Return Theory (RT) — Empirical: Multi-Carrier Radial Closure

This standalone empirical study tests Return Theory (RT) at its multi-carrier observational boundary: whether one frozen reconstructed radial history H_rec(z_obs) can satisfy heterogeneous null-cone carriers without being refitted to any target dataset. RT does not alter physical spacetime, gravitational dynamics, null propagation, catalog redshift, gravitational-wave strain, lens records, posterior coordinates, or physical distance. It enters only through conditional radial-inference predictions. The required distance operators are defined under an explicit transport card comprising homogeneity and isotropy, fixed spatial curvature, standard metric propagation, unchanged observed redshift, and Etherington reciprocity. A matched direct-history representation accompanies the factorized representation so that a Reference-Card coordinate cannot acquire empirical significance through parameterization alone. The active compact test uses the public DESI DR1 Lyα full-shape Alcock–Paczynski measurement D_M/D_H = 4.525 ± 0.071 at z_eff = 2.33. Across the frozen continuation family, the upper plateau histories are displaced by approximately 4.04σ, whereas the edge and tangent histories lie near 1.00σ and 0.17σ, respectively. Changing the lower-redshift continuation changes the pull by less than 0.022σ. The carrier therefore adjudicates the high-redshift continuation family but does not identify a particular reference factorization. This measurement is ruler-independent, not model-independent, and is not combined with Lyα BAO from the same sample without a joint covariance. Executable likelihood-integrity adapters are supplied for GWTC-5.0 standard sirens and the eight-lens TDCOSMO-2025 release. Published gravitational-wave posterior coordinates remain unchanged, and the original priors are removed only when constructing a likelihood representation. For time-delay cosmography, D_Δt is retained as the primary carrier, while the deflector angular-diameter distance D_d is not treated as an independent compact datum outside the full lensing-plus-kinematics likelihood. A separate seeded closure for 0 < z < 0.07 demonstrates the role of coherent peculiar-velocity covariance. In the registered 24-host synthetic realization, the full covariance increases the inferred H₀ uncertainty by a factor of 1.97 relative to a diagonal approximation. This is a covariance-closure diagnostic, not a public-event H₀ measurement. The DESI Lyα AP likelihood is active. The GWTC-5.0 and TDCOSMO-2025 modules are adapter-ready but remain guarded until their exact official priors, selection functions, covariances, and likelihood assets are frozen. The low-redshift module remains synthetic-only. The release includes the manuscript, executable operators and adapters, activation and dependency contracts, provenance registries, tests, machine-readable results, figures, licences, and an archive-integrity manifest.