Residual Dynamics (RD) A Finite-State Formal System that Treats Failure as a First-Class Object

This work introduces Residual Dynamics (RD), a finite-state formal system that treats failure as a first-class object rather than as noise, error, or exception. Unlike traditional formal systems and dynamical models that focus on how a system successfully generates a world, Residual Dynamics shifts the focus to how deviations from global consistency are produced, accumulated, propagated, and under what conditions they can be eliminated completely. The framework defines a finite residual state space together with a closed residual composition structure, allowing multiple failure fragments to be accumulated in a formally well-defined way. Failure injection is represented through a set of drive elements, and system evolution is described by inference rules that operate entirely within the residual space. As a result, Residual Dynamics is independent of the external generating system’s states, rules, and semantic interpretations. Within this framework, the “world” is not generated by evolution but is defined as the zero-residual slice, and success is a terminal criterion rather than a process. The paper includes a fully exhaustive finite-system example, providing checkable verification of residual reachability, zero-residual orbits, and the effective failure state space. Residual Dynamics is presented as an auditable, reproducible, and falsifiable formal framework for studying failure structures in finite systems.