Projective Conjugate Differentiation Across Registration, Selection, and Markets: A Scale-Free Character Theorem, an Exact RT Retention Law, and the Limits of Domain Universality

This mathematical research paper investigates whether a common, scale-independent principle of observable change arises across distinct dynamical systems, rather than being imposed separately on each phenomenon. Motivated by Registration Theory (RT) and its proposed conjugate relationship between generative differentiation and retained organization, the paper establishes a conditional characterization of logarithmic ratios as natural measures of relational change. Under common-scale invariance, additive composition, and conjugate reversal, the logarithm of a positive ratio is selected up to an overall multiplicative constant. The analysis then applies this principle to the native conservative daughter-registration model in RT. Its exact conservation law transfers unretained organizational variation into retained historical structure. The corresponding logarithmic ratio decreases monotonically under nontrivial daughter interactions, providing a mathematically determined, scale-independent measure of retentive reorganization. Independent comparisons with evolutionary replicator dynamics and relative financial price changes demonstrate that analogous logarithmic relationships arise naturally from the equations of distinct domains. The paper distinguishes mathematical universality from physical universality. It establishes a shared relational structure under explicit assumptions but does not claim that financial markets, biological evolution, and physical spacetime are governed by one experimentally demonstrated Trinity mechanism. The results provide a rigorous foundation for the broader Registration Theory conjecture that observable change arises through differential conjugate organization, while common underlying progression may remain inaccessible to internal measurement. The accompanying research archive includes editable LaTeX source, exact mathematical verification, computational tests, and supporting documentation.