PAPER-DCQ4: A Thermodynamic Reading of the Morse–Thimble Structure in the Discrete–Continuous–Quantum Correspondence

This paper proposes a thermodynamic reading of the Morse–thimble structure that appears in the Discrete–Continuous–Quantum (DCQ) programme. The starting point is the six-dimensional compact symplectic core N ≃ (CP1)3, together with a Morse function f : N → R whose minima reproduce the 64 distinguished discrete states identified in the earlier DCQ construction. The main purpose of the paper is narrow and structural. We do not claim a firstprinciples derivation of microscopic thermodynamics from an underlying physical Hamiltonian. Rather, we show that once f is interpreted as an effective energy landscape and the Lefschetz-thimble decomposition is read through a positive thermal weighting prescription, one may define a consistent partition function and the associated equilibrium quantities of statistical mechanics. In this sense, the paper should be read as an interpretive bridge between the Morse geometry already present in the DCQ framework and a thermodynamic layer naturally associated with it. Two consequences are emphasized. First, in the low-temperature regime, the partition function is dominated by the 64 minima of f, leading to a residual entropy of the form S ∼ kB ln 64 up to fluctuation corrections. Second, a simplified two-level thimble model already exhibits a Schottky-type anomaly in the heat capacity, illustrating how excited critical sectors may become thermally relevant at intermediate scales. The paper therefore isolates a mathematically controlled thermodynamic reading of the Morse–thimble structure, while leaving broader gravitational, cosmological, or particlephysics interpretations to future work.