PAPER-DCQ4: Temperature as an Emergent Concept: From Morse Critical Points to Statistical Thermodynamics via Thermodynamic Construction in the Discrete–Continuous–Quantum

This paper introduces temperature as a naturally emergent thermodynamic concept within the discrete–continuous–quantum correspondence framework established in DCQ1 and DCQ2. We demonstrate that by reinterpreting the Morse function as an energy function and utilizing the statistical weights provided by the Lefschetz thimble decomposition, a rigorous partition function can be defined, leading to the emergence of thermodynamic quantities such as internal energy, entropy, and free energy. We address the positivity of statistical weights through analytic continuation and discuss the physical conditions underwhich phase oscillations can be neglected. We further explore the geometric coupling between temperature and Berry curvature, and analyze the physical behavior in both highand low-temperature limits. Notably, the model naturally reproduces the ln 64 form of black hole entropy at low temperatures, and reveals indications of thermodynamic phase transitions induced by instanton tunneling at finite temperatures. This construction provides a paradigm for the emergence of complete thermodynamic laws from discrete combinatorial structures, offering a new perspective on the relationship between information, geometry, and thermodynamics.