From Sphere Geometry to Bilateral Resonant Invariance: A Comprehensive Investigation of Conjugate Registration, Retained History, Reciprocal Frames, and an RT Re-Derivation of Invariant c_RT

This comprehensive research manuscript documents a foundational investigation in Registration Theory (RT), proceeding from spherical geometry and generative registration frontiers through recursive daughter constitution, retained history, host-relative measurement, and bilateral resonant invariance.Registration Theory treats space-facing, time-facing, and energy-facing expressions as conjugate manifestations of continuing physical registration rather than independently primitive quantities. Its minimum physically meaningful invariant is a relationship between two constituted systems, each providing its own host-relative frame of registration. This relationship is described in RT as resonant registration or synchronicity.The manuscript preserves the full conceptual and mathematical progression of the investigation, including exact conservation identities for a recursive daughter model, retained-history production, host-accessible succession, reciprocal current–retention mechanisms, and the limitations of attempts to derive an invariant propagation rate from an isolated system's frontier and internal clock.Building on previously developed RT conjugate-involution and hyperbolic-invariant results, the analysis re-derives the established generalized Lorentz/Galilei transformation family within RT's bilateral registration framework. Historical mathematical priority is explicitly attributed to Ignatowsky, Frank–Rothe, Lévy-Leblond, and Mermin.Under stated conditions of linearity, reciprocity, parity, continuity, and compositional consistency, the bilateral transformations preserve a quadratic relational invariant characterized by a universal parameter K. For the positive-K branch, this yields a finite invariant conversion rate c_RT = 1/√K.The result is a conditional mathematical reconstruction of relativistic correspondence within RT's proposed ontology, not a claim that the historical transformation mathematics is new. The manuscript distinguishes established model results, conditional derivations, counterexamples, and unresolved foundational questions, including selection of the Lorentzian branch and physical calibration of the invariant rate from primitive Trinity dynamics.The accompanying supplementary archive contains the LaTeX manuscript source, figures, and mathematical verification code.