The Calculus of Persistence

This paper presents the Calculus of Persistence (CoP), a substrate-independent formal framework for modelling how bounded adaptive systems maintain functional continuity under environmental perturbation and incomplete information. Persistence is defined not as static identity or material conservation, but as recoverable viability-preserving adaptive transformation under partial observability. The framework introduces: a viability region with an absorbing collapse boundary; an instruction tensor of coherent transformation operators; a sigmoid-regulated persistence selection equation; an inverse recoverability operator; and a recursive persistence update rule that provably converges toward stable persistence manifolds under stationary environmental geometry. The limit of a continuous operator library recovers the Hamilton–Jacobi–Bellman optimality condition for the persistence value functional, grounding CoP within established optimal control theory. Emergent properties of the framework—including homeostatic self-regulation, modular operator clustering, and anti-fragile recoverability—are derived analytically and illustrated with a simulation in which an agent navigates a hostile environment, providing informal empirical evidence of the framework's predictive validity. CoP is shown to scale naturally across biological, computational, ecological, and economic domains.