CDS-DF: Conservative–Dissipative–Stochastic Decomposition Framework: A Modular Architecture for Rigorous Stochastic Dynamics

The Conservative-Dissipative-Stochastic Decomposition Framework (CDS-DF) This work presents the Conservative-Dissipative-Stochastic Decomposition Framework (CDS-DF), a rigorous, modular, and self-contained mathematical architecture designed for the systematic analysis and verification of stochastic dynamics in both finite and infinite-dimensional state spaces H. The framework is organized around a canonical structural decomposition of the drift term b(q) into a conservative component X(q), a dissipative gradient-type component G(q) GradJ(q), and a stochastic noise term Sigma(q) dW_t. The architecture is partitioned into five functional layers: Formal Core: Defines the CDS-DF object, admissible solution concepts (strong, mild, and variational), and the structural constraints of Strict Conservativity (grad J(q), X(q))_H = 0 and FDT-Certification Sigma Sigma^* = 2 beta^{-1} G. Analytical Foundation: Establishes existence, uniqueness, and well-posedness regimes for SDEs and SPDEs (monotone/coercive), including a dedicated route for Nonsmooth Inclusions via stochastic variational inequalities. Ergodic and Geometric Structure: Formalizes the long-time behavior through invariant measures, irreversibility-to-action mappings, and Information-Geometric stability. It certifies the Gibbs measure pi propto exp(-beta J) as an invariant distribution under specific structural couplings. Metastability and Topology: Rigorously defines metastable sectors, transition rates via Wentzell-Freidlin large deviations, and the emergence of classical Newtonian trajectories dq/dt = X - G grad J in the low-noise limit epsilon -> 0. Operational Closure: Provides an auditable protocol for model verification through "Dossiers," ensuring that any dynamical instance satisfies the framework's internal consistency checks and closure ledgers. By decoupling the structural decomposition from specific geometric formalisms, CDS-DF provides a neutral yet robust platform for the study of complex stochastic systems, ranging from molecular dynamics to quaternionic vorticity and dissipative field theories. The framework ensures mathematical reproducibility through a strict tag-based dependency map and an authoritative reference spine. Keywords: Stochastic Dynamics, SDE/SPDE, Structural Decomposition, Metastability, Information Geometry, Fluctuation-Dissipation Theorem, Large Deviations.