Asymmetry from Consequential History and Recursive Rule Constitution in Registration Theory: Host-Resolved Harmony, Saturation, Differentiated Closure, Inherited Identity, and Emergent Algebra

This preprint investigates a proposed mechanism in Registration Theory (RT) by which reciprocal interactions between already constituted systems can create both new consequential registrations and new, inheritable rules for further registrations. RT takes undistinguished, ever-dynamic Trinity as the ontological starting point and treats familiar physical and mathematical structures as targets for reconstruction.The paper proves a symmetry-selection obstruction for deterministic equivariant successors and constructs a finite bilateral-history selector that permits differentiated outcomes without an asymmetric governing rule. Under explicit compatibility-interface and closure-preserving transport assumptions, a recursive rule-constitution operator preserves inherited synchronization, lineage, and rule records through arbitrarily many finite levels of higher-order integration. The model also exhibits an exact nonproduct host-consequence quotient and distinct algebraic properties of closed-host continuation and formation.A complementary investigation distinguishes retained source identity from host-resolved harmonic identity, proves the preservation of frequency ratios under uniform scaling, examines a conditional phase-locking threshold, and analyzes native ACRF-4 retained-coupling redistribution and a locally contracting differentiated registration sector. A separate interval-enclosure verifier supports the restricted local stability proof.The paper explicitly distinguishes exact conditional mathematical results from unproved claims about universal physical daughter formation, primitive Trinity-level selection, and the physics of atoms. Supplementary executable Python checks and LaTeX source are provided for independent inspection. This is a preprint and has not undergone external peer review.,