The Emerging Mathematics of Registration Theory:Trinity, Host-Relative Equivalence, Coarse-Graining, Persistence, and ResidualMotion

This paper develops the first compact mathematical grammar for the new foundational layer of Registration Theory (Physical Foundations). Building from ontological zero, the Irreducible Rule of Distinction (IRD), and Trinity, it introduces host-relative equivalence, coarse-graining, persistence, residual distinction, tandem invariance, and residual motion as candidate elements of an emerging Registration Calculus. Its central governing principle is: What survives coarse-graining becomes persistence; what fails to survive equivalence becomes distinguishable change. A registering host is represented through a coarse-graining map. Two underlying physical realizations are host-equivalent when the host’s coarse-graining cannot distinguish them. When no distinction survives coarse-graining, the relation is represented by a host-relative relational zero. When equivalence fails, a distinguishable residual remains. The paper develops the consequences of this structure for persistence, rest, duration, and motion. Rest is interpreted as host-relative tandem invariance rather than absence of underlying dynamics, while motion emerges as a successive distinguishable residual after common-mode evolution has been coarse-grained away. It also proposes a cross-scale Trinity interpretation in which dynamically space-facing organization may support time-facing persistence at a coarser scale, while persistent time-supported organization may in turn appear as invariant spatial structure. These ideas are then connected to persistent individuation, countability, geometry, and symmetry. Finally, canonical ACRF-4 is identified as a first explicit downstream mathematical realization of the same structural pattern through its exact decomposition into a common mode and a three-dimensional relative-distinction sector. The paper does not claim a completed Registration Calculus or a derivation of conventional spacetime, velocity, relativity, quantum mechanics, or probability. Its contribution is to isolate a coherent mathematical bridge from the primitive ontology of distinction and Trinity to quotient structure, persistence classes, residual motion, and executable dynamical representation.