"""v476 -- GRAV.ENTROPIC.COMPRESS.01: the AP2 compression conjecture of the entropic-action bridge (v473) made WELL-POSED on an exactly solvable model -- the naive operator-side reading is shown to be ill-posed, the state-side reading is forced, and the mismatch between the two is exactly quadratic in the cross-cut correlations and gap-suppressed. AP2 itself stays [O]; this module fixes WHAT the conjecture must say, it does not prove it. Background. v473 recorded the bridge conjecture as P_Sigma (G~ g~^-1) P_Sigma = Delta_Sigma^{1/2}: compress Bianconi's relative metric operator to the seam and get the TFPT modular operator. On a quasi-free (Gaussian) system everything is computable: a state is its one-particle covariance C, the boundary restriction is the compression C_A = P C P (v155), and the boundary modular data is Peschel's K = ln((1-C_A)/C_A), i.e. the one-particle modular operator is Delta_A = f(C_A) with f(x) = x/(1-x). For a THERMAL bulk the bulk analog is exact: f(C_T) = e^{-h/T} -- the Bianconi-shaped "exponential of the modular Hamiltonian". The conjecture's two possible readings differ by an order of operations: f(P C P) (build the modular operator FROM the compressed state) vs P f(C) P (compress the bulk modular operator). [N] 1. PURE GAPPED BULK. Staggered-mass hopping chain (N = 16), ground state covariance C = projection onto negative modes: C^2 = C to 1e-12, gap 2m verified. The physical seam situation: the bulk is PURE, the boundary is thermal (Bisognano-Wichmann shape). [N] 2. COMPRESSION IS THERMAL (v155 in the bridge context). C_A = P C P has spectrum strictly inside (0,1); Peschel K finite; entanglement entropy > 0 -- the compression of a projection is a strict contraction, the boundary state is genuinely KMS-like. [N] 3. THE ORDER OBSTRUCTION (the reading DECIDED). On the pure bulk, spec(C) = {0,1}, so f(x) = x/(1-x) is SINGULAR on the spectrum: P f(C) P -- the literal v473 operator-side reading -- is ILL-POSED, while f(P C P) is finite. Hence the compression identity can only be read STATE-SIDE: build Delta_Sigma from the COMPRESSED relative metric. This matches Bianconi's own construction (her G~ is a LOCAL field, i.e. already the compressed datum) -- AP2 is retyped, not weakened. [E] 4. BLOCK ALGEBRA: THE MISMATCH IS EXACTLY SECOND ORDER. For block matrices, [C^2]_AA - (C_AA)^2 = C_AB C_BA and [C^3]_AA - (C_AA)^3 = C_AA C_AB C_BA + C_AB C_BA C_AA + C_AB C_BB C_BA (symbolic, exact): for every analytic f the two readings agree up to SECOND order in the cross-cut block -- the modular (entanglement) content lives exactly in that quadratic correction; first order vanishes identically. [N] 5. GAP CONTROL AT THE BOUNDED LEVEL (the TFPT regime). The clean comparison lives at the COVARIANCE level, where both readings are bounded in [0,1]: state-side [Fermi(h/T)]_AA (compress the state) vs operator-side Fermi(h_AA/T) (build the state from the compressed generator). Thermal bulk (T = 1): the bulk identity f(C_T) = e^{-h/T} holds to 1e-10; the mismatch obeys the quadratic law d <= x^2 (d/x^2 in [0.4, 1.0], bounded) and falls with the gap in the gapped regime (m >= 2), the cross-cut norm x falls monotonically; the decoupled cut ([h,P] = 0) agrees EXACTLY. The unbounded modular operator Delta = e^{-h/T} shows the same quadratic structure with exponentially amplified absolute norms -- recorded: the compression identity must be stated at the bounded (state/covariance) level, its modular content follows spectrally. [C] 6. WHAT THIS DOES AND DOES NOT DO. AP2 is now WELL-POSED: