The Verification Boundary: Exact Limits of Verification by Bounded Systems: Identification, Generation, Computation, Implementation, Correspondence — and the Forced Routing at Each Limit

Abstract What can a bounded system justifiably verify from the evidence and resources available to it? This paper organizes the question around distinct boundaries of identification, generation, computation, implementation and correspondence. For a declared space of possibilities and an evidence map, exact verification is possible precisely when the demanded claim is constant on each evidence fiber. A local version characterizes sound answers at a particular observation and separates unresolved evidence from observations outside the declared model. Computation or recursive generation based solely on unchanged evidence cannot distinguish possibilities that evidence leaves indistinguishable. The paper separates semantic support from effective computation and resource-bounded attainment. In explicitly presented finite classes, it develops adaptive acquisition, relational candidate selection, safe revision composition and finite-machine attainment results under declared operational contracts. It distinguishes pointwise availability from uniform implementation and identifies conditions under which abstraction and execution preserve the guarantees being claimed. Probabilistic continuation is treated through explicit probability assumptions and attainable guarantee regions; ideal convex mixing is distinguished from mixtures realizable under hard resource limits. Contextual equivalence for declared continuation systems preserves both demanded guarantees and metadata. These results support a structured verification record that distinguishes established answers, insufficient evidence, computational exhaustion, unavailable implementations, unresolved authority and model inconsistency. The routing protocol is sound under its stated assumptions rather than a universal classification of every possible failure. The framework also separates correctness within a model from evidence that the model represents a concrete system, including the verifier itself. Its contribution is a synthesis of exact conditions, impossibility witnesses and attainable constructions that makes the boundaries of justified verification explicit without converting internal coherence, agreement or successful execution into an unsupported claim about reality.