Circle and Tangent: A Model Hypothesis on Thinking Boundaries and Problem Entry Points

This bilingual preprint proposes the “Circle and Tangent” model as a conceptual hypothesis for describing the relationship among thinking boundaries, problem constraints, and problem entry points. In this model, the “circle” represents the relatively stable target structure behind a problem, while the “tangent” represents the constraints that allow an initially divergent thinking trajectory to contact that target structure. The paper argues that high-degree cognitive extension may correspond to a higher potential ceiling, but such potential does not automatically appear as stable ability. Without sufficient constraints, extended thinking may become scattered, unstable, or disconnected from the target structure. With effective constraints, however, the same extension may be compressed, calibrated, and transformed into problem entry, structural fit, and transferable reasoning. The model is derived from the author’s learning experience, previous theoretical writing on curiosity and systematic learning, and reflections on GPT-assisted thinking. Related literature on scaffolding, cognitive load, problem representation, productive failure, creativity under constraints, and learning transfer is used as adjacent theoretical background and external calibration. These references do not constitute the direct source or empirical proof of the model.