Registration Theory and the Mass-Gap Problem I: From Persistent Closure to Relational Attractor Selection

This first paper in an investigative series examines whether Registration Theory (RT) can generate distinct persistent excitation organizations through native successor dynamics, retained relational history, and conjugate closure, without assuming particles, mass, or physical excitation energy as fundamental primitives.Using the canonical four-registration ACRF-4 model, the investigation reconstructs and tests a previously established nonzero, successor-invariant closed registration. Linear stability analysis identifies two transversely unstable modes, demonstrating that persistence within a closed sector does not guarantee stability under broader host–system interaction.Numerical investigation identifies sixteen locally attracting fixed registrations in three symmetry-related families. A reproducible experiment involving 1,000 admissible initial conditions approaches one of these configurations in every trial. Evaluation of the native receiver-boundary consequence operator identifies twelve distinct host-resolved readouts, with a minimum nonzero pairwise separation of approximately 0.04797143 in model units.These findings suggest that unstable registration organizations may reorganize into alternative persistent structures and that distinct manifestations may depend on retained relational organization and host-relative accessibility.The investigation motivates a further hypothesis: synchronization and tuning between recursively constituted registering systems may influence which persistent organization is selected and maintained.The paper documents the research path, mathematical methods, numerical findings, limitations, and outstanding derivations. It does not claim a solution to the Yang–Mills mass-gap problem or a derivation of physical particle masses. Instead, it establishes a reproducible finite-model foundation for subsequent investigations into relational attractor selection, synchronization, and the possible emergence of a physically meaningful excitation gap.