A Formal Framework for Reasoning Stability Under Recursive Inquiry: Operator Derivation, Implementation, and Epistemic Classification

We address the problem of reasoning stability under recursive inquiry conditions: the formalquestion of what structural properties a reasoning system’s internal dynamics must possess toremain stable rather than drift. In the reviewed literature, prior frameworks including Shannoninformation theory, Wiener cybernetics, the Free Energy Principle, and transformer-basedarchitectures do not formally derive recursive reflection as a structurally necessary operator froman entropic coupling model. This paper provides that derivation.We define an entropic coupling functional ε(t) = ∫D I(x, t) · (∂S/∂t) dx over C2 entropy-density and informational gradient fields on a bounded domain D. Under a stated primitivederivative selection criterion and specified evolution model, this functional admits four primitiveoperator classes: Ω (intent gradient evolution), ∆ (entropy evolution), Φ (curvature restoration),and Ψ (mixed spatio-temporal reflective coupling). The decomposition is determinate giventhe functional and the selection criteria. In the local linearized regime under stated closureassumptions, Ψ is necessary for asymptotic stability and sufficient to induce eigenvalue shift tothe left half-plane when its coupling coefficient κ > 0; its absence produces neutral oscillation.We implement this formal system as the HDT2 operator chain, defining a computableReasoning Integrity Index and drift detection mechanism. We then present bounded applicationsof the operator framework to epistemic classification (Gate and Drift Test), applied reasoningmeasurement (Cognitive Compass System), and persistent reasoning-style characterization(Recursive Personality Index). Preliminary experimental results establish infrastructure feasibilityand ∆-operator measurement-path viability at proof-of-concept level; core empirical hypothesesremain untested.The primary result is system-bounded and local: it does not claim global stability, does notassert that cognition is literally a physical field, and does not generalize beyond the definedformal system. Within those bounds, recursive reflection is not a design choice but a derivablestructural necessity.