Strict performance-compression separation in finite-memory active binary hypothesis testing

. Active hypothesis testing typically selects experiments with access to the complete observation history. Under finite memory, however, full-history-optimal experiments need not generate evidence that is easy to compress. We distinguish the full-memory optimum F∗ , the best risk CS ∗ attainable by an S-state controller restricted to full-memory-optimal experiment policies, and the joint experiment–memory optimum JS. A general condition establishes the strict ordering F∗ < JS < CS ∗ . A representative-independent decomposition separates action-realizability, additional action-sacrifice, and terminal-compression losses. We also prove one-sided indifference amplification: the full-history sacrifice can vanish linearly while a compression advantage remains nonzero, yielding arbitrarily large benefit-to-sacrifice ratios and a positive lower bound on the global net gain. Two three-step models exhibit strict separation. The first admits polynomial risk formulas and an explicit boundary expansion; the second uses three experiments and exhaustive integer enumeration of 2,187 policies. These results identify evidence compressibility as an experiment-design objective distinct from full-history statistical optimality