state-side reading at the bounded level, mismatch quadratic in cross-cut correlations, gap-suppressed. NOT proven: the continuum seam statement (the Calderon compression of the actual relative metric operator equals the actual seam modular operator). AP2 stays [O]; no marker moves. NET TYPING: [E] the block-algebra identity (sympy, symbolic blocks); [N] the model exhibits (numpy, gapped chain); [C] the retyped conjecture; [O] AP2 itself. Extends GRAV.ENTROPIC.ACTION.01 (v473). Python-only by nature (matrix functions on a lattice model); the exact block identity is trivial algebra, flagged in the Wolfram README. """ import numpy as np import sympy as sp from tfpt_constants import check, summary, reset N = 16 # chain length (even) NA = 6 # seam block size def chain_h(m, n=N): """Single-particle Hamiltonian: hopping + staggered mass (gap 2m).""" h = np.zeros((n, n)) for i in range(n - 1): h[i, i + 1] = h[i + 1, i] = 1.0 for i in range(n): h[i, i] = m * (-1) ** i return h def ground_cov(h): """Covariance of the filled negative-energy sea: projection.""" w, v = np.linalg.eigh(h) neg = v[:, w < 0] return neg @ neg.T def thermal_cov(h, T): w, v = np.linalg.eigh(h) occ = 1.0 / (1.0 + np.exp(w / T)) return (v * occ) @ v.T def matfun(C, fun): w, v = np.linalg.eigh(C) return (v * fun(w)) @ v.T def run(): reset() print("v476 GRAV.ENTROPIC.COMPRESS.01: the AP2 compression conjecture made well-posed (toy exhibit)") # 1. pure gapped bulk (moderate mass: gapped AND visibly entangled boundary) m0 = 0.5 h = chain_h(m0) w = np.linalg.eigvalsh(h) gap = w[w > 0].min() - w[w < 0].max() C = ground_cov(h) purity = float(np.abs(C @ C - C).max()) check("PURE GAPPED BULK [N]: staggered-mass chain (N = %d, m = %.1f), spectral " "gap %.2f >= 2m = %.1f; ground-state covariance is a PROJECTION, " "|C^2 - C| = %.1e -- the physical seam situation: pure bulk, thermal " "boundary (Bisognano-Wichmann shape)" % (N, m0, gap, 2 * m0, purity), purity < 1e-12 and gap >= 2 * m0 - 1e-9) # 2. compression is thermal CA = C[:NA, :NA] lam = np.linalg.eigvalsh(CA) dist = float(min(lam.min(), (1 - lam).min())) ent = float(-(lam * np.log(np.clip(lam, 1e-300, 1)) + (1 - lam) * np.log(np.clip(1 - lam, 1e-300, 1))).sum()) K = matfun(CA, lambda x: np.log((1 - x) / x)) check("COMPRESSION IS THERMAL [N] (v155 in the bridge context): C_A = P C P has " "spectrum strictly inside (0,1) (distance %.1e > 0), the Peschel modular " "Hamiltonian K = ln((1-C_A)/C_A) is finite (|K| = %.1f), entanglement " "entropy S = %.4f > 0 -- the boundary state is genuinely KMS-like" % (dist, float(np.abs(K).max()), ent), dist > 1e-14 and ent > 1e-8 and np.isfinite(K).all()) # 3. the order obstruction: P f(C) P ill-posed on the pure bulk spec_dist = float(np.min(np.minimum(np.abs(np.linalg.eigvalsh(C)), np.abs(np.linalg.eigvalsh(C) - 1)))) delta_state = matfun(CA, lambda x: x / (1 - x)) check("ORDER OBSTRUCTION [N] (the reading DECIDED): the pure bulk has spec(C) " "= {0,1} to %.1e, so f(x) = x/(1-x) is SINGULAR on the spectrum -- the " "literal operator-side reading P f(C) P of the v473 conjecture is " "ILL-POSED, while the state-side f(P C P) is finite (|Delta_A| = %.1f): " "the compression identity can only be read STATE-SIDE (build Delta_Sigma " "from the COMPRESSED relative metric) -- matching Bianconi's own local-" "metric construction; AP2 retyped, not weakened" % (spec_dist, float(np.abs(delta_state).max())), spec_dist < 1e-12 and np.isfinite(delta_state).all()) # 4. block algebra: mismatch exactly second order in the cross block (symbolic) nA, nB = 2, 2 A = sp.MatrixSymbol('A', nA, nA) B = sp.MatrixSymbol('B', nA, nB) Bt = sp.MatrixSymbol('Bt', nB, nA) D = sp.MatrixSymbol('D', nB, nB) Cb = sp.BlockMatrix([[A, B], [Bt, D]]) sq = sp.block_collapse(Cb * Cb) cu = sp.block_collapse(Cb * Cb * Cb) mis2 = sp.expand(sq.blocks[0, 0] - A ** 2) mis3 = sp.expand(cu.blocks[0, 0] - A ** 3) check("BLOCK ALGEBRA [E] (symbolic): [C^2]_AA - (C_AA)^2 = %s and [C^3]_AA - " "(C_AA)^3 = %s -- every term carries the cross block TWICE, first order " "vanishes identically: for ANY analytic f the two readings agree up to " "SECOND order in the cross-cut correlations; the modular (entanglement) " "content lives exactly in that quadratic correction" % (mis2, mis3), mis2 == sp.expand(B * Bt) and mis3 == sp.expand(A * B * Bt + B * Bt * A + B * D * Bt)) # 5. gap control at the bounded (covariance) level T = 1.0 f = lambda x: x / (1 - x) deltas, crosses, ratios = [], [], [] bulk_id_err = 0.0 masses = (0.5, 1.0, 2.0, 4.0, 8.0) for m in masses: hm = chain_h(m) CT = thermal_cov(hm, T) wm, vm = np.linalg.eigh(hm) bulk_mod = (vm * np.exp(-wm / T)) @ vm.T # e^{-h/T} rel = float(np.abs(matfun(CT, f) - bulk_mod).max() / np.abs(bulk_mod).max()) bulk_id_err = max(bulk_id_err, rel) state_side = CT[:NA, :NA] # compress the state op_side = thermal_cov(hm[:NA, :NA], T) # state of the compressed generator d = float(np.linalg.norm(state_side - op_side, 2)) x = float(np.linalg.norm(CT[:NA, NA:], 2)) deltas.append(d) crosses.append(x) ratios.append(d / x ** 2) x_mono = all(crosses[i] > crosses[i + 1] for i in range(len(masses) - 1)) d_gapped = deltas[2] > deltas[3] > deltas[4] # falls in the gapped regime quad_law = max(ratios) <= 1.1 # d <= x^2 (quadratic law) # decoupled control: [h, P] = 0 => exact agreement hd = chain_h(2.0) hd[NA - 1, NA] = hd[NA, NA - 1] = 0.0 d_dec = float(np.linalg.norm(thermal_cov(hd, T)[:NA, :NA] - thermal_cov(hd[:NA, :NA], T), 2)) check("GAP CONTROL AT THE BOUNDED LEVEL [N] (the TFPT regime): thermal bulk " "(T = 1) obeys the Bianconi-shaped bulk identity f(C_T) = e^{-h/T} to " "relative %.1e; at the covariance level the mismatch d = ||[Fermi(h/T)]_AA - " "Fermi(h_AA/T)|| obeys the QUADRATIC law d <= x^2 (d/x^2 = %s) and falls " "with the gap in the gapped regime %s while the cross-cut norm x falls " "monotonically %s; decoupled cut => EXACT agreement (%.1e) -- the two " "readings converge in the gap-dominated regime, where TFPT operates " "(Delta_eff > 0, v76/v337)" % (bulk_id_err, ["%.2f" % r for r in ratios], ["%.1e" % d for d in deltas], ["%.1e" % x for x in crosses], d_dec), bulk_id_err < 1e-10 and x_mono and d_gapped and quad_law and d_dec < 1e-12) # 6. honest scope check("SCOPE [C/O]: AP2 is now WELL-POSED -- state-side reading forced, " "mismatch exactly quadratic in cross-cut correlations, gap-suppressed; " "NOT proven: the continuum seam statement (Calderon compression of the " "actual relative metric = the actual seam modular operator). AP2 stays " "[O]; no status marker moves", True) return summary("v476 GRAV.ENTROPIC.COMPRESS.01: AP2 made well-posed -- [N] pure gapped " "bulk (C^2 = C), compression thermal (Peschel K finite, S > 0), ORDER " "OBSTRUCTION: P f(C) P ill-posed on the pure bulk => state-side reading " "FORCED (matches Bianconi's local construction); [E] block algebra: " "mismatch exactly second order in the cross block; [N] gap control: " "f(C_T) = e^{-h/T} exact, mismatch quadratic + falling with the gap, " "decoupled cut exact; [C] retyped conjecture; [O] AP2 open, no marker moves") if __name__ == "__main__": raise SystemExit(1 if run() else 0)