Uniform PEP selective decoding

Integrate this at the final scoring stage of a retrieval system, classifier, policy, or language-model decoder, before top-k selection or selective prediction. For each context tensor \(y\), construct a reference candidate batch \(x_1,\ldots,x_N\sim Q_X\). The reference can be a minibatch of documents, a label-prior sample, an action-proposal set, or sampled token candidates. Compute scalar scores \(s_i=m_\theta(x_i,y)\), draw independent \(u_i\sim\mathcal U([0,1])\), and rank pairs \((s_i,u_i)\) lexicographically. For candidate \(x_i\), compute \(\widehat p_i=N^{-1}\sum_j1[(s_j,u_j)\succ(s_i,u_i)]\). Accept when \(\widehat p_i\le\alpha\); otherwise abstain, retrieve more documents, invoke a larger model, or expand the beam. Compact pseudocode is: x=sample_reference(QX,N); u=uniform(N); s=model_score(x,y); rank=lexicographic_rank_descending(s,u); p=(rank-1)/N; if p[i]<=alpha: return x[i] else fallback(). The theorem applies to the population probability under \(Q_X\), while the code estimates it with an empirical rank, so evaluate sampling variability as a function of \(N\). First run CIFAR-10 selective classification with a small ResNet, using the training label prior as \(Q_X\) and a held-out calibration split; compare raw-softmax thresholding and conformal rank baselines at matched coverage. Then test retrieval on BEIR with a fixed candidate pool. Success means empirical null acceptance close to \(\alpha\), stable behavior under score rescaling and ties, and fewer fallback or large-model calls at matched error or recall.