BU111 Proof Admissibility under B_U BU111|B_U 锚定下的证明准入

Abstract

BU111, “Proof Admissibility under B_U,” formulates proof admissibility as a B_U-anchored downstream formal reconstruction. Its object is the class of situations in which a formal proof, proof object, theorem derivation, machine-checked artifact, verification certificate, consistency argument, model-theoretic result, or proof-carrying structure claims decisive standing. The paper places this claim under the B_U chain of boundary consistency, dynamic completeness, admissible path space, structured pruning, residual gating, feedback, and settlement standing. Proof is therefore treated as a local formal readout whose strength depends on whether it declares its object boundary, preserves its admissible domain, clears its residual obligations, respects semantic transport, and enters the B_U settlement chain.

The central judgment is direct: proof completion is a formal event; proof admissibility is a structural event. A proof can be syntactically valid, internally consistent, mechanically checked, or accepted by a formal verifier while still leaving its standing boundary unsettled. Under B_U, a proof becomes admissible only when it proves the correct object under a declared boundary, operates inside an explicitly admissible proof domain, preserves the relevant quotient or invariant core, avoids proof-package drift, exposes unresolved gaps, and returns its result to a settlement-compatible chain. Proof admissibility therefore separates proof appearance from proof standing. A proof object may certify a local derivation; B_U adjudicates whether that derivation can enter retained structure.

BU111 develops this handshake through a sequence of formal positions: proof presentation, proof boundary, admissible proof domain, proof object, proof transport, proof-equivalence class, proof gap, verifier, admissibility judgment, and settlement-compatible proof standing. These definitions translate the B_U source-axis discipline into a vocabulary familiar to mathematical logic, proof theory, formal verification, theorem proving, type theory, model theory, and mechanized proof systems. The paper’s local formal shell follows a clear chain: proposition → proof presentation → admissible proof domain → proof object → proof equivalence → proof transport → proof gap → verifier judgment → settlement-compatible standing.

The theorem skeletons establish the main structure. The Declared Proof Boundary Theorem states that proof admissibility begins with a declared object and boundary. The Admissible Proof Domain Theorem requires the proof to operate inside a valid domain of rules, semantics, assumptions, and transformations. The Proof Object Standing Theorem distinguishes local derivability from retained standing. The Proof Transport Stability Theorem requires proof status to remain stable under admissible representation changes. The Proof Gap Obstruction Theorem identifies hidden assumptions, semantic drift, verifier overreach, and residual obligations as admissibility failures. The Settlement-Compatible Proof Theorem returns proof to the B_U ledger by requiring boundary declaration, residual clearing, feedback compatibility, and same-world-sheet settlement.

The final judgment is that formal proof is powerful when its position is correct. Proof can define, derive, verify, normalize, formalize, and mechanize. These operations acquire durable force when anchored in B_U. A proof becomes admissible when it proves the right object, under the right boundary, through an admissible path, with residuals exposed and cleared, and with settlement conditions preserved. Proof without admissibility remains local derivation. Proof with B_U-compatible admissibility becomes a retained standing condition.