PAPER-DCQ10: A Thermodynamic Reading of the Morse–Thimble Structure in the Discrete–Continuous–Quantum Correspondence

This paper develops a thermodynamic reading of the adapted Morse landscape arising in the Discrete–Continuous–Quantum correspondence. The starting point is the compact six-dimensional phase-orbit manifold

N ≃ (CP1)3,

together with an adapted Morse function

f : N → R

whose prescribed local minima include the 64 finite phase-sector states inherited from the six-bit configuration space

H6 = {±1}6.

The main purpose of the paper is deliberately limited. We do not claim a first-principles derivation of microscopic thermodynamics from a physical Hamiltonian, nor do we assume that DCQ2 has already supplied a rigorous Picard–Lefschetz convergence theorem. Instead, we show that once the adapted Morse function is interpreted as an effective energy landscape,
the compact real integral

Z(β) =󰁝Ne−βf dμN

defines a finite equilibrium partition function. The formal thimble language is used only as a semiclassical sector bookkeeping device for organizing contributions from critical points.

Two consequences are emphasized. First, in the low-temperature regime, the 64 prescribed minima contribute a discrete ground-sector degeneracy term

kB ln 64,

after separating non-universal local fluctuation-volume factors. Second, a simplified twolevel critical-sector model exhibits a Schottky-type heat-capacity peak, illustrating how excited Morse sectors may become relevant at intermediate temperature scales.

Thus the paper isolates a mathematically stable thermodynamic layer attached to the DCQ Morse landscape while leaving stronger dynamical, gravitational, and phenomenological interpretations to later work.