BU114, “Formal Verification under B_U,” formulates formal verification as a B_U-anchored downstream formal reconstruction. Its object is the class of situations in which a specification, verifier, proof system, model checker, theorem prover, proof-carrying artifact, certified program, hardware protocol, AI safety claim, or guaranteed system asserts that a behavior has been formally verified. The paper places this assertion under the B_U chain of boundary consistency, dynamic completeness, admissible path space, structured pruning, residual gating, feedback, and settlement standing. Formal verification is therefore treated as a local certification procedure whose structural force depends on whether the specification boundary is declared, the verified behavior is correctly identified, the verifier operates within an admissible domain, residual obligations are exposed, execution conditions are preserved, and the verified result can enter a settlement-compatible chain. The central judgment is direct: verification proves compliance with a declared formal boundary; B_U adjudicates whether that verified compliance carries durable standing. A system may satisfy a specification, pass a model checker, produce a machine-checked proof, carry a proof object, or execute under a certified protocol. These events establish local formal verification. They acquire B_U-compatible force only when the formal boundary, runtime environment, implementation channel, hardware substrate, human use context, feedback loop, and settlement consequences remain aligned. A verified artifact can be correct inside its formal shell while residual risk remains in the surrounding world. BU114 therefore separates verified compliance from settlement standing. The file develops this handshake through a sequence of definitions: specification boundary, formal verifier, admissible verification domain, verified behavior, proof-carrying artifact, implementation transport, runtime condition, verification gap, residual obligation, feedback exposure, and settlement-compatible verification. These definitions translate the B_U source-axis discipline into a vocabulary familiar to formal methods, program verification, theorem proving, model checking, proof-carrying code, hardware verification, runtime assurance, AI safety verification, and guaranteed system design. The local formal shell follows a clear chain: specification → verifier → proof artifact → admissible domain → implementation transport → runtime condition → verification gap → residual audit → settlement-compatible standing. The theorem skeletons establish the main structure. The Declared Specification Boundary Theorem states that verification begins with an explicit specification boundary. The Verifier Domain Theorem requires the verifier to operate within a declared admissible verification domain. The Proof-Carrying Artifact Theorem states that proof artifacts certify only the behavior covered by their declared boundary. The Implementation Transport Theorem requires verified properties to remain stable when transported from formal model to executable system, hardware substrate, or operational environment. The Verification Gap Obstruction Theorem identifies specification incompleteness, model mismatch, runtime drift, hardware residuals, adversarial context, and feedback failure as validity gaps. The Settlement-Compatible Verification Theorem returns formal verification to the B_U ledger through residual clearing, execution feedback, friction-cost awareness, and same-world-sheet settlement. The final judgment is that formal verification is one of the strongest local tools in the formal stratum. Its durable force depends on correct placement. Verification can certify a specification, a proof object, a protocol, or an implementation path. B_U-compatible verification certifies that the verified behavior survives boundary declaration, implementation transport, runtime exposure, residual audit, feedback, and settlement. Formal verification gains standing when its certified result remains valid on the same boundary and settles on the same world-sheet.