CDS-DF: Conservative–Dissipative–Stochastic Decomposition Framework. A rigorous, modular, and self-contained mathematical framework for stochastic dynamics (SDE/SPDE)

Conservative–Dissipative–Stochastic Decomposition Framework (CDS-DF) 0.1 Purpose and scope This work defines the Conservative–Dissipative–Stochastic Decomposition Framework (CDS-DF) as a rigorous, modular, and self-contained mathematical framework for stochastic dynamics (SDE/SPDE) organized around a structural decomposition into: a conservative component, a dissipative (gradient-type) component, and a stochastic component (noise). The framework is presented as a set of formal objects, hypotheses, and theorems that can be audited and revised when completeness or internal consistency requires it. The name is intentionally descriptive and structurally neutral: it indicates a decomposition into conservative, dissipative, and stochastic components, without committing to any specific geometric formalism (Hamiltonian, gradient-flow, metriplectic, etc.) beyond what is explicitly proved. Narrative or interpretive claims are strictly separated from the structural layer. 0.2 Editorial and rigor conventions P0 — Canonical modularity. Each Part has one canonical role (definition layer / theorem layer / long-time and invariance layer / metastability-topology layer / operational layer). P1 — Typed claims. Every claim is labeled as one of: [DEF] Definition [LEM] / [THM] Lemma / Theorem (with a complete proof, or a precise citation whose hypotheses are explicitly verified) [COR] Corollary [EX] Example [REM] Remark [CONJ] Conjecture [PROG] Program / proof sketch (explicitly non-closed) [INT] Interpretation (non-structural; cannot alter definitions, hypotheses, or results) P2 — Closure criterion. A claim is CLOSED only if: it has a complete internal proof, or it cites an external result with a fully checkable mapping from CDS-DF hypotheses to the cited hypotheses; and it is placed within the CDS-DF pipeline (inputs → checks → outputs). P3 — Structure/interpretation firewall. The mathematical structure is presented independently of interpretation. Interpretive content is allowed only in [INT] blocks and is never permitted to modify the formal layer. P4 — Notation discipline. Mathematical expressions are written in plain text using inline code for symbols and operators (no LaTeX). All symbols are defined before use. P5 — Sealed/public status policy. In any sealed/public document, every ledger entry must use only the status labels CLOSED or OPEN. [PROG] is a claim type (proof sketch / program). It must appear only inside items whose ledger status is OPEN, and must never be used as a status label. [CONJ] is a claim type. Conjectures are recorded in ledgers as OPEN (never as a third status). If any section is draft/internal only, it must be explicitly labeled [DRAFT/INTERNAL] and is excluded from sealed/public releases. 0.3 Part architecture The work is organized into five canonical Parts (PART I–PART V). Each document maintains its own closure ledger (Section 0.5). PART I — Formal core (Definitions and the CDS-DF object) Canonical role: fix the language. [DEF] The CDS-DF object, including: state space (typically a separable Hilbert space), conservative drift, dissipative drift, an energy/functional, a metric/dissipation operator, and noise, solution notion (mild / variational / strong), filtration/adaptedness, and the primary stochastic convention (Itô by default; alternative conventions only under explicit admissibility conditions). [DEF] A minimal set of checkable hypotheses (domains, regularity, monotonicity/coercivity, measurability, growth bounds, and any model-specific constraints). [DEF] “CDS-DF compatibility”: explicit conditions under which a model qualifies as an instance of CDS-DF. Output of PART I: a single authoritative specification of the CDS-DF object, its admissible class, and its verification checklist. PART II — Core theorems (Well-posedness, semigroups, generators) Canonical role: turn the core into executable mathematics. [THM] Existence/uniqueness by regime (SDE / variational SPDE / mild SPDE), with explicit hypotheses and solution concepts. [THM] Markov property and semigroup construction; appropriate continuity properties (Feller-type or stated variants); generator characterization and/or the martingale problem. [THM] Fundamental identities in the appropriate functional setting (Itô-type identities and energy estimates stated with explicit domains). [EX] Prototype models with full verification (at least two). Output of PART II: a closed base layer enabling invariant measures, mixing, time-reversal, and asymptotic analysis. PART III — Long-time theory (invariant measures, ergodicity, and rates) Canonical role: provide the long-time analysis toolbox (finite- or infinite-dimensional) downstream dossiers can invoke. [THM] Construction and verification of invariant measures (Krylov–Bogoliubov / tightness routes; generator-based verification when applicable). [THM] Uniqueness and ergodicity routes (e.g., Doob-type mechanisms: strong Feller + irreducibility; and Harris-type mechanisms: Lyapunov + minorization). [THM] Quantitative long-time structure: rates of convergence / mixing (route-dependent; declared explicitly). [REM] Reversible vs non-reversible structure is treated only insofar as it is needed for the exported long-time claims. Output of PART III: exported long-time claims (E3.) plus canonical internal route IDs (P3.) usable by PART V dossiers. PART IV — Metastability and topology (structural “quantization”) Canonical role: formalize and close the metastability/topology layer. [DEF] Metastable sectors, relevant topological invariants, and an operational notion of sector persistence under noise. [THM] and/or [PROG] Bridge results: from noisy dynamics to rare transitions and, where available, quantitative rate statements (large-deviation or Kramers-type asymptotics) under explicitly stated assumptions. [EX] A toy model (e.g., an XY-type chain or an alternative candidate) with closed, checkable claims; quantitative results are included whenever hypotheses can be verified. Output of PART IV: a publishable definition set and CDS-DF-level criteria for the emergence and persistence of sectors. PART V — Operational manual and verified model library Canonical role: make CDS-DF usable as a framework. PART V is split into two canonical documents: PART V-A: operational manual + dossier template + verification protocol. PART V-B: global reference spine §5.11 (theorem-level pointer pinning) and the publication release checklist. A single operational pipeline: Input → Checklist → Outputs. A decision tree indicating which route to use depending on noise structure, degeneracy, regularity, and dimension. A library of verified models presented as “CDS-DF dossiers” (hypotheses, verification steps, invoked theorems, and derived outputs). Extension rules: how to add a new model or add a new lemma/theorem module without compromising closure. Output of PART V: a practical guide, a globally auditable reference spine (§5.11), and a set of fully verified demonstrations of CDS-DF. 0.4 Dependency map PART I is upstream of all other Parts. PART II depends on PART I. PART III depends on PART I–PART II. PART IV depends on PART I–PART II and partially on PART III (when infinite-dimensional rate theory is invoked). PART V depends on PART I–PART IV. 0.5 Closure ledger (accountability) Each Part maintains a local closure ledger containing: a list of claims with stable IDs (e.g., P2.T3 for Theorem 3 in PART II), a status label: CLOSED / OPEN (Conjectures are always recorded as OPEN and typed [CONJ]), (with any unfinished item—including a [PROG] sketch—recorded as OPEN), explicit dependencies on earlier claims, evidence: internal proof or a precise citation plus a verified hypothesis-mapping, minimal actions required to move a claim to CLOSED. 0.6 Writing conventions (publication-ready style) Mathematics first; interpretation is restricted to [INT] blocks. Every symbol is defined before use; definitions are collected where appropriate. Each theorem includes: checkable hypotheses, statement, proof or citation, and a minimal application note describing where it is used downstream. Each example includes: a complete checklist and explicit pointers to where each condition is verified (Part/Section/Claim IDs).