Unified Valence Framework: Categorical, Metric, Dynamical, Modular, and Complexity-Theoretic Structure

The Unified Valence Framework introduces a cross-disciplinary structural invariant designed to measure, compare, and transport mathematical, computational, and geometric information across heterogeneous domains. Valence organizes systems into layered categorical, metric, dynamical, and modular components, enabling consistent structural comparison, deformation tracking, and semantic compression. The framework is intentionally architecture-agnostic and applies to algorithms, networks, differential systems, probabilistic models, and algebraic structures. Examples and applications illustrate how Valence can detect bifurcations, collapse high-dimensional behavior into interpretable signatures, and quantify computational or geometric complexity. The goal is not to resolve specific conjectures, but to provide a unifying measurement language with general theoretical and practical utility.