Decomposed Mean-Field State Layer

1. Integration point: insert the layer between recurrent or SSM updates and the task readout. Maintain $R$ hidden particles with shared parameters and shared input $u_t$, but use different initial states or small fixed latent seeds. Apply the same neural transition to every particle; pool particles only when producing the output or a loss statistic. A practical first version uses an MLP transition $F_\theta(x,u)=x+\Delta t\,f_\theta(x,u)$ with spectral normalization on Jacobian-sensitive layers. 2. Pseudocode: initialize x[r] = mu0 + L * epsilon[r]; for each time $t$, compute x[r] = F_theta(x[r], u[t]) for all $r$; compute mean = mean_r(x[r]), cov = mean_r((x[r]-mean)(x[r]-mean)^T), and third = mean_r(outer3(x[r])); decode from mean, cov, or concatenated low-rank statistics; backpropagate through all particles. Add an optional diagnostic loss comparing third to the Gaussian/cumulant closure above, rather than enforcing it. 3. Computed versus estimated: the transition is learned; moments are empirical averages; the $O(R^{-1/2})$ sampling law is the paper-inspired quantitative prediction, not a learned parameter. Estimate multimodality using the fraction of particles on either side of a learned symmetry coordinate. 4. First cheap experiment: train a 2D GRU and this particle layer on sequences generated by a bistable nonlinear system, such as $\dot z_1=z_2$, $\dot z_2=z_1-z_1^3-0.2z_2+u(t)$, using identical mean initial state but a bimodal initial distribution. Compare GRU, Gaussian moment closure, and $R\in\{4,16,64,256\}$. Measure long-horizon trajectory error, third-moment error, and branch occupancy. The falsifiable signatures are third-moment RMSE proportional to $R^{-1/2}$ and preservation of bimodality when the Gaussian closure collapses it; doubling $R$ should reduce sampling error by approximately $\sqrt{2}$, while deterministic single-state baselines should lose branch occupancy after rollout.