Invariant Identity and Expanding Registration Capacity: An Exact Independence Result, a Conditional Branching Theorem, and the Boundary to Physical Mathematical Infinity

This mathematical research note advances the Emergent Mathematics branch of Registration Theory by testing the hypothesis that a retained mathematical constant necessarily entails an indefinitely expanding conjugate class of representations. The universal implication is refuted by exact finite and host-coarse-grained countermodels: a value such as 39 may persist across arbitrarily many updates without an increase in host-distinguishable registration capacity, even when underlying states continually change. A complementary conditional theorem supplies a sufficient mechanism for growth. If q ≥ 2 admissible, repeated, host-context-compatible successor alternatives preserve one numerical identity and the joint parent-choice map is injective on reachable host-consequence classes, exactly qⁿ distinct classes exist at depth n. With root exclusion, the cumulative number through depth n is (q⁽ⁿ⁺¹⁾ − 1)/(q − 1). A finite host capacity M obstructs unbounded exact discrimination, requiring qⁿ ≤ M, while a finite nonreplenished processing budget E₀ and a positive per-step cost floor ε constrain a physically realized run to n ≤ ⌊E₀/ε⌋. These are exact, conditional mathematical results and countermodels, accompanied by 13 deterministic finite model checks. They do not establish the required successor dynamics from primitive Trinity, a universal rise in physical entropy, a physically measured computational energy cost, actual infinity, or the physical speed of light. The note explicitly separates the analyst's use of numerical metalanguage from physically generated arithmetic and identifies the remaining dynamical, accessibility, retention, and calibration gates needed for stronger Registration Theory claims.