Five Conjugate-Mixing Directions from Native ACRF-4 Response: An Exact Retained-History Rank Theorem, Lie-Algebra Completion, and Conditional Recursive Propagation in Registration Theory

Registration Theory (RT) describes physical organizations as dynamically constituted through bilateral consequential registration, conjugate relationships, retained history, and recursively inherited host rules. This paper investigates whether the five conjugate-mixing generator directions needed to complete the SU(3) Lie algebra are mathematically accessible from native registration dynamics.Building on the previously established SU(3) conjugate-closure theorem and history-sensitive ACRF-4 successor, the investigation derives an exact five-dimensional response-rank theorem. Independent variations of the native retained bilateral couplings map onto the five-dimensional space of imaginary-symmetric traceless SU(3) generators through a defined skew-adjoint, complex-structure-preserving host-resolution operation. An exact determinant calculation establishes that the map has full rank.The commutators of these five generator directions span the remaining three real rotational directions, completing the eight-dimensional Lie algebra su(3). The result establishes full infinitesimal Lie-algebra accessibility within the specified conjugate response resolution, without introducing new parameters into the native ACRF-4 successor.The paper also proves exact finite-time persistence of two retained-history modes that remain invisible to uniform current observations but are distinguishable through suitable future relational responses. A conditional recursive propagation theorem establishes sufficient conditions under which SU(3) response organization, inherited identity, and consequential distinctions remain available through arbitrarily deep finite hierarchies of constituted hosts.These results strengthen the mathematical accessibility of Yang–Mills-like non-Abelian organization within Registration Theory. They do not require SU(3) to emerge inevitably from Trinity. The physical realization of SU(3) inter-host transport, independent local gauge covariance, continuum Yang–Mills dynamics, and the quantum mass gap remain open derivations.The principal contribution is an exact native-response accessibility theorem connecting retained bilateral history, conjugate closure, Lie-algebra generation, and conditional recursive consequence propagation.