Common-Frontier / Distinct-Closure Theorem: Causal Propagation Through Layered Historical Fibers in Registration Theory

This paper establishes a conditional Common-Frontier / Distinct-Closure Theorem within an abstract Registration Theory successor model. It demonstrates that distinct historical-fiber organizations can share an identical outward causal propagation structure while supporting different internal persistence and closure modes. The theorem establishes a sufficient mathematical architecture for a common propagation frontier under explicitly stated structural conditions. It does not establish that Trinity's full recursive daughter-registration dynamics necessarily generate this architecture. Deriving the common-frontier structure directly from canonical bilateral and daughter-registration rules remains an open mathematical objective. An exact two-step daughter-registration counterexample demonstrates that connected causal paths need not produce detectable host consequences because of cancellation. The paper distinguishes primitive causal reachability, finite successor-hop propagation, host-independent reconstructed speed, and emergent physical mass. It identifies the additional mathematical conditions required to derive a universal characteristic speed without presupposing physical space, physical time, or the speed of light. The work connects Trinity's bilateral registration and recursive daughter formation to the ongoing ACRF-4 consequence-reconstruction program, including the investigation of whether admissible bilateral and boundary responses can recover all consequential distinctions. Version 0.1 is a timestamped mathematical research note containing exact conditional results, explicit counterexamples, and clearly identified open derivations.