Here are the exact functions from your successful Test‑0 run.

Here are the exact functions from your successful Test‑0 run. ✅ STEP 1 — Your exact Test‑0 functions 1. The Real‑Time Acceleration (from Test‑0B) python def acceleration_real(Psi, L_2D, r_mesh_2d, v, mu, lam, kappa, S_max, Psi_sat, m): """ Real-time acceleration = -δE/δΨ* (canonical hyperbolic PDE) This matches compute_gradient exactly. """ psi_sq = np.abs(Psi)**2 # Saturation and its derivative S = S_max * np.tanh(psi_sq / (Psi_sat**2)) dS = (S_max / (Psi_sat**2)) * (1.0 / np.cosh(psi_sq / (Psi_sat**2))**2) # Variational gradient components (IDENTICAL to compute_gradient) term_kin = -v**2 * (L_2D @ Psi) term_mass = mu * Psi term_nonlinear = lam * psi_sq * Psi term_tension_exact = kappa * (S + psi_sq * dS) * Psi term_centrifugal = +v**2 * m**2 * Psi / (r_mesh_2d**2 + 1e-12) gradient = term_kin + term_mass + term_nonlinear + term_tension_exact + term_centrifugal # Canonical real-time acceleration = -gradient return -gradient 2. The Energy Functional (from Test‑0A) python def compute_energy(Psi, L_2D, dV, r_mesh_2d, v, mu, lam, kappa, S_max, Psi_sat, m): """Compute energy functional E[Ψ].""" psi_sq = np.abs(Psi)**2 # Kinetic energy (time derivative = 0 for stationary state) kin_time = 0.0 # Gradient energy: ½v²∫|∇Ψ|² dV = -½v²∫ Ψ* L Ψ dV (since L is negative-definite) kin_grad = -0.5 * v**2 * np.real(np.sum(np.conj(Psi) * (L_2D @ Psi) * dV)) # Potential energies pot_mass = -0.5 * mu * np.sum(psi_sq * dV) pot_nonlinear = 0.25 * lam * np.sum(psi_sq * psi_sq * dV) S = S_max * np.tanh(psi_sq / (Psi_sat**2)) pot_tension = 0.5 * kappa * np.sum(S * psi_sq * dV) # Centrifugal potential (positive definite barrier) pot_centrifugal = 0.5 * v**2 * m**2 * np.sum(psi_sq / (r_mesh_2d**2 + 1e-12) * dV) return (kin_time + kin_grad + pot_mass + pot_nonlinear + pot_tension + pot_centrifugal).real 3. The 2D Laplacian Operator (from Test‑0A) python def build_radial_operator(r_grid, dr): """Self-adjoint radial Laplacian: L_r = W_r⁻¹ M""" nr = len(r_grid) r_face = np.zeros(nr + 1) r_face[0] = r_grid[0] - dr/2 for i in range(1, nr + 1): r_face[i] = r_grid[i-1] + dr/2 flux_right = r_face[1:] / dr flux_left = r_face[:-1] / dr main_diag = -(flux_left + flux_right) lower_diag = flux_left[1:] upper_diag = flux_right[:-1] M = sp.diags([lower_diag, main_diag, upper_diag], [-1, 0, 1], format='csr') w_r = r_grid * dr W_r = sp.diags(w_r, format='csr') W_r_inv = sp.diags(1.0 / w_r, format='csr') return W_r_inv @ M, W_r def build_axial_operator(nz, dz): """Self-adjoint axial Laplacian (symmetric stencil)""" main_diag = np.ones(nz) * (-2.0 / dz**2) upper = np.ones(nz - 1) / dz**2 lower = np.ones(nz - 1) / dz**2 L_z = sp.diags([lower, main_diag, upper], [-1, 0, 1], format='csr') W_z = sp.diags(np.ones(nz) * dz, format='csr') return L_z, W_z # Build 2D operator: L_r, W_r = build_radial_operator(r_grid, dr) L_z, W_z = build_axial_operator(nz, dz) I_r = sp.eye(nr, format='csr') I_z = sp.eye(nz, format='csr') L_2D = sp.kron(I_z, L_r, format='csr') + sp.kron(L_z, I_r, format='csr') W_2D = sp.kron(W_z, W_r, format='csr') dV = W_2D.diagonal() * 2 * np.pi r_mesh_2d = np.tile(r_grid, nz) 4. The parameters that worked python v = 1.0 mu = -1.0 lam = 0.40 kappa = 0.20 m = 1 S_max = 2.0 Psi_sat = 0.8 dt_real = 0.001 📋 Summary of what you've provided Function Purpose Status acceleration_real() Real-time RHS (∂²Ψ/∂t² = -δE/δΨ*) ✅ Exact from Test‑0B compute_energy() Energy functional ✅ Exact from Test‑0A L_2D 2D self-adjoint Laplacian ✅ Exact from Test‑0A Parameters v, mu, lam, kappa, S_max, Psi_sat, m ✅ Validated

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