Prescribed-Order Equilibrium Vector Field

(1) Integrate the construction into the drift function of a Neural ODE, latent world model, or residual recurrent block. Choose a known equilibrium or anchor state c and compute z = x - c. Enumerate A_k = {alpha in N^n : |alpha| = k}. The coefficient network outputs one n-dimensional vector g_alpha(z) for each alpha. The final vector field is the sum of z^alpha times these coefficient vectors. For k >= 1 this guarantees f(c) = 0; for k >= 2 it also guarantees vanishing Jacobian at c, and generally all derivatives of total order below k vanish. (2) Pseudocode: z = x - c; coeff = MLP(z); split coeff into vectors g[a]; set f = zeros_like(x); for each a in A_k, compute monomial = product(z[i] ** a[i]) and update f += monomial * g[a]; return f. Use x_next = x + dt * f(x) in a residual block or pass f to an ODE solver. (3) The mathematical part is the fixed monomial basis and the exact vanishing-order guarantee. Estimate the empirical local order by sampling unit vectors u and fitting the slope of log(||f(c + r u)||) versus log(r) over decreasing radii r; the slope should approach k. For large n or k, use a factorized polynomial feature map or a sampled subset of multi-indices to control the O(n choose k) basis size. (4) First experiment: train a small recurrent latent dynamics model or Neural ODE on damped-oscillator trajectories with the known rest state as c. Compare an unconstrained MLP against k = 1, 2, and 3 versions at matched hidden width and approximately matched FLOPs. Measure one-step prediction error, long-horizon rollout error, drift from c under zero-input integration, Jacobian norm near c, and gradient stability. A successful result is lower long-horizon error or much smaller equilibrium drift at equal training loss; a failure is a substantial parameter and compute increase without improved local or rollout behavior.