Dynamic-programming Doob sampler for exact rare-event conditioning

1. Exact integration point: place the module between a neural transition model and its rollout sampler. The base model may be an SSM, GRU, transformer world model, or discrete diffusion transition model with normalized kernel $P_\theta$. The target can be a terminal class, reachability set, or exact number $m$ of threshold crossings. 2. Pseudocode: initialize terminal values $h_T$ from the constraint; for $t=T-1$ down to zero, compute $h_t(z)$ by summing over candidate next states, or use dynamic programming over a discretized latent state and event count; at rollout time sample $z'$ from logits $\log P_\theta(z'|z)+\log h_{t+1}(z')-\log h_t(z)$; disallow states with zero feasibility. For large latent spaces, train a value network $v_\psi(z,t,c)\approx\log h_t(z,c)$ and use normalized logits $\log P_\theta(z'|z)+v_\psi(z',t+1,c')$. 3. Computed from the paper's mechanism: backward dynamic programming and rejection-free boundary enforcement. Estimated empirically: message tables or value-network estimates, normalization residuals, terminal-constraint violations, and KL divergence from exact conditioned paths. 4. First cheap experiment: construct a 10-state neural Markov model with horizon 50 and terminal-event probability $10^{-3}$. Compare rejection sampling, classifier-guided sampling, and the Doob sampler. Check terminal-event frequency, path log-likelihood, and effective sample size. The predicted signature is zero constraint violations for exact messages and a cost per accepted trajectory that remains approximately constant as the unconditioned event probability decreases, versus approximately $1/p_C$ forward rollouts for rejection sampling. For approximate messages, the KL gap to exact conditioned paths should decrease monotonically as the value-network message error decreases.