Scale-Resolved State Dynamics: Foundational Closure Specification

Scale-Resolved State Dynamics (SRSD) is a mathematical framework for exact quotient-residual resolution of finite-dimensional real coordinate states at declared positive scales. For a positive coordinate scale vector, SRSD decomposes each real coordinate uniquely into a lattice-valued coarse quotient and an exact within-cell residual. The resulting resolution map is a bijection with an explicit inverse, so exact resolution preserves every distinction present in the pre-resolution coordinate map. This yields a precise fiber-theoretic characterization of distinguishability and identifies where information loss can and cannot occur. The framework formally separates four distinct reconstruction problems: reconstruction of the pre-resolution value, reconstruction of the source state, reconstruction using branch information, and selection of a representative from a noninjective fiber. It also establishes exact criteria for information loss under later record projection and for reconstruction when the underlying representation or dynamics is noninjective. The foundational theory accommodates nonuniform and variable scales, products, restricted coordinate images, injectively coordinatized spaces, manifolds, periodic variables, and pass-through factors without enlarging its primitive mathematical core. Dynamics, topology, probability, stability theory, cocycles, lattice constructions, computation, physical systems, and quantum systems are treated as derived structures, specializations, or applications rather than additional foundational primitives. This specification defines and freezes the ontology, primitive definitions, structural assumptions, terminology, and universal theorem package of SRSD. Its closure claim is explicitly relative to SRSD-expressible systems: it establishes the sufficiency of the frozen core for the framework defined here, without asserting that future mathematics cannot motivate a separately versioned theory.