Thermal Origin of the Repulsive Logarithmic Schrödinger Free-Energy Term: Exact Occupancy Dependence, Neutron Bounds, and a Cold-Atom Test of the Counting Class

This paper derives the thermal origin of the repulsive (acoustic-branch) logarithmic term of the logarithmic Schrödinger equation (LogSE) at theequilibrium free-energy level; the focusing matter branch, and the promotion ofthe free-energy term to a conservative phase term in the dynamical equation, areexplicitly reserved for the dynamical part of the program. Maxwell–Boltzmann/Gibbs counting of n quanta over M micro-nodes yields an exact per-cell entropicpotential V (n) = kB·T·ψ₀ (n+1) − ln M; the discrete insertion potential is ΔF (n) =kB·T·ln (n+1) /G. Both prescriptions give the logarithm with coefficient kB·T athigh occupancy — exact at the equilibrium free-energy level, with all cell-sizeambiguity absorbed by the reference scale — and both reduce to a weak contactlikeresponse at low occupancy, differing only by O (1) coefficients: the lowoccupancyswitch-off is prescription-independent. We define the local logarithmiccoefficient bₗoc (n) = kB·T·n·ψ₁ (n+1) (trigamma), with bₗoc → kB·T for n ≫ 1and bₗoc ≈ (π²/6) kB·T·n for n ≪ 1. Single-particle interferometry bounds onfundamental LogSE terms (Shull et al. 1980; Gähler, Klein in the Bose class thelogarithmic response saturates at high occupancy. A BEC measurement of the ln ρform and its density dependence therefore tests both the thermal coefficient andthe underlying counting class. Monte Carlo simulations with single-quantumtransfer confirm the logarithmic high-occupancy equation of state (slope −1/T;residual 2–13% deviations are systematic K/T coherence corrections, statisticalerrors 0. 5–1%), directly measure the discrete insertion free energy via occupancyhistogramratios (the Gibbs n! fingerprint R· (n+1) = const vs. the Bose fingerprintR = const), and demonstrate the counting-class equation-of-state discriminator insilico (MB slope −1/T at all occupancies; Bose slope − (1+n̄) /T).