq) rho = gauss_rho(GAB, cs) P = parity_first(la // 2 if la % 2 == 0 else (la + 1) // 2, nq) rhoP = (P[:, None] * rho) * P[None, :] vn.append((N, ed_S(0.5 * (rho + rhoP)) - ed_S(rho))) print(" ED vN touching deficits: " + " ".join( "N=%d: %.5f" % t for t in vn)) check("S2.5: vN DIRECTION [N] -- exact-diagonalisation von Neumann " "deficits rise with N (%.4f -> %.4f): the bounded-defect witness " "is not a Renyi-2 artefact" % (vn[0][1], vn[-1][1]), all(vn[i + 1][1] > vn[i][1] for i in range(len(vn) - 1))) # gap-continuity: touching = m/N -> 0 limit of the disjoint family print(" gap continuity D(touch) - D(gap m = 1):") diffs = [] for N in [512, 1024, 2048, 4096]: G = gamma_analytic(N) la = N // 4 D0, _ = deficit2_flip(sub(G, list(range(2 * la))), la) ia = list(range(la)) ib = list(range(la + 1, 2 * la + 1)) D1, _ = deficit2_flip(sub(G, ia + ib), la) diffs.append((N, D0 - D1)) print(" N = %5d : %.5f" % (N, D0 - D1)) check("S2.6: GAP CONTINUITY [N] -- |D(touch) - D(one-site gap)| shrinks " "monotonically (%.4f -> %.4f): the touching witness is the " "continuous x -> 1 endpoint of WP5d-alpha's (v501) disjoint " "family, not a separate lattice artefact" % (diffs[0][1], diffs[-1][1]), all(diffs[i + 1][1] < diffs[i][1] for i in range(len(diffs) - 1))) # U(1) contrast: unbounded growth ~ (1/2) ln Var Q print(" U(1) full sector sum, touching halves (exact quadrature):") rows = [] for L in [64, 128, 256, 512, 1024]: D, varQ = u1_touch_deficit(L) rows.append((L, D, varQ)) print(" L = %5d : D_U1 = %.6f, Var Q_A = %.5f" % (L, D, varQ)) xs = np.array([0.5 * np.log(r[2]) for r in rows]) ys = np.array([r[1] for r in rows]) slope = np.polyfit(xs, ys, 1)[0] lnL = np.array([np.log(r[0]) for r in rows]) kl = np.polyfit(lnL, [r[2] for r in rows], 1)[0] burst = [r[0] for r in rows if r[1] > LN2] check("S2.7: U(1) DIVERGENCE [N] -- D_U1(touch) grows %.4f -> %.4f with " "NO ceiling: it bursts the finite-group bound ln 2 from L = %d on, " "tracks (1/2) ln Var Q_A (slope %.3f ~ 1) with Var Q_A = " "(1/pi^2) ln L + const (Klich-Levitov slope %.5f vs 1/pi^2 = " "%.5f): infinite index -- the number-conserving subnet FAILS " "strong additivity, exactly the known continuum failure of the " "current net (the preregistered negative control has teeth)" % (rows[0][1], rows[-1][1], burst[0] if burst else -1, slope, kl, 1 / np.pi ** 2), len(burst) > 0 and burst[0] <= 256 and all(rows[i + 1][1] > rows[i][1] for i in range(len(rows) - 1)) and 0.7 < slope < 1.1 and abs(kl - 1 / np.pi ** 2) < 0.002) # =========================================================================== # S3 -- split property: nuclearity witnesses # =========================================================================== def section3(): print(" -- S3: split property (singular-value ladders of the coupling " "block)") N = 4096 G = gamma_analytic(N) ell = 128 print(" N = 4096, ell = 128: pair-mean ladder rate vs elliptic nome " "pi K(1-x)/K(x):") ratios = [] rates = [] tnorms = [] pair_degen = [] for d in [16, 32, 64, 128, 256, 512]: cuts = (0, ell, ell + d, 2 * ell + d) x, _ = cross_ratio(cuts, N) ia = list(range(ell)) ib = list(range(ell + d, 2 * ell + d)) sv = np.linalg.svd(G[np.ix_(ia, ib)], compute_uv=False) tnorms.append(float(sv.sum())) pairs = [] for i in range(6): if sv[2 * i + 1] > 1e-13: pairs.append(np.sqrt(sv[2 * i] * sv[2 * i + 1])) pair_degen.append(sv[0] / sv[1]) ks = np.arange(len(pairs)) rate = -np.polyfit(ks, np.log(pairs), 1)[0] pred = np.pi * ellipk(1 - x) / ellipk(x) rates.append(rate) ratios.append(rate / pred) print(" d = %4d : x = %.5f, rate = %.4f, pred = %.4f, " "ratio = %.3f, ||C||_1 = %.4f, s1 = %.3e" % (d, x, rate, pred, rate / pred, sv.sum(), sv[0])) check("S3.1: NOME LADDER [N] -- sigma_k pairs fall exponentially in k " "with rate matching pi K(1-x)/K(x) to %.1f%%..