Claim-Level Verification for AI-Assisted Mathematics: A Kernel-Closed Hamilton Classification and a Mixed-Boundary Rado Study

AI-assisted mathematical research can produce conjectures, programs, proof sketches, solver instances, and formal proof fragments faster than those objects can be assessed. This paper introduces a claim-level verification framework in which each load-bearing claim is bound to its representing object, checking procedure, scope, remaining assumptions, and stable version identifier. The first case gives the complete Hamilton cycle and path classifications for the noncrossing partition refinement graph under a kernel-only Lean 4 boundary. The second case assigns distinct analytic, SAT, witness, Lean, and conjectural boundaries to a Rado-number study. Together the cases demonstrate both kernel-closed and mixed-boundary closure profiles. The framework makes verification adequacy claim-relative and requires documented, monotone transitions to stronger status. It gives readers direct procedures for locating and checking the evidence attached to each mathematical claim.