Hashed Local-Density Particle Layer

Integrate this at the drift evaluation of a particle-based neural SDE, score-based sampler, or latent particle world model. The network supplies a potential or vector field, while the histogram supplies a cheap density-dependent scalar gate. At each Euler-Maruyama step, use inputs particles Y with shape [N,d], bandwidth h, number of random shifts L, and learned or fixed functions Phi and Xi. For each shift U_l sampled uniformly from [0,h)^d, compute integer keys c_i=floor((Y_i-U_l)/h), insert the keys into a hash table, and count occupancies n_l[key]. Set rho_i=(1/(N*h^d))*average_l n_l[c_i], clip rho_i to [rho_min,rho_max], compute drift_i=-Xi(rho_i)*gradPhi(Y_i), and update Y_i <- Y_i + dt*drift_i + sqrt(2*dt)*Normal(0,I). If the model has a learned drift f_theta(Y_i,t), use f_theta(Y_i,t)*Xi(rho_i) or concatenate rho_i to the drift network input; the multiplicative version directly tests the paper's mechanism. Estimate rho_min and rho_max from training quantiles or set conservative fixed values to prevent exploding gains. The paper-derived bandwidth proxy is R(h)=h^2(1+|log h|)+(h^{-d}+log N)/N; initialize h=(N log N)^(-1/(d+2)) and test nearby multiplicative values. Occupancy counts and runtime are computed exactly; Gaussian-envelope assumptions should instead be checked empirically by monitoring maximum occupancy, clipping frequency, and agreement with brute-force KDE at small N. First experiment: use a 2D or 4D score-based diffusion sampler with N=1,024--16,384 particles, comparing exact KDE, ordinary fixed-bandwidth KDE, and hashed histograms at equal wall-clock budget. Measure drift approximation error, sample quality, runtime scaling, and quality versus integration steps. Success means near-linear runtime growth, at least 5x speedup over KDE at N=16k, and no material degradation in final sample quality; random shifts should also reduce grid-alignment artifacts relative to a fixed histogram.