Fixed-Support Fold Geometry in Finite\\ Orthogonal-Polynomial Ensembles

Fix distinct one-body energies e₁<⋯<eₘ and positive weights g₁,…,gₘ. In the r-particle orthogonal-polynomial ensemble, the unnormalized mass of a configuration I⊆[m], $|I|=r$, is aI(g)=Δ(eI)²∏i∈ Igᵢ, and only the total energy Hᵣ(I)=∑i∈ Ieᵢ is observed. We study the resulting fixed-support inverse problem: when does the complete energy-resolved pushforward determine the one-body weights, and what is the exact geometry of its failures? For every total-energy atom x, let qₓ be the conditional microscopic law inside the fiber :Hᵣ(I)=x\. We prove that the Jacobian of the logarithmic pushforward map with respect to the logarithmic one-body weights is exactly the matrix of conditional occupation expectations Bₓᵢ= Eqₓ[\1_∈ I\], and that its component Hessians are conditional occupation covariance matrices. Thus B is the exact tangent observability matrix of the coarse-grained thermodynamic experiment. Singleton energy fibers give a linear incidence system; in the pair sector its graph recovers all weights whenever every connected component is non-bipartite. This implies, in particular, fixed-support rigidity of the generic Boolean parity supports used in an earlier support-changing non-identifiability construction. We then give a complete global classification when the pair-sector singleton graph is connected and bipartite. Singleton atoms leave one alternating weight direction. Every collided energy fiber restricts this direction by a positive Laurent equation. For genuine pair-sum supports with at least three levels, the formally possible continuous invisible case cannot occur. The fixed-support fiber is therefore either a singleton or a two-point set. A two-point fiber occurs exactly when all active collision fibers share one same-side mass ratio; the observation map then has a common quadratic fold. Away from the fold center, both realizations are locally identifiable despite being globally indistinguishable. The smallest positive example has four levels. On the support $(0,1,2,3)$, the weights (1,1,1,1), (1/3,3,3,1/3) have identical two-particle total-energy partition functions e-β+4e-2β+10e-3β+4e-4β+e-5β, but exchange the two microscopic masses in the resonant fiber $0+3=1+2$. We prove minimality and classify every four-level fixed-support fiber. A separate eight-level example exhibits the same fold mechanism simultaneously across six collided energy fibers. These results identify a new coarse-grained singular geometry at the intersection of finite OPEs, log-linear models, additive pair-sum structure, and inverse statistical mechanics.