Total-Energy Pushforwards of Finite Orthogonal-Polynomial Ensembles

Let

$Z^g(\beta)=\sum_{j=1}^{m}g_j e^{-\beta e_j},$

$\qquad e_1<\cdots<e_m,\quad g_j>0,$

and let

$\tau_r^g(\beta)$

$=\sum_{|I|=r}\left(\prod_{i\in I}g_i\right)$

$\Delta(e_I)^2 e^{-\beta\sum_{i\in I}e_i}$

be its order-\(r\) compound partition function. After normalization this

is a finite orthogonal-polynomial ensemble, hence a rank-\(r\) projection

determinantal point process. It is simultaneously a one-parameter

canonical Gibbs family relative to the fixed positive reference masses

\(a_I^g=(\prod_{i\in I}g_i)\Delta(e_I)^2\), with Hamiltonian

\(H_r(I)=\sum_{i\in I}e_i\); its total-energy pushforward is therefore a

finite weighted density of states. We determine exactly what the scalar

observation \(\tau_r\), and therefore its curvature

\(C_r=(\log\tau_r)''\), retains and forgets.

The function \(\tau_r\) is the bilateral Laplace transform of the positive

measure obtained by pushing the unnormalized configuration measure through

the total-energy map. Equality of compound curvatures is equivalent to

equality of these pushforwards up to a positive mass factor and a

translation. The Gibbs law disintegrates over equal-total-energy fibers:

temperature changes only the law of the total energy, while the conditional

law inside every fiber is independent of temperature. At the level of arbitrary positive configuration masses on a fixed

configuration space, the entire ambiguity is a product of open simplices,

of dimension equal to the number of configurations minus the number of

distinct total-energy atoms. Conversely, on a known one-body support with

an injective \(r\)-subset-sum map, \(\tau_r\) recovers all one-body weights.

For the canonical projection kernel \(K_r(\beta)\), every pair of linear

statistics satisfies the exact covariance identity

$\operatorname{Cov}\!\left(\sum_i f_i\eta_i,\sum_i h_i\eta_i\right)$

$=\sum_{i<j}(f_i-f_j)(h_i-h_j)|K_r(i,j;\beta)|^2.$

In particular, compound curvature is simultaneously the variance of total

energy, the Fisher information of the inverse-temperature family, and the

Dirichlet energy of the energy coordinate. When \(\beta=(k_{\mathrm B}T)^{-1}\)

has its physical meaning relative to this fixed reference measure, the heat

capacity is \(k_{\mathrm B}\beta^2 C_r\). The same bilinear form gives the

full finite-temperature linear response to an energy tilt.

We then analyze a positive Boolean parity family whose coefficients are the

classical Hirota modified-KdV multi-soliton coefficients. An elementary

alternant proof gives equality of the even and odd two-particle

pushforwards. Under injectivity of the \(\{0,1,2\}^d\) total-energy map, we

prove the converse: every positive parity solution is necessarily of this

pair-product form. Finally, a fully explicit eight-level example gives two

non-translate finite orthogonal-polynomial ensembles with identical

\(\tau_2\) and identical total-energy laws for every inverse temperature,

but inequivalent microscopic particle laws. Thus complete observation of

one thermodynamic linear statistic can remain strictly weaker than

microscopic determinantal identifiability.