Spectrum-preserving conditional binary graph sampler

(1) Exact integration point: insert this sampler as a stochastic graph-refinement block in a graph VAE, graph world model, or discrete graph denoiser. Let node features be $u_v=f_\theta(x,G)_v$ and let the network output bath logits $\ell_v$ for $v\in B$. Define $p_v=\sigma(\ell_v)$ and $h_v=(1-p_v,p_v)$. Maintain a binary state vector $z\in\{0,1\}^{|V|}$ and apply $K$ Markov events before decoding or computing the task loss. Keep exchange rates $r_e$ and bath rates $q_v$ fixed across all examples. (2) Pseudocode: initialize $z$ from encoder Bernoulli samples or a deterministic binary seed; form an event list containing each edge $e$ with weight $r_e$ and each bath $v$ with weight $q_v$; repeat $K$ times, sample one event proportionally to its weight, swap $(z_u,z_v)$ if the event is edge $(u,v)$, and otherwise sample $z_v\sim\operatorname{Bernoulli}(p_v)$ for a bath event. Decode the final $z$. Use straight-through Bernoulli samples for a first differentiable prototype, or optimize a variational objective with the sampler treated as a discrete latent transition. (3) Computed from the mathematics: the exact local operators $S_e,H(v)$, the generator, and the fixed event-rate schedule. Estimated empirically: the second-largest eigenvalue modulus on small graphs, integrated autocorrelation time, effective sample size, and downstream loss. Explicitly build $P_h$ for several different $p_v$ settings and verify that its eigenvalues remain unchanged up to numerical error. (4) First cheap experiment: train a binary graph-VAE on 8- to 32-node cycles, grids, and random geometric graphs. Compare against independent Bernoulli resampling and ordinary single-site Gibbs updates at equal event count and wall-clock cost. Sweep bath probabilities from 0.01 to 0.99 while holding rates fixed. Success requires nearly unchanged autocorrelation or relaxation curves across probability settings, plus better conditional reconstruction log-likelihood or sample quality than independent resampling at equal sampling cost. Also test whether the fixed spectrum makes the sampler more stable than a learned transition-rate baseline.