When does an oscillation count as a return? Entropy budgets in an isolated thermomechanical model

Abstract

Preprint (version 1.0), not peer reviewed. Repeated crossings of a coordinate can conceal irreversible changes in the rest of a system. We develop an upper-undergraduate computational activity that distinguishes crossings, finite-amplitude returns and recurrence of the complete thermodynamic state. A nonlinear structural coordinate exchanges energy with a finite internal heat bath; damping produces entropy while conserving total energy. Eliminating the bath temperature gives an entropy ceiling at each coordinate, so that increasing entropy progressively excludes previously accessible configurations. This construction predicts when a specified return becomes impossible and connects the return problem to stability under thermal constraints. For one example, 25 coordinate crossings correspond to only one return between prescribed state regions and no complete-state recurrence. An additional example is unstable at fixed temperature but locally stable at fixed total energy. Independent integrations using different state variables and numerical methods check the energy, entropy and event budgets. A worksheet and reproducible code accompany the derivations. The contribution is an instructional synthesis with explicit diagnostics, rather than a new thermodynamic law; learning outcomes have not been measured. Files: manuscript PDF and a supplementary archive containing simulation scripts, validation results, numerical figure data, environment requirements, and a teaching worksheet with solutions. All reported data are numerical outputs of the stated model; no experimental or student data were collected. AI disclosure: OpenAI Codex (GPT-6) assisted research ideation, algebraic development, literature discovery and comparison, code generation, numerical analysis, data-based plotting, manuscript drafting and translation, and teaching exercises. The archive documents computational checks and provenance. Assistant reviews are not human peer review. These checks do not replace the named author's critical review and responsibility. Funding: no specific grant. Competing interests: none declared.