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 equilibrium thermodynamics of a populated collective occupation field under Maxwell–Boltzmann/Gibbs counting and identifies which of its natural occupancy potentials governs a deterministic collective field. The treatment is ensemble-exact: for cell weights W(n) = Gⁿ/n! the grand partition function is Ξ = exp(Gz), the mean occupation is n̄ = Gz, and the Legendre free energy of the mean is F_MB(n̄) = k_BT[n̄ ln(n̄/G) − n̄], giving μ_MB(n̄) = k_BT ln(n̄/G) for every n̄ > 0 — the logarithmic coefficient at the equilibrium free-energy level is exactly k_BT, with no Stirling approximation and no low-occupancy flattening. Three occupancy potentials that coincide at high occupation are shown to answer different questions: the ensemble chemical potential of the fluctuating mean (the object that governs a collective field), the discrete insertion cost ΔF(n) = k_BT ln[(n+1)/G] (exact for jump kinetics), and the sharp-occupation continuation k_BT[ψ₀(n+1) − ln G] (a derivative of an analytic continuation, with no ensemble carrier at low occupancy). The domain statement follows from the same ensemble: Var(n) = n̄, so relative fluctuations scale as n̄⁻¹ᐟ², and a deterministic collective field requires large occupation under a specified coarse graining. At occupation of order unity and below this paper predicts no nonlinear single-particle Schrödinger equation; the few-quantum sector must be derived from a discrete stochastic or Fock-space description. Neutron interferometry bounds therefore remain decisive against a universal one-particle logarithmic term, while the present mechanism is compatible with them as an explicitly collective effective closure. (This version supersedes v1.0, whose low-occupancy switch-off claim conflated the three potentials above; the complete revision record is Appendix B.) Monte Carlo verification (three independent seeds per temperature, energy bookkeeping validated against the full Hamiltonian; reproducibility details in Appendix B) gives interacting-chain slopes d ln⟨n⟩/dV = −2.2509 ± 0.0075, −1.0597 ± 0.0084, −0.5131 ± 0.0053 at T = 0.5, 1, 2, approaching the ideal −1/T as K/T decreases, and occupancy histograms with the distinct fingerprints R(n)(n+1) ≈ const (Maxwell–Boltzmann/Gibbs) versus R(n) ≈ const (uniform-configuration “Bose”) sampling. These tests validate the equilibrium counting implementation; they do not derive the promotion of the free-energy term into conservative phase dynamics. For Bose counting, μ_BE = k_BT ln[n̄/(n̄+G)] = k_BT ln[ν/(1+ν)], with ν = n̄/G, so the response agrees with the dilute Maxwell–Boltzmann form but saturates at high occupancy. A cold-atom search for a temperature-proportional ln ρ contribution therefore remains a conditional test of the counting class, subject to three explicit assumptions: a populated collective FDS field is present, its bath temperature is locked to the measured gas temperature, and the equilibrium free-energy derivative enters the effective phase dynamics. Failure of any one assumption changes the experimental interpretation.