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 justify from its evidence, computation and memory, and what follows when those resources cannot settle its question? This paper develops a mathematical account of verification boundaries under explicit task, evidence and resource contracts. An evidence-based verifier cannot correctly decide a claim that takes different truth values within one evidence fiber. Generating hypotheses or recursively processing the same evidence does not remove that obstruction without an additional informative channel or premise. Related results separate semantic determination, effective computation and bounded implementation; show why some finite resource classes have incomparable attainable capabilities rather than one greatest evaluator; and characterize conditional limits on self-verification. Positive results complement these obstructions. Finite query-tree constructions characterize adaptive acquisition and relational answer selection under declared costs. Explicit finite-machine models provide attainable implementation families with accounted resources and transport bounds. For finite permitted procedure classes, maximal antichains characterize terminal guarantee feasibility; their limitations under continuation distinguish terminal comparison from compositional preservation. Probability is treated through separate objects: posterior belief, environmental stochastic assumptions and private algorithmic randomization. Failure of a deterministic guarantee supplies no premise-free probability estimate. The results support a typed verification protocol: accepted certificates justify scoped assertions, bounded dispatch returns unresolved outcomes when necessary, and multiple obstruction causes remain visible beneath any summary label. Soundness, termination, diagnostic completeness and implementation correspondence are separate obligations. The paper establishes conditional boundaries and attainable constructions for its specified classes, while leaving general model correspondence and unrestricted justified self-improvement as distinct questions. This public edition updates the previously deposited v6.3 release. It corrects editorial status and import-scope wording, clarifies that probabilistic continuation requires additional declared premises, and incorporates the reference correction. Mathematical statements and proofs are preserved. Earlier versions remain part of the publication history.