A Formal Framework for Reasoning Stability Under Recursive Inquiry: Operator Derivation, Local Stability, and Exploratory Applications

Abstract We address the problem of reasoning stability under recursive inquiry conditions: the formal question of what structural properties a modeled reasoning system's internal dynamics must possess to remain locally stable rather than drift. In the reviewed literature, prior frameworks including Shannon information theory, Wiener cybernetics, the Free Energy Principle, interrogative models of inquiry, transformer architectures, and entropy-regularization methods provide important accounts of uncertainty, feedback, inference, inquiry, attention, and entropy control. They do not, in their standard formulations, derive a reflection-like operator from the specific uncertainty-gradient coupling functional defined in this manuscript and tie that operator's presence or absence to a local linearized stability condition. This paper addresses that narrower formal gap. We define a model-specific uncertainty-density field S (x, t) and an informational gradient field I (x, t), both over a bounded domain 𝒟, and introduce the coupling functional ε (t) = ∫_𝒟 I (x, t) · (∂S/∂t) dx. Here S (x, t) is not identified with Shannon entropy, thermodynamic entropy, von Neumann entropy, Kolmogorov complexity, or any other established entropy measure. It is a formal uncertainty-density field representing unresolved informational constraint within the model. Under a stated primitive derivative-selection criterion, specified evolution model, and first- and second-order closure criterion, the framework retains four stability-relevant operator classes: Ω (informational-gradient evolution), Δ (uncertainty-density evolution), Φ (curvature restoration), and Ψₘath (mixed spatio-temporal reflective coupling). This basis is not claimed to be complete or minimal over all possible derivative operators. It is a sufficient, non-redundant, and closure-minimal stability-relevant operator basis within the declared formal system. In the local linearized regime under stated closure assumptions, Ψₘath = ∂²I/∂x∂t is necessary for the specified asymptotic local stability condition and sufficient to induce eigenvalue shift when its coupling coefficient κ > 0. This result is system-bounded, local, and conditional. The manuscript does not establish global stability, nonlinear Lyapunov stability, or universal stability for reasoning systems. The parameter κ is treated as a formal coupling parameter; no empirical identification procedure for κ is provided here. We then present the HDT² operator chain, the Reasoning Integrity Index, the Gate and Drift Test, the Cognitive Compass System, and the Recursive Personality Index as model-guided exploratory applications of the formal vocabulary. These sections are not presented as stability-preserving discretizations or mathematical continuations of the PDE formalism. In particular, the computational analogue Ψcomp is distinguished from the formal operator Ψₘath; no preservation proof is supplied showing that Ψcomp inherits the stability role of Ψₘath. Finally, the empirical hypotheses H1–H4 are presented as untested theoretical predictions, not validated findings. The preliminary implementation exercises reported in this manuscript establish only infrastructure feasibility and partial measurement-path viability. Because the Ω measurement path failed in the reported implementation, H1–H4 could not be validly tested. Future empirical validation requires preregistration, independent annotation, inter-rater reliability assessment, a priori power analysis, declared inclusion and exclusion criteria, and separation of confirmatory from exploratory analyses. The primary contribution is therefore bounded: this manuscript derives a stability-relevant operator structure within a defined formal system and establishes a local linearized stability result for that system, while explicitly separating formal derivation from exploratory computational and empirical applications. Keywords: reasoning stability, uncertainty-gradient coupling, recursive reflection, operator derivation, local linearized stability, Reasoning Integrity Index, exploratory measurement, inquiry dynamics