Endogenous Configuration Capacity in Registration Theory: Expansion, Consequence Equivalence, and the Limits of an Entropy Derivation

This research note investigates whether continuous Trinity expansion in Registration Theory (RT) necessarily increases the number of independent, consequentially distinguishable physical configurations. Drawing on the canonical RT axioms, the Universal Expansion Principle (UEP), and prior investigations of daughter registration, historical retention, and consequence equivalence, the study distinguishes expanding spatial extent, registration multiplicity, consequential configuration capacity, and host-relative thermodynamic entropy. Four finite mathematical models demonstrate that expansion may preserve, increase, or accompany a reduction in consequential configuration capacity, depending on the governing dynamics. A conditional consequence-class embedding theorem establishes sufficient conditions for monotonic capacity growth and identifies the additional requirement for strict growth. The central result is that increasing consequential configuration capacity is not presently derivable from the canonical RT commitments or the established UEP shell-model results alone. It requires additional dynamical structure governing independent registration alternatives, consequence preservation, and daughter formation. The paper establishes a precise boundary between Trinity's generative expansion and conventional thermodynamic entropy, identifying the mathematical conditions that must be satisfied before an entropy-growth theorem can be derived within RT.