Table of Contents:
  • <p> Modern large language models demonstrate impressive competence, yet they typically operate in a reactive mode: they produce plausible answers without sustaining autonomous problem investigation. This paper proposes a Dynamic Operator Model (DOM) that formalizes cognitive modes as stable trajectory types in a conceptual state space. We introduce a training field as a stochastic feedback environment that reshapes transition topology and keeps the system in metastable exploration, and a thinking initiator as a symmetry‑breaking perturbation that pushes the system out of the reactive attractor. Using a Koopman/operator perspective, we show how mode transitions become measurable through spectral properties of the evolution operator and propose a computable indicator of the thinking mode, I = S(ψ₁)·|arg(μ₁)|, combining entropy of the leading eigenvector with the phase of the leading eigenvalue. We hypothesize a threshold (phase‑transition‑like) dependence of I on the intensity of the training field λ, and outline a minimal verification protocol that connects the theory to modern engineering approaches (Coconut, Soft Thinking, GFlowNets).</p>