Return Theory (RT): Truth and Reference in Embedded Observation — Geometry, Mathematics, and Physics — A Companion to Foundations

This sector-independent companion to RT Foundations integrates truth, reference, observational entitlement, mathematical architecture, and physical instantiation within the geometry of embedded observation. It distinguishes underlying situations, admitted records, validation and representation references, reconstruction, observational images, and the claims those images license. Twenty-two proved mathematical results address fibre-based identification, measurable descent, scalar ambiguity, common post-processing, matched predictive measures, Lorentzian temporal attribution, coherent phase, joint carriers, reference gauges, exact transverse-record criteria, unresolved factorisations, consistent clock changes, and first-order nuisance projection. A delimited philosophical chapter reads the architecture alongside correspondence, coherence, deflationary, conventionalist, and indeterminacy-of-reference positions without presenting RT as a theory of truth or mental content. RT Thoughts (DOI: 10.5281/zenodo.23311766) is cited as the extended philosophical companion and supplies conceptual context, not mathematical premises. Standard mathematical results, RT role assignments, philosophical readings, conditional host consequences, and discriminating connection claims retain separate status. The document adds no foundational RT postulate or physical law and reports no empirical test verdict. It contains no sector inventory, dataset, fit score, active run, or publication schedule, allowing later applications to instantiate its stated assumptions without requiring routine revision of this companion. Genuine errors remain correctable. The complete LaTeX source and thirty exact illustrative verification checks accompany the PDF. The normative foundation is RT Foundations 1.0.0, DOI: 10.5281/zenodo.23286579.