Maximal Compound Curvature as a Complete Observable for Finite Partition Functions

Abstract

Let $Z^g(\beta)=\sum_{j=1}^{m}g_j e^{-\beta e_j},$ $\qquad 0<g_j,\qquad e_1<\cdots<e_m,$ be a primitive finite weighted partition function. Its Hankel compound partition functions are $\tau_r^g(\beta)$ $=\det\!\bigl(M_{a+b}(\beta)\bigr)_{a,b=0}^{r-1},$ $\qquad$ $M_n(\beta)=\sum_j g_j e^{-\beta e_j}e_j^n,$ and their logarithmic curvatures are \(C_r=(\log\tau_r)''\). We show that the single maximal proper curvature \(C_{m-1}\) is a complete exact observable for the original weighted spectrum, modulo common energy translation and common weight scaling. The key identity is a complement--time-reflection formula $\tau_{m-r}^g(\beta)$ $=GD^2e^{-\beta S}\tau_r^{g^\vee}(-\beta),$ $\qquad$ $g_j^\vee=\frac{1}{g_jq'(e_j)^2},$ where \(q(z)=\prod_j(z-e_j)\), \(S=\sum_j e_j\), \(G=\prod_jg_j\), and \(D^2=\prod_{a<b}(e_b-e_a)^2\). In particular, \(C_{m-1}^g(\beta)=(\log Z^{g^\vee})''(-\beta)\), and the complement map is involutive. After normalization, the same coefficient identity is an exact particle--hole duality between the associated finite discrete orthogonal-polynomial ensembles. A finite local jet of this one scalar therefore reconstructs not only the original weighted spectrum but every particle-number sector, its canonical determinantal correlation kernel, all occupation correlations, and all gap probabilities. We then resolve the scalar curvature into positive principal-minor spectral channels. For the leading \(r\times r\) Jacobi block, $\frac{\dd}{\dd\beta}\lambda_i^{(r)}$ $=-C_r\lvert v_i^{(r)}(r)\rvert^2<0,$ and the associated channel measure pushes forward under its moving eigenvalue to Lebesgue measure on one energy interval. Across all proper blocks, the channels are in bijection with all unordered pairs of energy levels. At the positive-temperature endpoint, the maximal block has velocity tails $-\lambda_i'(\beta)\sim B_i e^{-\delta_i\beta},$ $\qquad \delta_i=e_m-e_i,$ which form an exact one-sided coordinate chart: $e_i=e_m-\delta_i,$ $\qquad$ $\frac{g_i}{g_m}$ $=\frac{\delta_i^2}{B_i}$ $\prod_{k\ne i}\left(\frac{\delta_k}{\delta_k-\delta_i}\right)^2.$ The normalized endpoint coefficients form a multiplicative graph \(g_b/g_a\); its logarithmic slopes are exactly the inverse temperatures at which two Boltzmann contributions cross. We separate these thermodynamic compression, decomposition, and interpretation statements from the classical Toda ingredients: isospectrality, principal-minor monotonicity, norming constants, bidiagonal coordinates, scattering asymptotics, and tropical dominance envelopes. An exact four-level example and two independent proofs of the complement reflection are included. The results concern exact identifiability; stability under noisy differentiation or low-temperature fitting is not asserted.