Topological Protection Through Validated Memory: A Mathematical Framework for Identity Persistence in Adaptive Systems

We introduce the Coherence-Rupture-Regeneration (CRR) framework, a mathematicaltheory for adaptive systems that maintain identity through discontinuous transformations.The framework’s core innovation is a topological arrow of time that protects system iden-tity via distributed memory encoding whilst permitting local plasticity through rupture-regeneration cycles.Three operators define the dynamics: coherence integration C(t) = R L(x, τ )dτ , rupturedetection via degradation signatures, and validated memory regeneration R[χ] = R ϕ(τ ) exp(C(τ )/Ω)dτ .Critically, the scalar coherence C(t) can reset locally at ruptures whilst the field configura-tion {C(τ )}τ <t persists non-locally, creating topological protection analogous to gauge theoryholonomy or topological quantum states.This local-nonlocal separation generates a validation arrow of time: past states influencefuture reconstruction weighted by accumulated coherence C(τ ), not recency, frequency, orcurrent relevance. This principle—validated memory—differs fundamentally from existingtemporal weighting schemes and generates testable predictions across neuroscience, ecology,and continual learning.We show that CRR’s temporal boundary grows monotonically (topological irrevers-ibility), show how identity persists through field topology whilst local dynamics remainplastic, and demonstrate finite Landauer costs despite apparent information erasure. Aproof-of-concept implementation shows 51% reduction in catastrophic forgetting comparedto baseline, though distinguishing validated from recency weighting requires further experi-mentation.Connections to the Free Energy Principle show CRR as a temporal governance architec-ture that uses, rather than modifies, active inference. The framework introduces memorysignatures (fragile, resilient, oscillatory, chaotic, dialectical) as potential topological phaseswith testable transition dynamics. We identify specific mathematical challenges, empiricaltests, and cross-domain predictions required to validate or refute the framework.Keywords: Topological Protection, Validated Memory, Temporal Dynamics, ActiveInference, Free Energy Principle, Catastrophic Forgetting, Adaptive Systems. Claude Sonnet 4.5 (Anthropic) was used for final Latex Proof and equation checkingContact: [email protected] for enquiries