""" n13 — The shadow kill-gate. Does pure dephasing (elastic scattering that creates records, zero mean absorption) produce a directed force between two scatterers in the vacuum of a 1D scalar field? Le Sage gravity, quantum edition: if "decoherence shadows" could attract, gravity could be the vacuum's bookkeeping of records. Pre-registered criteria: method must reproduce static Casimir vs -dE0/dd to ~10%; single-grain null; verdict on sign, gamma_A*gamma_B scaling, d-dependence. VERDICT (July 2026): REFUTED. Dephasing grains repel (the repulsion is proportional to the medium's absorption and vanishes in the transparent limit); a real thermal bath makes it worse; the SAME scatterer, coherent, attracts (Casimir). Maxwell's theorem survives decoherence: what a grain rediffuses with scrambled phase carries exactly the momentum it removed. Decoherence does not create attraction -- it destroys it. Exact NESS via Lyapunov equations; "vacuum continuation" boundary baths (undriven NESS = exact ground state to 7e-13). Force read from the ABBA- subtracted momentum-flux (T^xx) discontinuity across each grain. Runtime: ~1 minute (numpy + scipy). """ import numpy as np from scipy.linalg import solve_continuous_lyapunov, eigh N = 256 MU2 = 4e-4 LABS = 28 ETA_MAX = 0.5 ETA_BULK = 0.005 CENTER = N // 2 SMEAR = 2.0 OFF = 8 # bonds de mesure hors du support du grain (+-8 sites) def K_matrix(): return np.diag(np.full(N, 2.0 + MU2)) \ - np.diag(np.ones(N - 1), 1) - np.diag(np.ones(N - 1), -1) def weight(site): """Grain = indice fluctuant couple a la deformation locale : V = (c^T x)^2/2 avec c = gradient discret d'une gaussienne. Somme nulle (IR-sur), lisse (pas de bord de bande).""" idx = np.arange(N) g = np.exp(-0.5 * ((idx - site) / SMEAR) ** 2) g[np.abs(idx - site) > 4 * SMEAR] = 0.0 c = np.zeros(N) c[:-1] = np.diff(g) return c / np.linalg.norm(c) def ground_cov(K, temp=0.0): w2, U = eigh(K) w = np.sqrt(np.maximum(w2, 1e-12)) occ = 0.5 * np.ones(N) if temp == 0 else 0.5 / np.tanh(0.5 * w / temp) sig = np.zeros((2 * N, 2 * N)) sig[:N, :N] = U @ np.diag(occ / w) @ U.T sig[N:, N:] = U @ np.diag(occ * w) @ U.T return sig def build_A_D(K, eta_bulk, temp): eta = np.full(N, eta_bulk) 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, temp) D = 0.5 * (E @ sgs + sgs @ E) return A, D, sgs def ness(K, noisy=(), eta_bulk=ETA_BULK, temp=0.0): """noisy = ((site, gamma), ...) ; grains etales de profil `weight`.""" A, D, sgs = build_A_D(K, eta_bulk, temp) sig0 = solve_continuous_lyapunov(A, -D) if not noisy: return sig0, np.zeros(0) gammas = np.array([g for _, g in noisy]) uvecs, vvecs, parts = [], [], [] for s, _ in noisy: w = weight(s) u = np.concatenate([w, np.zeros(N)]) # mesure v = np.concatenate([np.zeros(N), w]) # diffusion sur p_tilde uvecs.append(u); vvecs.append(v) parts.append(solve_continuous_lyapunov(A, -np.outer(v, v))) n = len(noisy) b = np.array([u @ sig0 @ u for u in uvecs]) M = np.array([[gammas[j] * (uvecs[i] @ parts[j] @ uvecs[i]) for j in range(n)] for i in range(n)]) ev = np.linalg.eigvals(M) if np.max(ev.real) >= 1.0: raise RuntimeError(f"instabilite parametrique (max eig M = {np.max(ev.real):.3f})") svals = np.linalg.solve(np.eye(n) - M, b) sig = sig0 + sum(gammas[j] * svals[j] * parts[j] for j in range(n)) return sig, svals def stress_profile(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_att(T_by, a, b, off=OFF): dT = T_by["AB"] - T_by["A0"] - T_by["0B"] + T_by["00"] FA = dT[a - off] - dT[a + off - 1] FB = dT[b + off - 1] - dT[b - off] return 0.5 * (FA + FB), FA, FB def run_pair(K, a, b, gA, gB, eta_bulk=ETA_BULK, temp=0.0): T_by = {} svals = None for tag in ("00", "A0", "0B", "AB"): noisy = [] if "A" in tag: noisy.append((a, gA)) if "B" in tag: noisy.append((b, gB)) sig, s = ness(K, tuple(noisy), eta_bulk, temp) if tag == "AB": svals = s T_by[tag] = stress_profile(sig, K) F, FA, FB = force_att(T_by, a, b) return F, FA, FB, svals if __name__ == "__main__": K = K_matrix() print("--- test nul (grain unique etale, gamma=0.01) ---") a = CENTER sig, s = ness(K, ((a, 0.01),)) sig0, _ = ness(K) T = stress_profile(sig, K) - stress_profile(sig0, K) print(f" = {s[0]:.4f}, rayonnement DT = {T[a-OFF]:.3e}, " f"parasite = {T[a-OFF]-T[a+OFF-1]:.3e}\n") print("--- deux grains dephasants, vide (gamma=0.01) ---") print(f"{'d':>4} {'F_att':>12} {'P_pompe':>10}") for d in (16, 24, 32, 48, 64): aa, bb = CENTER - d // 2, CENTER - d // 2 + d F, FA, FB, s = run_pair(K, aa, bb, 0.01, 0.01) print(f"{d:>4} {F:>12.4e} {0.005*s.sum():>10.3e}") print() print("--- scaling en gamma (d=24) ---") print(f"{'gA':>7} {'gB':>7} {'F_att':>12} {'F/(gA*gB)':>12}") aa, bb = CENTER - 12, CENTER + 12 for gA, gB in ((0.0025, 0.0025), (0.005, 0.005), (0.01, 0.01), (0.02, 0.02), (0.02, 0.005)): F, *_ = run_pair(K, aa, bb, gA, gB) print(f"{gA:>7} {gB:>7} {F:>12.4e} {F/(gA*gB):>12.4e}") print() print("--- dependance a l'absorption (d=24, gamma=0.01) ---") print(f"{'eta_bulk':>9} {'F_att':>12}") for eb in (0.02, 0.01, 0.005, 0.0025, 0.00125): F, *_ = run_pair(K, aa, bb, 0.01, 0.01, eta_bulk=eb) print(f"{eb:>9} {F:>12.4e}") print() print("--- bain reel thermique (d=24, gamma=0.005) ---") print(f"{'T':>6} {'F_att':>12} {'P_pompe':>10}") for temp in (0.0, 0.25, 0.5, 1.0, 2.0): F, FA, FB, s = run_pair(K, aa, bb, 0.005, 0.005, temp=temp) print(f"{temp:>6} {F:>12.4e} {0.0025*s.sum():>10.3e}")