%.1f%% over " "d = 16..512 (the free-fermion two-interval elliptic nome; pair " "near-degeneracy s1/s2 <= %.3f): Buchholz-Wichmann-type " "nuclearity of the coupling operator, quantitatively CFT-pinned" % (100 * (1 - max(ratios)), 100 * (1 - min(ratios)), max(pair_degen)), all(0.95 < r < 1.02 for r in ratios) and max(pair_degen) < 1.15) check("S3.2: DEEPER SPLIT WITH DISTANCE [N] -- the ladder rate grows " "monotonically %.3f -> %.3f as d widens (x falls): wider " "separation = faster eigenvalue decay = 'more split', as the " "nuclearity picture demands" % (rates[0], rates[-1]), all(rates[i + 1] > rates[i] for i in range(len(rates) - 1))) # sigma_1 ~ x^{1/2} ell = 64 xs, s1s = [], [] for d in [256, 384, 512, 768, 1024, 1536]: cuts = (0, ell, ell + d, 2 * ell + d) x, _ = cross_ratio(cuts, N) ia = list(range(ell)) ib = list(range(ell + d, 2 * ell + d)) sv = np.linalg.svd(G[np.ix_(ia, ib)], compute_uv=False) xs.append(x) s1s.append(sv[0]) sl = np.polyfit(np.log(xs), np.log(s1s), 1)[0] check("S3.3: LEADING COUPLING POWER LAW [N] -- sigma_1 ~ x^p with fitted " "p = %.4f vs 1/2 (the dimension-1/2 Majorana two-point function; " "at ell << d << N this is sigma_1 ~ sqrt(l_A l_B)/d): the leading " "coupling falls as a CFT power in the separation, exponentially " "in k -- witness vs proof separated in S6" % sl, abs(sl - 0.5) < 0.05) check("S3.4: TRACE-NORM DECAY [N] -- ||C||_1 = %.4f -> %.4f falls " "monotonically over d = 16..512: the coupling operator is trace " "class with shrinking norm (nuclearity proxy summable)" % (tnorms[0], tnorms[-1]), all(tnorms[i + 1] < tnorms[i] for i in range(len(tnorms) - 1))) # orbifold inheritance: identical sigma_k (S0.6) + compound (S0.5) + MI ell = 64 rows = [] for d in [256, 512, 1024, 1536]: cuts = (0, ell, ell + d, 2 * ell + d) x, _ = cross_ratio(cuts, N) ia = list(range(ell)) ib = list(range(ell + d, 2 * ell + d)) GA, GB = sub(G, ia), sub(G, ib) GAB = sub(G, ia + ib) I2f = S_R2(GA) + S_R2(GB) - S_R2(GAB) D2, _ = deficit2_flip(GAB, ell) rows.append((x, I2f, I2f - D2)) print(" orbifold Renyi-2 MI vs fermionic (ell = 64): " + " ".join( "x=%.4f: %.2e/%.2e" % (x, o, f) for (x, f, o) in rows)) ok_inh = all(o < 0.03 * f for (_, f, o) in rows) and \ all(rows[i + 1][2] < rows[i][2] for i in range(len(rows) - 1)) check("S3.5: ORBIFOLD INHERITANCE [E-machine + N] -- the sector-summed " "state has EXACTLY the fermionic sigma_k ladder (P_A flips " "C -> -C, S0.6) and its even-bilinear coupling Gram is Lambda^2 C " "(sigma_a sigma_b, S0.5) -- same ladder, doubled rate; " "numerically the orbifold Renyi-2 MI is < 3 percent of the " "fermionic one and falls with x: split scaling is INHERITED, as " "finite index demands (Longo heredity, [C])", ok_inh) # dimerised control: exponential sigma_1(d) Gt = ground_gamma(ring_A(512, t_weak=0.2)) ds = np.array([2, 4, 6, 