C 2 = C: Recoverable Continuity as a Structural Condition for Mathematical Identity, Reference, and Transformation

This paper develops This paper develops C² = C as a mathematical formulation of recoverable continuity: the condition that mathematical identity, reference, proof, equivalence, and transformation remain coherently attributable across formal change. The paper argues that formal validity, output stability, and recoverable attribution are not identical. A transformation may be syntactically valid, output-admissible, or even idempotently stable while still failing to preserve attribution from source to result. To formalize this distinction, the paper introduces continuity systems, recoverability relations, C-admissibility, proof-chain continuity, attribution gaps, recoverability deficits, quotient attribution risk, and regime-transfer failure. The central mathematical claim is that continuity must remain continuous for mathematical objects, symbols, operations, proofs, equivalences, and domains to remain identifiable as themselves across transformation. C² = C is therefore presented not as a replacement for existing mathematical foundations, but as a structural condition required for coherent mathematical attribution. as a mathematical formulation of recoverable continuity: the condition that mathematical identity, reference, proof, equivalence, and transformation remain coherently attributable across formal change. The paper argues that formal validity, output stability, and recoverable attribution are not identical. A transformation may be syntactically valid, output-admissible, or even idempotently stable while still failing to preserve attribution from source to result. To formalize this distinction, the paper introduces continuity systems, recoverability relations, C-admissibility, proof-chain continuity, attribution gaps, recoverability deficits, quotient attribution risk, and regime-transfer failure. The central mathematical claim is that continuity must remain continuous for mathematical objects, symbols, operations, proofs, equivalences, and domains to remain identifiable as themselves across transformation. This paper develops C² = C as a mathematical formulation of recoverable continuity: the condition that mathematical identity, reference, proof, equivalence, and transformation remain coherently attributable across formal change. The paper argues that formal validity, output stability, and recoverable attribution are not identical. A transformation may be syntactically valid, output-admissible, or even idempotently stable while still failing to preserve attribution from source to result. To formalize this distinction, the paper introduces continuity systems, recoverability relations, C-admissibility, proof-chain continuity, attribution gaps, recoverability deficits, quotient attribution risk, and regime-transfer failure. The central mathematical claim is that continuity must remain continuous for mathematical objects, symbols, operations, proofs, equivalences, and domains to remain identifiable as themselves across transformation. C² = C is therefore presented not as a replacement for existing mathematical foundations, but as a structural condition required for coherent mathematical attribution. is therefore presented not as a replacement for existing mathematical foundations, but as a structural condition required for coherent mathematical attribution.