TYPED AMPLITUDE COMPOSITION FOR REUSABLE INFERENCE: SEMANTICS, ERROR BOUNDS, AND EXACT CORRECTION

Probabilistic representations are often built before the eventual query or constraint is known. We study inference-time composition: compositional inference that modifies an already represented law while preserving its intended semantics and reusing prior computation. Algebraically natural operations can encode the wrong law even in exact arithmetic, and approximation, target mismatch, and Monte Carlo error belong to different layers. We develop one contract that keeps these layers explicit. Typed semantics fix the intended law for products of complete laws, factors relative to a shared base, labeled mixtures, event projectors, and integer tempering; explicit matrix-product-state (MPS/TT) constructions realize it; deterministic bounds propagate compression residuals, and a dimension-free amplitude-to-law inequality converts them into total-variation control of the represented law; exact density-ratio correction targets a separately declared law under support assumptions, and one scored pool answers later conditions without new target evaluations. On held-out known-answer families, composition recovers the requested law to total variation 7.4 × 10−16 and propagated bounds cover every observed error. In RNA design, where later sequence constraints are imposed on candidates compatible with a target structure, one compiled proposal per puzzle is reused for later conditions, composed with exact conditions, and repays all measured setup by 20,000 samples on a workload with 2.6 × 1017 compatible states. At 256 requested ViennaRNA scores it estimates a prespecified conditional endpoint more accurately than tested annealed SMC (equal-puzzle difference +0.142, interval [0.040, 0.244]), and its favorable pool quality transfers to EternaFold.