Total-Energy Pushforwards of Finite Orthogonal-Polynomial Ensembles

Let Zᵍ(β)=∑ⱼ₌₁ᵐgⱼ e-β eⱼ, e₁<⋯<eₘ, gⱼ>0, and let τᵣᵍ(β) =∑|I|=r(∏i∈ Igᵢ) Δ(eI)² e^-β∑i∈ Ieᵢ 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 \(a_I^g=(∏i∈ Ig_i)Δ(e_I)^2\), with Hamiltonian \(H_r(I)=∑i∈ Ie_i\); its total-energy pushforward is therefore a finite weighted density of states. We determine exactly what the scalar observation \(τ_r\), and therefore its curvature \(C_r=(logτ_r)''\), retains and forgets. The function \(τ_r\) 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, \(τ_r\) recovers all one-body weights. For the canonical projection kernel \(K_r(β)\), every pair of linear statistics satisfies the exact covariance identity Cov\!(∑ᵢ fᵢηᵢ,∑ᵢ hᵢηᵢ) =∑i<j(fᵢ-fⱼ)(hᵢ-hⱼ)|Kᵣ(i,j;β)|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 \(β=(kBT)⁻¹\) has its physical meaning relative to this fixed reference measure, the heat capacity is \(kBβ^2 C_r\). 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 \(τ_2\) 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.