8, 10, 12, 16]) s1d = [] for d in ds: ia = list(range(64)) ib = list(range(64 + d, 128 + d)) s1d.append(np.linalg.svd(Gt[np.ix_(ia, ib)], compute_uv=False)[0]) coef = np.polyfit(ds, np.log(s1d), 1) resid = np.abs(np.log(s1d) - np.polyval(coef, ds)).max() check("S3.6: MASSIVE CONTROL [N] -- dimerised chain: sigma_1(d) falls " "EXPONENTIALLY, rate %.3f per site (s1: %.1e -> %.1e over " "d = 2..16, log-linear residual %.2f): a gapped seam would split " "trivially and kill every CFT power law above -- the critical " "power laws are a property of the seam, not of the method" % (-coef[0], s1d[0], s1d[-1], resid), -coef[0] > 0.5 and resid < 0.3 and s1d[-1] < 1e-6) # =========================================================================== # S4 -- Pimsner-Popa / Longo-Rehren, exact # =========================================================================== def section4(): print(" -- S4: Pimsner-Popa bound and Longo-Rehren index, exact") rng = np.random.default_rng(4991) nq = 4 dim = 2 ** nq cs = jw_majoranas(nq) Pv = parity_first(nq, nq) def E2(y): return 0.5 * (y + (Pv[:, None] * y) * Pv[None, :]) a = rng.normal(size=(dim, dim)) + 1j * rng.normal(size=(dim, dim)) a = a @ a.conj().T f = monomial(cs, (0, 1), dim) g = monomial(cs, (2, 3), dim) dev_struct = max( np.abs(E2(np.eye(dim)) - np.eye(dim)).max(), abs(np.trace(E2(a)) - np.trace(a)), np.abs(E2(E2(a)) - E2(a)).max(), np.abs(E2(f @ a @ g) - f @ E2(a) @ g).max()) check("S4.1: E IS A CONDITIONAL EXPECTATION [E-machine] -- unital, " "trace-preserving, idempotent, even-bimodule (E(f a g) = " "f E(a) g for even f, g): max dev %.1e" % dev_struct, dev_struct < 1e-12) PaP = (Pv[:, None] * a) * Pv[None, :] dev_id = np.abs(E2(a) - 0.5 * a - 0.5 * PaP).max() eig_min = np.linalg.eigvalsh(E2(a) - 0.5 * a).min() check("S4.2: PP BOUND, ONE LINE [E] -- E(a) - a/2 = P a P / 2 " "IDENTICALLY (dev %.1e), and PaP >= 0 whenever a >= 0 (min eig " "%.2e): E(a) >= (1/2) a for ALL positive a -- the Pimsner-Popa " "inequality with lambda = 1/[F : F_even] = 1/2, exact" % (dev_id, eig_min), dev_id < 1e-12 and eig_min > -1e-12) # exact integer sweep, N = 8 Majoranas: all (even, odd) monomial pairs def exact_monomial(S): M = np.eye(dim, dtype=complex) for s_ in S: M = M @ cs[s_].toarray() return M evens = [c for r in range(0, 9, 2) for c in combinations(range(8), r)] odds = [c for r in range(1, 9, 2) for c in combinations(range(8), r)] mono_e = {S: exact_monomial(S) for S in evens} mono_o = {T: exact_monomial(T) for T in odds} I16 = np.eye(dim) n_ext = n_triv = n_bad = 0 for S in evens: cS = mono_e[S] for T in odds: cT = mono_o[T] M = cS.conj().T @ cT + cT.conj().T @ cS if not M.any(): n_triv += 1 continue ok = (np.array_equal(M @ M, 4 * I16) and np.trace(M) == 0 and np.array_equal((Pv[:, None] * M) * Pv[None, :], -M)) if ok: n_ext += 1 else: n_bad += 1 check("S4.3: ATTAINMENT, EXACT INTEGERS [E] -- sweep of ALL %d " "(even, odd) monomial pairs x = c_S + c_T at N = 8: %d pairs give " "x*x = 2 + M with M^2 = 4, Tr M = 0, PMP = -M EXACTLY (spectrum " "{+2, -2} balanced => E(x*x) - lam x*x loses positivity for any " "lam > 1/2), %d anticommuting pairs give x*x = 2 (ratio 1), 0 " "violations: lambda_PP = 1/2 EXACTLY = 1/index" % (len(evens) * len(odds), n_ext, n_triv), n_bad == 0 and n_ext + n_triv == 16384 and n_ext > 0) # N = 12: all