Positive Reflected Bellman Layer

1. Integration point: use the layer as the spatial propagation block in a neural operator, value-function network, or latent controlled-dynamics model. The input is a field V at time n+1 on grid or point-cloud locations x_i; the output is the propagated field at time n. Do not append an unconstrained signed spatial mixer if the certificate is important. 2. Pseudocode: for every x_i, control a_j, and Rademacher branch xi_q, compute y = x_i + mu_ij Delta_t + sigma_ij sqrt(Delta_t) xi_q. If y is outside the domain, compute an oblique projection along gamma, set D to twice the overshoot, and calculate reflected location ybar. Interpolate V at ybar. Accumulate z_ij = mean_q of exp(-r_ij Delta_t) times [exp(-kappa_ijq D)V(ybar) + D exp(-kappa_ijq D/2)g_ijq], then add Delta_t f_ij. Return max_j z_ij. Use temperature-controlled log-sum-exp over controls only during warmup if hard maxima cause optimization difficulty. 3. Predict mu, sigma, r, kappa, g, and f with neural heads; enforce r and kappa nonnegative with softplus and keep branch probabilities exactly 1/P. Use bilinear interpolation on a grid or a positive kernel interpolator on point clouds. Reflection geometry is computed deterministically; learned coefficients are empirical. 4. First experiment: train on a two-dimensional reflected controlled diffusion with an analytic or high-resolution reference on a 32 by 32 grid. Compare four-branch and sixteen-branch layers against an MLP neural operator and an unconstrained learned convolution with matched parameter count and FLOPs. Use one-step and twenty-step rollout losses with nonnegative terminal and boundary sources. Measure rollout error, minimum prediction, monotonicity violations after positive input perturbations, and gradient norms. Success is zero or near-zero positivity violations and at least two times lower long-horizon error at equal FLOPs than the unconstrained baseline.