Proof-Valid Caching under Premise Erasures: Local Structural Limits and Shared-Workload Gains
We study reliable query recovery under independent premise erasures in semantically transparent caching systems, where every cached object must be a logical consequence of the premise base. Recovery succeeds only when the query remains derivable from surviving premises and the cache. Under a deterministic canonical-witness regime, we prove a query-local projection theorem and an exact residual-leaf law: recovery fails exactly when an erased base leaf retains a cache-free path to the query. Single-query design becomes weighted partial path interception. For shared workloads, we introduce semantic modules and derive exact reliability laws under joint and maximal-error criteria. The shared-module cache is exactly optimal under exact module routing and homogeneous costs, whereas optimal selection in general derivation DAGs is NP-complete at depth two. Against a coded benchmark recovering workload-relevant leaf payloads, MDS parity caching is optimal up to one packet. Leaf-only transparency incurs a first-order overhead inversely proportional to the erasure rate; shared modules multiply that inverse-erasure-rate scaling by the module-to-leaf cost ratio divided by the number of protected leaves. A Datalog instance and Monte Carlo checks illustrate the theory. For derivation-structured content, the results provide exact stochastic-erasure counterparts of function-correcting storage and an exact distributional quantification of maximal recoverability.