One Law, Two Resolutions: Conservative Recursive Daughter Registration and Consequential Retention in Registration Theory

This technical research note advances the Registration Theory (RT) program by constructing an explicit conditional model of recursive daughter registration with conservative transfer and consequential retained organization. A single autonomous update generates successive energy-bearing daughter states from bilateral interactions while maintaining an ancestry network. The paper proves exact conservation of total transferable model capacity, the capacity-weighted organizational mean, and a combined external-plus-internal organizational second moment. Differences among parent organizational states are incorporated into daughters' retained internal moments, and the retained variables influence future daughter allocations, making memory dynamically consequential. Two host-level mathematical projections of this same evolution display an advancing ancestry frontier and a retained organizational description, with a precise criterion and counterexample for dynamical closure. A competing persistent-parent successor and its fixed-support capacity-decay result are also assessed. The paper situates the work alongside Whitehead's process philosophy, energetic causal sets, quantum causal histories, group field theory, relational quantum mechanics, and tensor-network coarse-graining. The mathematical laws here are explicit RT-motivated constitutive constructions, not derivations of measured energy, spacetime, mass, clocks, or universal constants. The paper gives reproducible exact-rational tests and identifies the native bilateral daughter-selection and host-accessibility laws as next research targets.