Information-budgeted Gibbs router

Integrate this at the router or portfolio layer, not inside the expert networks. Let K be the number of MoE experts, LoRA adapters, augmentations, or candidate optimizer updates. At step t, obtain the per-example or minibatch loss vector c[0:K] by evaluating each candidate, or use detached proxy losses when full evaluation is too expensive. Maintain cumulative scores C[i] and a positive prior pi[i]. Use this loop: (1) choose eta from a logarithmic grid, or by bisection over eta if losses are clipped to [0,1], selecting the largest eta whose running information estimate is at most budget B; (2) compute logits[i]=log(pi[i])-eta*C[i] and p=softmax(logits); (3) execute the weighted output y=sum_i p[i] y_i; (4) after observing the candidate losses, compute q=sum_i p[i] exp(-eta*c[i]) and I=log(q)+eta*dot(p,c); (5) update C[i]+=c[i] and the running information accumulator. Use logsumexp for stable computation and clip or normalize losses before exponentiation. If eta changes, recompute p from current C rather than multiplying old probabilities; also record the retempering potential Phi=-(1/eta)*logsumexp(log(pi)-eta*C), which is the exact scale-dependent state variable. The mathematics determines the Gibbs update, nonnegative information gap, and KL-prior interpretation; B, the eta grid, and clipping range are empirical hyperparameters. First test on CIFAR-100 with a small four-expert ResNet or four LoRA adapters, comparing fixed-temperature softmax, standard fixed-eta Hedge, and the proposed method at equal expert FLOPs. Measure validation accuracy, router entropy, cumulative regret to the best expert, and loss spikes after synthetic distribution shifts. Success means lower best-expert regret, fewer routing collapses, or faster recovery after a corruption shift at equal compute.