hermitianised odd monomials + random pencil sweep nq6 = 6 dim6 = 64 cs6 = jw_majoranas(nq6) Pv6 = parity_first(nq6, nq6) I64 = np.eye(dim6) n_ok = 0 for r in range(1, 13, 2): for T in combinations(range(12), r): M = np.eye(dim6, dtype=complex) for s_ in T: M = M @ cs6[s_].toarray() if not np.array_equal(M.conj().T, M): M = 1j * M ok = (np.array_equal(M.conj().T, M) and np.array_equal(M @ M, I64) and np.trace(M) == 0 and np.array_equal((Pv6[:, None] * M) * Pv6[None, :], -M)) n_ok += int(ok) def E2_6(y): return 0.5 * (y + (Pv6[:, None] * y) * Pv6[None, :]) mins = 1.0 for _ in range(200): ve = rng.normal(size=dim6) + 1j * rng.normal(size=dim6) vo = rng.normal(size=dim6) + 1j * rng.normal(size=dim6) ve *= (Pv6 > 0) vo *= (Pv6 < 0) v = ve / np.linalg.norm(ve) + vo / np.linalg.norm(vo) mins = min(mins, pp_lambda(np.outer(v, v.conj()), E2_6)) for _ in range(40): r = rng.integers(1, dim6 + 1) X = rng.normal(size=(dim6, r)) + 1j * rng.normal(size=(dim6, r)) mins = min(mins, pp_lambda(X @ X.conj().T, E2_6)) check("S4.4: N = 12 EXACT + PENCIL SWEEP [E/N] -- all 2048 hermitianised " "odd monomials h_T satisfy h^2 = 1, Tr h = 0, PhP = -h exactly " "(%d/2048): each x = 1 + h_T attains lambda = 1/2; random pencil " "sweep (200 balanced rank-ones + 40 Wisharts, dim 64): " "min lambda = %.10f, never below 1/2" % (n_ok, mins), n_ok == 2048 and abs(mins - 0.5) < 1e-10) # two-interval group {1, P_A, P_B, P_AB}: lambda = 1/4 = 1/mu PA = parity_first(2, 4) PB = Pv * PA def E4(y): return 0.25 * (y + (PA[:, None] * y) * PA[None, :] + (PB[:, None] * y) * PB[None, :] + (Pv[:, None] * y) * Pv[None, :]) a4 = rng.normal(size=(dim, dim)) + 1j * rng.normal(size=(dim, dim)) a4 = a4 @ a4.conj().T eig4 = np.linalg.eigvalsh(E4(a4) - 0.25 * a4).min() secs = [] for (sa, sb) in [(1, 1), (1, -1), (-1, 1), (-1, -1)]: v = rng.normal(size=dim) * ((PA == sa) & (PB == sb)) secs.append(v / np.linalg.norm(v)) v4 = 0.5 * sum(secs) lam4 = pp_lambda(np.outer(v4, v4), E4) check("S4.5: TWO-INTERVAL GROUP [E] -- E_4 over {1, P_A, P_B, P_AB}: " "bound E_4(a) - a/4 >= 0 (min eig %.2e, identity = average of " "three positive conjugates) and the uniform 4-sector vector " "attains lambda = %.12f = 1/4 = 1/mu(SO(16)_1-class step): the " "Longo-Rehren index of the two-interval orbifold inclusion, " "exact on the lattice" % (eig4, lam4), eig4 > -1e-12 and abs(lam4 - 0.25) < 1e-12) # index consistency: entropic offset (alpha route) == 1/lambda_PP G512 = gamma_analytic(512) ia = list(range(0, 254)) ib = list(range(256, 510)) d16, _ = deficit2_flip(sub(G512, ia + ib), 254, layers=SEAM_LAYERS) check("S4.6: INDEX CONSISTENCY, TWO ROUTES [N/E] -- the WP5d-alpha " "(v501) entropic offset gives exp(Delta_inf) = %.9f (|Delta - " "ln 2| = %.1e at N = 512, %d layers) and the PP pencil gives " "1/lambda = 2 exactly; squared: exp(2 Delta) = %.6f = 1/lambda_E4 " "= 4 = mu: [F : F_even] = 2 per step, witnessed ENTROPICALLY and " "OPERATOR-ALGEBRAICALLY, independently" % (np.exp(d16), abs(d16 - LN2), SEAM_LAYERS, np.exp(2 * d16)), abs(d16 - LN2) < 1e-9 and abs(np.exp(2 * d16) - 4) < 1e-8) # U(1) control: lambda = 1/(m+1) -> 0 lams = [] for m in [2, 3, 4]: dimm = 2 ** m occ = np.array([bin(i).count("1") for i in range(dimm)]) def EU(y, occ=occ, dimm=dimm, m=m): out = np.zeros((dimm, dimm), complex) for q in range(m + 1): pq = (occ == q).astype(float) out += (pq[:, None] * y) * pq[None, :] return out v = np.zeros(dimm) for q in range(m + 1): v[np.where(occ == q)[0][0]] = 1 / np.sqrt(m + 1) lams.append((m, pp_lambda(np.outer(v, v), EU))) aU = rng.normal(size=(dimm, dimm)) aU = aU @ aU.T assert np.linalg.eigvalsh(EU(aU) - aU / (m + 1)).min() > -1e-12 check("S4.7: U(1) PP CONTROL [E] -- charge dephasing on m modes has " "m + 1 sectors: lambda = %s vs 1/(m+1) = %s (uniform-spread " "attainment, dephasing bound E(a) >= a/(m+1) verified): " "lambda -> 0 with region size -- INFINITE index, no uniform PP " "bound, the operator-algebraic twin of the S2.7 divergence" % (["%.6f" % l for (_, l) in lams], ["%.6f" % (1 / (m + 1)) for (m, _) in lams]), all(abs(l - 1 / (m + 1)) < 1e-11 for (m, l) in lams)) # =========================================================================== # S5 -- negative controls + finite-size honesty # =========================================================================== def section5(): print(" -- S5: negative controls and finite-size honesty") vals = [] for N in [256, 1024]: Gt = ground_gamma(ring_A(N, t_weak=0.2)) la = N // 4 GAB = sub(Gt, list(range(2 * la))) d2, R = deficit2_flip(GAB, la) SA = S_vN(sub(Gt, list(range(la)))) SB = S_vN(sub(Gt, list(range(la, 2 * la)))) SAB = S_vN(GAB) vals.append((N, d2, SA + SB - SAB, -np.log((1 + R ** SEAM_LAYERS) / 2))) drift = abs(vals[1][1] - vals[0][1]) mi_drift = abs(vals[1][2] - vals[0][2]) check("S5.1: TRIVIAL PHASE [N] -- dimerised chain: touching deficit " "Delta_2 = %.6f, N-INDEPENDENT (drift %.1e over 256 -> 1024, no " "approach to ln 2), touching MI = %.4f bounded (drift %.1e vs " "the critical ln2/6 = 0.1155 per doubling), 16-layer deficit " "%.4f << ln 2: every strong-additivity indicator is trivial in " "the gapped phase -- the witnesses measure the CRITICAL seam" % (vals[1][1], drift, vals[1][2], mi_drift, vals[1][3]), drift < 1e-9 and mi_drift < 1e-9 and vals[1][1] < 0.01) ds = [u1_touch_deficit(L)[0] for L in [8, 12, 16]] check("S5.2: SMALL-N HONESTY, U(1) [N] -- D_U1 at L = 8/12/16 = " "%.4f/%.4f/%.4f is NON-monotone (filling parity artefacts): " "below L ~ 16 the lattice cannot distinguish the nets; all " "scaling claims here start at L = 64 (documented, not hidden)" % tuple(ds), ds[1] > ds[0] and ds[1] > ds[2]) G4096 = gamma_analytic(4096) d2, _ = deficit2_flip(sub(G4096, list(range(2048))), 1024) rel = (LN2 - d2) / LN2 check("S5.3: SMALL-N HONESTY, Z2 [N] -- even at N = 4096 the touching " "deficit sits %.1f%% below ln 2 (the N^{-1/4} Ising approach is " "SLOW): the bounded-vs-divergent verdict rests on the fitted " "exponent + bound, not on visual saturation -- stated as " "witness, not proof" % (100 * rel), 0.10 < rel < 0.20) s = -np.ones(2048) GAB = sub(G4096, list(range(2048))) dev = np.abs(GAB * np.outer(s, s) - GAB).max() check("S5.4: WRONG PRESCRIPTION [E-machine] -- averaging the touching " "pair over {1, P_AB} only (2 of 4 structures) leaves rho " "INVARIANT (dev %.1e): deficit exactly 0 instead of -> ln 2 -- " "the bookkeeping is falsifiable, as in WP5d-alpha (v501) S6.3" % dev, dev < 1e-15) # =========================================================================== # S6 -- honest typing + verdict # =========================================================================== def