Registration Theory and the Mass-Gap Problem III: Relational Host Constitution and Simultaneous Manifestation

This third paper in the Registration Theory (RT) mass-gap investigative series examines whether persistent host organizations can emerge through the mutual regulation, synchronization, and retained relational history of continuing consequential registrations, without assuming independently existing particles, masses, or predefined host boundaries.Beginning with RT's undivided Trinity ontology and recursively constituted hosts, the investigation consolidates earlier results on bilateral manifestation, relational attractor selection, and stable mutual coexistence. In canonical four-registration ACRF-4, bilateral consequence measurements distinguish sixteen coordinate-labeled attracting configurations, compared with twelve distinguishable one-sided readouts. Symmetry analysis further demonstrates that coordinate distinctions must not automatically be identified with physically distinct organizations.The investigation then extends the canonical successor's functional structure to an adaptive six-registration network. A stable configuration emerges with two strongly connected relational groups of three registrations, rather than the three host pairs previously assigned by the experimental design. Retained coupling strengths are approximately 2.352 within groups and 0.099 between groups, demonstrating substantial relational differentiation within one continuing, mutually interacting network.The complete network exhibits local attraction, with a maximum admissible linear stability multiplier of approximately 0.875. In 450 targeted perturbation trials, the system returned to its persistent organization. A further 240 randomized simulations identified four broad families of persistent registration patterns under the same successor law.An additional symmetry result establishes that exactly undifferentiated initial conditions cannot spontaneously develop inequivalent registration roles under the deterministic, permutation-symmetric successor without a distinguishing condition or further mechanism.These findings support the investigation of host constitution as an emergent consequence of dynamically maintained relational organization rather than the external grouping of independently existing objects. They also clarify that different hosts may resolve distinct manifestations of one ongoing registration organization without requiring exclusive selection or elimination of alternative configurations.The paper distinguishes demonstrated relational grouping from the stronger, still-open derivation of fully constituted hosts capable of recursively participating in further registrations. The six-registration extension is explicitly identified as a conditional mathematical model, not a uniquely derived primitive Trinity successor law.The results provide a foundation for investigating whether distinct persistent synchronization regimes can generate physically meaningful excitation classes and, ultimately, a positive mass gap. No derivation of physical particle masses or solution to the Yang–Mills mass-gap problem is claimed.The accompanying reproducibility package contains numerical data, computational methods, stability analyses, and figures supporting the reported findings.