Binomial-thinning (s,S) capacity controller

Integrate the controller into a sparse MoE layer or a conditional channel layer at inference time. Let a_t be the number of currently provisioned experts, let d_t be the token demand in the next batch, and let y_t be the number provisioned after a possible activation. First collect a calibration set and measure f_t(z): mean task loss or perplexity degradation plus latency and dropped-token penalty when exactly z experts are available. Estimate gamma_t as the empirical availability rate, gamma_t=(number of expert slots that successfully accept work)/(number provisioned slots), optionally conditioned on batch size. Construct F_t(y) from the paper's formula by Monte Carlo sampling Z from Binomial(y,gamma_t) and averaging the lookup-table values f_t(Z). Enforce discrete convexity by isotonic regression on the first differences F_t(y+1)-F_t(y). Use the fixed-charge objective C_t(y|a_t)=F_t(y)+c max(y-a_t,0)+K 1[y>a_t]+h max(a_t-y,0), where h penalizes unavailable capacity or dropped tokens. For each capacity y in a small grid, compute C_t and select the minimizing target S_t; define s_t as the largest current capacity for which activating to S_t is cheaper than retaining capacity. At runtime, measure a_t and demand d_t; if a_t<s_t, activate S_t-a_t experts or channel groups; otherwise keep the current set. Re-estimate gamma and f periodically, not per token. The paper supplies the binomial convexity calculation; gamma, f, K, c, and h are estimated empirically. The first experiment should use a 4- or 8-expert Switch-Transformer-style layer on WikiText-103 or C4, comparing this controller with greedy top-k routing plus per-batch capacity resizing. Measure perplexity, expert activation count, kernel launches, p50/p99 latency, and dropped-token rate. Success is lower activation and launch overhead at matched perplexity, with fewer oscillations in active capacity and no increase in overflow rate.