Vauban Research The x402 protocol ([X402-V2]) defines three wire messages for HTTP- native payment flows but provides no composability model for payment receipts. A recipient holding a SettlementReceipt and a DelegationGrant cannot natively express that the settlement satisfies a delegated spending condition, or that a RefundClaim negates an earlier SettlementReceipt, without bespoke facilitator-side logic. As automated payment pipelines grow, the absence of a normative composability layer leads to fragmented, non-interoperable verification surfaces. This document specifies the Vauban Proof Stack Framework (VPSF) Claim Algebra for x402 payment receipts: a grammar of five composition operators (Conjunction, Implication, Aggregation, Selective Disclosure, Revocation; abbreviated ∧, →, ⊕, ▷, ¬) applied over the four canonical Payment Claim types (PaymentIntent, SettlementReceipt, RefundClaim, DelegationGrant) defined in [LIFECYCLE-FSM]. The algebra is chain-agnostic by invariant; the Starknet-based proof system described in [STARK-RECEIPTS] is the first reference implementation. Composability is grounded in the JCS canonical preimage discipline ([RFC8785]), ensuring operator results are deterministic and cross-implementation consistent. An open-source Rust implementation is provided in [VAUBAN-CRATE].