Finite Partition-Function Curvatures

Abstract

Let $Z(\beta)=\sum_{j=1}^{m}g_j e^{-\beta e_j},\qquad$ $0<g_j,\quad e_1<\cdots<e_m,$ be a primitive finite weighted canonical partition function, and let \(K=(\log Z)''\) be its log-partition curvature. Physically, \(K\) is the canonical energy variance and satisfies \(C_V=k_B\beta^2K\). We develop a curvature-only description of finite-level systems. From the derivatives of \(K\) at an arbitrary finite inverse temperature, we construct centered moments and a hierarchy of differential Hankel determinants. Their first rank loss detects the exact number of distinct energy levels. If that number is \(m\), the jet \(K,K',\ldots,K^{(2m-3)}\) reconstructs the centered energies, the local Gibbs weights, and hence the original weighted spectrum modulo the two unavoidable gauges: a common energy translation and a common weight factor. The next determinant has the form $\Delta_{m+1}=\Delta_m K^{(2m-2)}+Q_m(K,\ldots,K^{(2m-3)}),$ so each exact \(m\)-level stratum is an autonomous differential equation of order \(2m-2\). Conversely, on a connected interval, positivity of the lower differential Hankel determinants together with \(\Delta_{m+1}=0\) is sufficient for a unique positive finite-level realization. The proof combines a self-contained one-dimensional flat moment lemma with ordinary-differential-equation uniqueness. We also identify the compound determinants \(Z^s\Delta_s\) as positive subset partition functions and place their bilinear identity at the boundary with finite Toda theory. After normalization, these compound functions are finite discrete orthogonal-polynomial ensembles and hence fixed-size projection determinantal point processes. Consequently, the same ordinary-curvature jet reconstructs every particle-number sector, its canonical correlation kernel, all occupation correlations, and all gap probabilities, up to the induced common translation of the energy support. Explicit two- and three-level examples, an alternative two-level proof, and a nonpositive formal solution show why the positivity chamber is essential. The results are exact structural statements; numerical stability under noisy calorimetric differentiation is a separate problem.