section6(): print(" -- S6: honest typing") check("S6.1 [E]: exact ingredients -- the algebra spans/indices (shared " "site: full; disjoint: exactly 2; U(1): growing gap), the PP " "identities E(a) - a/2 = PaP/2 and E4(a) - a/4 >= 0, the integer " "attainment sweeps (16384 + 2048 monomials, 0 violations), " "lambda_E4 = 1/4, the U(1) lambda = 1/(m+1) family, the exact " "quadrature, the compound identity, sigma_k parity invariance", True) check("S6.2 [N]: scaling witnesses -- bounded Z2 touching deficit " "(< ln 2, exponent 1/4 approach) vs divergent U(1) deficit " "((1/2) ln Var Q growth, Klich-Levitov pinned); the nome ladder " "(within 2 percent of the elliptic prediction); sigma_1 ~ x^{1/2}; " "trace-norm decay; ED vN direction; gap continuity; the " "dimerised exponentials", True) check("S6.3 [C]/[O]: fences -- the DICTIONARY (bounded deficit = finite " "index = complete rationality is Longo-Xu; split heredity under " "finite index is Longo; the entropic order parameters are CHMP; " "the nome ladder is the free-fermion modular package) stays " "[C]; the continuum theorem 'finite-group orbifolds of " "completely rational nets are completely rational (hence split " "+ strongly additive)' is Xu's -- CITED, NOT CLAIMED; the " "continuum proof for THE seam quotient net and the interacting " "condensed (E8)_1 net stays [O] (WP5e / Costello-Li); " "SEAM.EQUIV.01 untouched, no marker moves", True) # =========================================================================== def run(): reset() print("v504 CELEST.WP5DB.01: split + strong additivity from the lattice " "(WP5d-beta of CELEST.SEAM.01)") print(" model: critical Majorana ring (NS), c = 1/2 per layer; orbifold " "= parity sector sum; U(1)/Dirac = negative control") section0() section1() section2() section3() section4() section5() section6() return summary("v504 CELEST.WP5DB.01: both remaining KLM legs witnessed for the " "orbifold prescription. STRONG ADDITIVITY: exact at the lattice " "level with the shared boundary point (Even(A) v Even(B) = " "Even(AuB), GF2 spans full, rank 32/32), index exactly 2 without " "it (the ln 2 bit of v501, localised at the split point); the " "entropic touching defect is BOUNDED by ln 2 with the Ising " "1/4-exponent approach (p = 0.2444) while the U(1) control " "DIVERGES ((1/2) ln Var Q, Klich-Levitov pinned, infinite index): " "bounded-vs-divergent is the lattice discriminator and the " "quotient net sits on the bounded side. SPLIT: sigma_k fall " "exponentially at the elliptic-nome rate pi K(1-x)/K(x) " "(1.3-2.0%), sigma_1 ~ x^{1/2}, trace norm summable; the orbifold " "inherits the SAME ladder exactly (C -> -C machine identity + " "compound Lambda^2 C) -- Longo heredity. PIMSNER-POPA: " "E(a) - a/2 = PaP/2 identically, integer attainment (16384 + " "2048 monomials, 0 violations), lambda = 1/2 = 1/[F:F_even], " "lambda_E4 = 1/4 = 1/mu, exp(Delta_inf) = 2 = 1/lambda -- the " "Longo-Rehren index pinned by two independent routes. With v501: " "all THREE KLM ingredients of complete rationality witnessed on " "the lattice. Continuum stays [C] (dictionary) / [O] (Xu's " "orbifold theorem CITED, not claimed; the seam quotient net and " "the condensed (E8)_1 net are WP5e/Costello-Li). " "SEAM.EQUIV.01 untouched") if __name__ == "__main__": raise SystemExit(1 if run() else 0)