Coherence, Rupture, and Regeneration: A Mathematical Framework for Temporal Identity Through Discontinuous Change

We present the Coherence-Rupture-Regeneration (CRR) framework, a non-Markovianmathematical formalism that emerged from phenomenological enquiry into how sys-tems maintain identity through transformative change. The framework comprisesthree fundamental operators: (i) a coherence function C(x, t) = R t0 L(x, τ ) dτ thataccumulates memory density non-locally across temporal history; (ii) rupture termsPi ρi(x)δ(t−ti) representing local discontinuous transitions that perturb the coher-ence field; and (iii) a regeneration operator R[χ](x, t) that reconstructs system statesby reading from the non-local coherence field through discontinuity. The central in-sight is that rupture is local whilst memory is non-local; this mathematicaldistinction enables identity to persist through catastrophic transformation. We es-tablish rigorous foundations, derive key mathematical properties, and demonstrateapplications across neural systems, quantum mechanics, ecology, and artificial in-telligence. For AI systems, CRR provides an architecture for syntactic agency withtemporal boundaries; systems that maintain internal coherence without claimingembodied understanding. We examine how current disembodied AI systems createpsychological risks through recursive self-mythologisation, and how CRR-based ar-chitectures reduce these risks whilst remaining transparent about their limitations.This work integrates process philosophy, active inference theory, and phenomeno-logy to offer both a mathematical calculus of adaptive temporality and a frameworkfor safer human-AI interaction. All claims are presented with explicit acknowledge-ment of what remains conjectural, requiring empirical validation across domains.The equations and final proof-reading were conducted with the support of ClaudeSonnet 4.5.