APTADINAMIA Formal Framework for Structural Viability with Fractional Memory, Accumulated Affection, and Categorical Collapse

Present APTADINAMIA, a unified mathematical framework for structural viability in complex systems with memory. Starting from five minimal axioms on the informational trajectory of a system, was construction of the structural tension accumulator as the causal convolution of instantaneous decoupling under a memory kernel, and derive a threshold-based viability criterion in which the critical boundary depends endogenously on the system's accumulated history. The contributions are: (I) global existence and conditioned uniqueness of the induced structural regime; (II) an exponential lower bound on accumulated affection — via a Gronwall-type differential inequality — establishing that collapse occurs in finite time with explicit rate μ = √(κθ_min) determined by operational system parameters; (III) a dual necessity theorem showing that interventions restoring only permissivity are provably insufficient for recovery when structural decoupling persists; (IV) formal non-reducibility to Aubin's viability theory, with Aubin's framework identified as the degenerate zero-memory case; (V) differentiation from damage mechanics (Chaboche–Lemaitre), where irreversibility is a constitutive postulate rather than a derived theorem; and (VI) categorical irreducibility with respect to stochastic categories with fixed object space. A numerical appendix illustrates the dynamics under exponential and power-law kernels, three threshold geometries, and the contrast between permissivity-only and decoupling-reduction recovery strategies.What distinguishes APTADINAMIA from existing viability frameworks is not the introduction of memory per se — non-Markovian dynamics are well-studied — but the endogenous deformation of the viability domain: the critical threshold Θ(λ(t)) contracts as the system's own accumulated affection degrades its permissivity. This self-referential mechanism produces irreversibility as a theorem, not as a constitutive assumption, and generates exponentially accelerating collapse with an explicit, falsifiable rate. The present paper establishes these results rigorously, positions them against the relevant literature, and illustrates them numerically.