VSEL: Verifiable Semantic Execution Layer

Contemporarycryptographicexecutionsystems—particularlythoseemployingzero-knowledge proofs—provide strong guarantees that a computation satisfies a given arithmetic circuit. However, satisfying a circuit is not equivalent to executing correctly with respect to the intended semantics of the system being proven. This paper identifies and formalizes the se- mantic gap: theclassoffailuresinwhichexecutionisprovablyvalidunderaproofsystemyet provably invalid under the system’s formal specification. We present the Verifiable Semantic Execution Layer (VSEL), a layered architecture that binds formal specification, execution, constraint derivation, proof generation, and verification into a single semantically coherent pipeline. VSEL models systems as deterministic labeled transition systems, defines explicit semantic mappings between concrete and formal artifacts, derives constraints mechanically from a semantic intermediate representation, and requires that every accepted proof attest not merely to constraint satisfaction but to membership in the formal language of valid execution traces. We define the proof obligations, invariant system, and refinement chain required for end-to-end semantic correctness; characterize the adversarial model including specification manipulation, underconstraint exploitation, and compositional failure; and es- tablish the conditions under which composition of independently correct systems preserves global correctness. The architecture integrates hybrid post-quantum cryptography to ensure long-term validity of proofs and commitments, and introduces a formal economic invariant layer that elevates economic semantics from informal domain knowledge to enforceable first- class predicates over states and execution traces. We provide a complete formal treatment of the system model, semantic preservation theorems, constraint soundness and complete- ness conditions, witness uniqueness requirements, economic admissibility conditions, and the assume-guarantee framework for safe composition.