Homometry and the Reconstruction of Relations: Admissibility, information, and stable reconstruction

Autocorrelation preserves pair differences while concealing how they fit together. Homometry and the Reconstruction of Relations proves that every nonempty finite Boolean relation can occur as the complete set of translation classes in a binary homometry fibre on an elementary abelian two-group. Each coordinate is recovered by a single normalized cubic Fourier observation. The construction combines a complete binary AND/OR fibre, four-copy correlation encoding, independent Fourier planes, and compatible translations. Every accepted input has exactly one class. Hidden labels realize every rational joint law of binary observations under uniform classes. A separate positive-integer alphabet construction gives explicit domain and dynamic-range bounds. A finite reduced reconstruction algebra describes admissible classes, observable multiplicities, Shannon and collision entropy, and error-erasure certificates. The main paper gives the sharp number of unrestricted monomial observations correcting one arbitrary error on a phase torus, a precise lattice criterion for stability across changing Fourier supports, and a cyclic information-precision law. An arithmetic criterion characterizes translation by local integrality of the spectral quotient. The companion, Relations, Geometry, and Controlled Observation, develops complementary cyclic rigidity criteria. All indexed p-adic Fourier valuations determine a perfect difference set up to translation. A short shift certificate excludes an order-five zero pattern without classification. A reduced lacunary divisor test, an additive-period theorem, and explicit overlap and energy bounds sharpen recognition after a zero run. An exact fourth-moment defect measures decoder inconsistency; a separate classical group-theoretic theorem makes one nonaffine collineation sufficient for Desarguesian recognition. Earlier trace-power criteria, selected-order analytical results, and quotient-moment proofs of the classical order-five and order-seven classifications remain included. Independent companion sections connect homometry to polygon shadows, winding, compatible memory, harmonic depth, ordered signatures, positive support certificates, passive masks, and response identifiability. The two papers keep their proof dependencies separate and preserve the scope of each observation model. General cyclic-plane rigidity and uniform lacunary rigidity remain open: the papers identify exact sufficient conditions without proving that all perfect difference sets satisfy them. Optimal binary realization size, efficient reconstruction, and stable physical acquisition also remain unresolved. The project includes both papers, editable sources, vector figures, reproducible verification, a separate corrections and scope report, and the preserved source collection. Analytical proofs, exact finite checks, and numerical illustrations are distinguished throughout.