The Law of the New Deductive closure and gated growth in certified knowledge stores

A store of certified knowledge (a proof library, a certificate cache, a database of checked derivations) can be modeled as a directed graph: each held certificate, once checked sound, is an edge from its claimed start to its claimed finish, and the content of the store is the reachability relation of that graph. Two motions are then possible. Advance chains held certificates end to end. Commit appends a certificate the store does not hold, through an admission gate. We prove that these motions differ in kind: Advance conserves. Only commit can create. Composition of held sound certificates never enlarges the reachability relation, in any certificate calculus, while admission of a sound certificate makes its claimed conversion reachable in one move; so, for stores whose content is claimed-endpoint reachability and whose held set grows only by gated admission of sound certificates, all content growth factors through admission. We further prove that the reachability packaging is the unique least reflexive, composition-closed content generated by the holdings; that the gate’s three checks (replay, exact price within a standing budget, refusal of relabels) are pairwise non-redundant, each refusing a counterfeit the other two accept; and that under a complexity ceiling the set of admissible certificates is finite. The decomposition is stable under adversarial extension: schema-filling joins, anti-unification, open terms, binders, deep and non-terminating rewriting, permutation rules, nondeterministic replay, rational cost weights, and purchased budget capacity each destroy a comfortable auxiliary property (termination, evaluation invariance, intrinsic price, cost-only finiteness) while the conservation and creation split survives. Closure idempotence is classical and is not claimed as new; the contribution is the two-stroke decomposition, the admission discipline, the finite purchasable frontier, and the adversarial invariance results, all machine-checked. The question was prompted by a measured saturation in a running theorem-proving system, in which 11,108 successful compositions over a certified library collapsed to 1,065 distinct results and re-ingestion of those results added nothing; under the modeling reading developed here, the plateau is exactly what advance predicts.