We introduce partial path interception on derivation DAGs (directed acyclic graphs), a combinatorial optimization problem arising from proof-valid caching: stored objects must be logical consequences of a premise base whose premises are erased independently, a query is recovered only while derivable from surviving premises and stored objects, and the queried object itself may not be stored, so reliability must come from premises or intermediate consequences. An exact residual-leaf law pins the recovery probability to a closed form in the exposed-leaf count and reduces design to weighted partial path interception (PPI) on the witness DAG. We classify PPI across its canonical regimes: it is NP-complete already on depth-two DAGs with leaf out-degree two and fixed-parameter intractable in the budget, ruling out efficient approximation schemes; it contains smallest T-edge subgraph as a special case, hence admits no approximation within the square root of the optimum under the Strongish Planted Clique Hypothesis, matched there by trivial leaf-only caching; whereas unique-path instances admit an exact dynamic program, bounded-treewidth instances are fixed-parameter tractable, and individual coverage admits a tight logarithmic greedy guarantee. Semantic-module caches are exactly optimal for shared workloads under exact module routing. An appendix prices transparency against a coded erasure benchmark.

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