""" n17 — The modular test: does the thermodynamic (Jacobson/Verlinde) form of "gravity = entanglement bookkeeping" survive the intervention that killed E_N and I as force carriers (n16)? MEASURED (July 2026), three verdicts: P1. The first law holds exactly: dS(A) = d to 0.1% (thermal and vacuum, ratio -> 1.000000 for infinitesimal perturbations). The Gaussian modular Hamiltonian is G = 2 arccoth(2 i Omega sigma) i Omega -- note the operator ORDER: our first implementation had i Omega on the left, wrong for noncommuting cases; caught against the exact thermal-chain case (G = blockdiag(K, I)/T, matched to 1e-15) and kept on the record. P2. No modular temperature exists even statically: T_eff = F/(dS_screen/dd) drifts monotonically x1.9 over d = 14-26. The entropic force law F = T dS/dx has no universal T in this arena. P3 + 5b. The last live thread -- the screen-entropy gradient flips sign WITH the true force under mid-channel noise (4/4) -- was executed by moving the noise outside the pair: sign concordance breaks (5/6 disagreements across the two outside geometries). One more common cause: both quantities tracked the radiation gradient, not each other. CONCLUSION: in this arena the force has no information-theoretic carrier at all. Entanglement (n16), total correlations (n16), and entropy gradients (here) are all non-causal shadows of the propagator that carries the force. What survives of book one: the first law as an exact identity, and whatever interacting/holographic regimes this Gaussian 1D arena cannot reach. Runtime: ~5 minutes. """ import numpy as np from scipy.linalg import solve_continuous_lyapunov, eigh N = 256 MU2 = 2.5e-3 LABS = 28 ETA_MAX = 0.5 ETA_B = 0.005 LAM = 1.0 A0 = 100 CUT = 111 # ecran = sites [0, CUT) MIDS = [109, 110, 111, 112] # ------------------------------------------------------------- modele def K_matrix(defects=()): K = np.diag(np.full(N, 2.0 + MU2)) \ - np.diag(np.ones(N - 1), 1) - np.diag(np.ones(N - 1), -1) for s in defects: K[s, s] += LAM return K def ground_cov(K): w2, U = eigh(K) w = np.sqrt(np.maximum(w2, 1e-14)) sig = np.zeros((2 * N, 2 * N)) sig[:N, :N] = U @ np.diag(0.5 / w) @ U.T sig[N:, N:] = U @ np.diag(0.5 * w) @ U.T return sig def ground_E0(K): w2 = eigh(K, eigvals_only=True) return 0.5 * np.sum(np.sqrt(np.maximum(w2, 0))) def ness(K, gamma=0.0): eta = np.full(N, ETA_B) for j in range(LABS): s = ((LABS - j) / LABS) ** 2 eta[j] += ETA_MAX * s eta[N - 1 - j] += ETA_MAX * s E = np.diag(np.concatenate([eta, eta])) A_H = np.zeros((2 * N, 2 * N)) A_H[:N, N:] = np.eye(N) A_H[N:, :N] = -K A = A_H - E / 2 sgs = ground_cov(K) D = 0.5 * (E @ sgs + sgs @ E) for i in MIDS: D[N + i, N + i] += gamma return solve_continuous_lyapunov(A, -D) # ------------------------------------------------------- info gaussienne def block_cov(sig, sites): idx = list(sites) + [N + s for s in sites] return sig[np.ix_(idx, idx)] def symp_eigs(sr): n = len(sr) // 2 Om = np.zeros((2 * n, 2 * n)) Om[:n, n:] = np.eye(n); Om[n:, :n] = -np.eye(n) ev = np.linalg.eigvals(1j * Om @ sr) nu = np.sort(np.abs(ev)) return nu[::2] def entropy(sr): S = 0.0 for v in symp_eigs(sr): v = max(v, 0.5 + 1e-12) S += (v + .5) * np.log(v + .5) - (v - .5) * np.log(v - .5) return S def modular_G(sr): """G tel que rho ~ exp(-1/2 X^T G X) : G = 2 i Omega arccoth(2 i Omega sigma).""" n = len(sr) // 2 Om = np.zeros((2 * n, 2 * n)) Om[:n, n:] = np.eye(n); Om[n:, :n] = -np.eye(n) M = 1j * Om @ sr lam, V = np.linalg.eig(M) lam = np.where(np.abs(lam) < 0.5 + 1e-10, np.sign(lam.real) * (0.5 + 1e-10), lam) f = np.arctanh(1.0 / (2 * lam)) # arccoth(2x) = artanh(1/(2x)) Gm = V @ np.diag(f) @ np.linalg.inv(V) return (2 * Gm @ (1j * Om)).real # G = 2 f(i Omega sigma) i Omega # ------------------------------------------------------- force (stress) def stress(sig, K): sxx = sig[:N, :N]; spp = sig[N:, N:] pp = 0.5 * (np.diag(spp)[:-1] + np.diag(spp)[1:]) dx2 = np.diag(sxx)[:-1] + np.diag(sxx)[1:] - 2 * np.diag(sxx, 1) xx = 0.5 * (np.diag(sxx)[:-1] + np.diag(sxx)[1:]) return 0.5 * pp + 0.5 * dx2 - 0.5 * MU2 * xx def force_on_B(d, gamma, off=5): """Force d'interaction sur B (>0 = vers A = attraction), ABBA.""" b = A0 + d T = {} for tag, dl in (("00", ()), ("A0", (A0,)), ("0B", (b,)), ("AB", (A0, b))): K = K_matrix(dl) T[tag] = stress(ness(K, gamma), K) dT = T["AB"] - T["A0"] - T["0B"] + T["00"] return dT[b + off - 1] - dT[b - off] # ===================================================================== if __name__ == "__main__": # ---- P1 : premiere loi, etat fondamental exact ---- print("--- P1. premiere loi dS(A) = dTr(G_A sigma)/2, bloc de 12 autour de A0 ---") Ablk = list(range(A0 - 6, A0 + 6)) d0 = 18 sig0 = ground_cov(K_matrix((A0, A0 + d0))) G_A = modular_G(block_cov(sig0, Ablk)) for dd in (1, 2): sp = block_cov(ground_cov(K_matrix((A0, A0 + d0 + dd))), Ablk) sm = block_cov(ground_cov(K_matrix((A0, A0 + d0 - dd))), Ablk) dS = (entropy(sp) - entropy(sm)) / 2 dK = 0.5 * np.trace(G_A @ (sp - sm)) / 2 print(f" delta={dd}: dS = {dS:+.6e}, d = {dK:+.6e}, ratio = {dS/dK:.4f}") print() # ---- P2 : statique — S_ecran(d) vs F(d) ---- print("--- P2. statique : F = T_eff dS_ecran/dd ? (A fixe, B mobile, coupure fixe) ---") print(f"{'d':>4} {'F (dE0/dd)':>12} {'dS/dd':>12} {'T_eff=F/(dS/dd)':>16}") screen = list(range(CUT)) stat = {} for d in (14, 16, 18, 20, 22, 24, 26): E = {dd: ground_E0(K_matrix((A0, A0 + dd))) for dd in (d - 1, d + 1)} F = (E[d + 1] - E[d - 1]) / 2 S = {dd: entropy(block_cov(ground_cov(K_matrix((A0, A0 + dd))), screen)) for dd in (d - 1, d + 1)} dS = (S[d + 1] - S[d - 1]) / 2 stat[d] = (F, dS) print(f"{d:>4} {F:>12.4e} {dS:>12.4e} {F/dS if abs(dS)>0 else float('nan'):>16.4e}") Teff = np.median([F / dS for F, dS in stat.values()]) print(f" T_eff (mediane) = {Teff:.4e}") print() # ---- P3 : intervention ---- print("--- P3. intervention : F_vraie vs F_mod = T_eff dS/dd (d=18) ---") print(f"{'gamma':>8} {'F_vraie':>12} {'dS/dd':>12} {'F_mod':>12} {'meme signe?':>12}") d = 18 for g in (0.0, 0.01, 0.03, 0.1): F = force_on_B(d, g) Ss = {dd: entropy(block_cov(ness(K_matrix((A0, A0 + dd)), g), screen)) for dd in (d - 1, d + 1)} dS = (Ss[d + 1] - Ss[d - 1]) / 2 Fm = Teff * dS print(f"{g:>8} {F:>12.4e} {dS:>12.4e} {Fm:>12.4e} {str(np.sign(F)==np.sign(Fm)):>12}") # ---- 5b : le dephaseur HORS de la paire (test de cause commune) ---- print("--- 5b. dephaseur exterieur : la concordance de signe casse ---") Teff = 1.7312e-2 for name, mids in (("gauche de A", [90, 91, 92, 93]), ("droite de B", [124, 125, 126, 127]), ("centre (rappel P3)", [109, 110, 111, 112])): globals()['MIDS'][:] = mids print(f" {name}:") for g in (0.01, 0.03, 0.1): F = force_on_B(18, g) Ss = {dd: entropy(block_cov(ness(K_matrix((A0, A0 + dd)), g), screen)) for dd in (17, 19)} dS = (Ss[19] - Ss[17]) / 2 print(f" gamma={g:<5} F={F:+.3e} T_eff*dS/dd={Teff*dS:+.3e} " f"meme signe: {np.sign(F) == np.sign(dS)}")