From Primitive Distinction to Emergent Mathematics: Registration, Persistence, and Recursive Structure

This paper inaugurates a foundational research branch of Registration Theory (RT) investigating whether mathematical identity, counting, composition, algebra, and premetric geometric relations can be reconstructed from primitive conjugate distinction, persistent consequence, and recursive daughter registration rather than being assumed as primitive physical entities. The paper distinguishes a host's physical registration of a consequential relation from a subsequent numerical representation of that relation. It formalizes admissible registration histories and host-contextual consequence equivalence. Under stated context-closure hypotheses, it proves that the equivalence is a congruence and that daughter operations descend to a well-defined algebra of consequential registration classes. A faithful-iteration theorem establishes a natural-number successor structure; with an additional physically coherent and faithful history-concatenation rule, the repeated registrations form a free monogenic monoid isomorphic to the natural numbers under addition. The work also establishes precise limitations. A weak-axiom counterexample proves that distinguishable initial change, iteration, and retained memory do not by themselves yield unbounded counting. The established reduced RT incorporation law is solved exactly for a fixed event and incorporation rate, showing saturation and vanishing successive state separation; this yields a quantified finite-resolution limitation for a specified Lipschitz host readout. An exact two-path daughter cancellation example separates causal ancestry from host visibility. A separate accessibility condition links the proposed history algebra to the ongoing universal boundary-consequence reconstruction problem. Directed successor incidence can yield combinatorial order and graph distance, but physical metric, operational time, algebraic field structure, and a universal propagation speed remain separate derivation targets. Version 0.1 contains exact conditional mathematical results, explicit counterexamples, disciplined proof obligations, and a staged reconstruction agenda. It does not claim that full arithmetic, abstract mathematics, or physical spacetime have already been derived universally from canonical Trinity dynamics. It establishes the timestamp baseline for a new foundational RT research branch.