Observable quotients and deterministic inverse tomography for characteristic-function interactions

We study inverse reconstruction from characteristic-function interactions and Laplace-type semigroup measurements, beginning with the observable quotient: the complete profile identifies precisely the pushforward of spectral energy under the joint generator-rate map, and no fixed finite set of scalar observations can identify an arbitrary positive spectrum. Under finite atomicity, the classical consecutive Prony protocol gives exact recovery from \(2k\) values for a positive \(k\)-atomic spectrum, or \(2k-1\) additional positive-time values for a probability spectrum; these counts are sharp for that specified protocol. The emphasis is on structural query design beyond classical Prony reconstruction. With known finite bipartite support and complete axis profiles, exactly \(\beta\) additional mixed Laplace queries are necessary and sufficient, where \(\beta\) is the support graph's cycle rank. A maximum-volume selection supplies a stable minimal-size design. For rational fixed-marginal grids, we prove a sharp deterministic number–precision exponent \(\Theta(N^{-\max\{\beta/m,1\}})\) for \(m\) fixed queries and give explicit decoders. A Gaussian example identifies the dependence defect from canonical-correlation geometry. Deterministic exact data, finite precision and statistical sampling are treated as distinct problems.