Canonical Predictive Quotients: A Theory of Prediction under Hidden Predictive State Uncertainty with ICL Implications

Many prediction problems involve a hidden predictive state: a latent task, environment, rule, or data-generating mechanism that changes the query-optimal act. For a fixed query, recovering the full hidden predictive state is often stronger than necessary. The relevant target is the Bayes-act distinction among hidden predictive states that preserves oracle-gain adaptation. This question is central for in-context learning (ICL): a prompt may reveal part of an unseen task or rule, but an ICL system needs only to recover the prediction-relevant quotient state label for the query. We formalize this target through the canonical predictive quotient (CPQ), a problem-relative Bayes-act quotient of hidden predictive states. The theory characterizes the value of adaptation, the quotient that preserves full oracle gain, the sharp zero/positive-gain boundary, and a Bayes barrier showing why context-only prediction cannot recover a positive oracle gain at the risk level. We give two exact examples. A finite-state example makes the quotient structure, sharp positivity boundary, and a strict-loss mechanism explicit. A linear-Gaussian prior-family witness gives closed forms for a collapsed Gaussian baseline and a state-aware oracle, a restricted zero-gain boundary, and an exact prediction-level disagreement identity for the restricted gain. Together, the results give a decision-theoretic target for ICL: not full latent complexity, but the prediction-relevant quotient state label. We support the theory with exact checks, realized bridge experiments over prediction-relevant quotient state labels, and external validity tests on released pretrained in-context predictors.