A Witness Logic for Semantic Composition
An Axiomatic Characterization of Witness Rank
Witness rank is the unique numerical invariant on semantic interface complexes in the exact regime that respects seam-independent disclosure as the primitive operation of compositional repair (Disclosure Characterization Theorem, Lean-verified)
A numerical invariant on the exact regime satisfying axioms A1–A4b but not equal to dim H¹, or a natural alternative primitive that yields a strictly stronger or strictly different theorem
Abstract
Classical program logics make local correctness compositional under a stable semantic frame. Open autonomous systems break that assumption: components satisfying their local contracts can produce globally incoherent behavior because the semantic frame itself fails to glue across interfaces. Working over the category of semantic interface complexes — finite diagrams of local semantic carriers with observable projections and latent seam dimensions — we state six natural axioms for an obstruction invariant and prove that, in the exact regime, the unique invariant satisfying them is the witness rank, equal to the dimension of the first cohomology of the seam complex. Consequences: a repair duality identifying minimum disclosure cardinality with the obstruction invariant; a disclosure normal form exhibiting every coherence certificate as a typed disclosure derivation; and a communication lower bound showing that protocols whose transcript carries fewer independent declarations than the witness rank cannot soundly certify coherence.