Fano-Calibrated Multi-User Watermark Budget

Modify the watermarking logit-bias or adapter-training component of an autoregressive transformer. Maintain an unwatermarked frozen reference model p_nat and a keyed model p_theta(.|prefix,u), where u is sampled uniformly from a registry of N user IDs. For every training example, compute the per-token KL between the keyed and reference next-token distributions and sum it over n positions: D_KL_hat = sum_t KL(p_theta(.|x_<t,u) || p_nat(.|x_<t)). Train a separate attribution decoder q_phi(u|x_1:n), and estimate key information using the variational classification expression in the third formula. The generator objective minimizes language-model loss and maximizes this estimated key information, while a hinge or augmented-Lagrangian term limits total KL to n times Delta_target. Set Delta_target from the Fano expression using a chosen target error epsilon; for example, with N=1024, n=128, and epsilon=0.05, the initial budget is roughly ((0.95 log 1024)-log 2)/128. Use an adaptive multiplier lambda: increase it when measured KL exceeds the budget and decrease it when below. Alternate q_phi and generator updates, and evaluate on fresh prompts and keys not seen in the minibatch. Compare against fixed-strength watermarking at matched perplexity. Measure top-1 attribution error as N grows, key confusion entropy, detection error without the key, and average token KL. The falsifiable prediction is that calibrated training reaches the target attribution error with lower average KL than fixed-strength baselines, and that error sharply degrades when n Delta falls below the Fano threshold. Because the bound is a converse rather than an achievability theorem, report the gap between empirical required information and the lower bound.