Observable quotients and deterministic inverse tomography for characteristic-function interactions

We study deterministic inverse problems generated by characteristic-function interactions and their Laplace-type smoothing profiles. The first question is not inversion but identification of the exact observable quotient. A full commuting-semigroup profile determines precisely the pushforward of spectral energy by the joint generator-rate map, while no fixed finite family of linear scalar queries identifies an arbitrary positive spectrum. Under finite atomicity, consecutive Prony data become sufficient: an arbitrary positive \(k\)-atomic spectrum is recovered from \(2k\) consecutive values including its mass, whereas a probability spectrum requires only \(2k-1\) additional positive-time values; these counts are worst-case sharp for the consecutive protocol. We give the corresponding confluent-Vandermonde Jacobians and an explicit positive collision family yielding the global Hölder ceiling near merging rates. For known finite supports, the information count changes. One generic positive ray recovers unknown weights, and after complete axis profiles on a bipartite support exactly the cycle rank \(β\) additional mixed Laplace queries are necessary and sufficient. A local maximum-volume selection gives a stable query set of this minimal size. On rational fixed-marginal grids we obtain the optimal number–precision exponent \(Θ(N^{-max\{β/m,1\}})\) for \(m\) fixed physical Laplace queries, together with an explicit full-rank solve-and-round decoder and a constructive compressed design with stated height bounds. A compact Gaussian example illustrates the difference between the quotient determining the extremal value and the finer information determining the maximizing set. Throughout, exact deterministic queries, measurement precision and statistical sampling are kept separate. For separated positive finite spectra, we give the classical Cuyt–Lee mixed-ray recovery scheme with \((r+1)k\) evaluations and identify the limits of this deterministic information count.