Bilateral Registration, Adaptive Holonomy Selection, and the Irreducible Response Susceptibility in Registration Theory: A Three-Registration ACRF Construction, Parameter-Elimination Audit, and the Remaining First-Principles Bridge

This timestamped research note records a bounded three-registration extension of Registration Theory (Physical Foundations) and an internal parameter-elimination audit of adaptive loop-holonomy selection. The contribution is not a new holonomy formalism, bifurcation theorem, or adaptive-network mechanism. Those mathematical structures are established in gauge theory, equivariant bifurcation theory, and adaptive-network dynamics. The Registration Theory contribution is the application of those tools to an independently developed bilateral-registration architecture and the isolation of exactly where the remaining magnitude-setting freedom enters the RT/ACRF model. Starting from the published bilateral circle invariant and unoriented compact phase structure of canonical ACRF-4, the paper derives a three-registration contextual-defect expression for triangle holonomy. A minimal inherited ACRF successor is then analyzed. Its circulation mode is linearly contractive with multiplier 0.90, while its distortion sector is expansive with multiplier 1.05, ruling out a direct S3-to-C3 chiral pitchfork in this minimal extension. Cubic nonlinearities nevertheless allow relational distortion to source circulation. Restoring the canonical adaptive retained-coupling update produces six symmetry-related, locally attracting S3-to-S2 fixed points carrying a persistent nonzero loop-holonomy magnitude. For the inherited canonical numerical parameters, the selected model value is |Theta*| approximately 0.91602417; the paper explicitly does not identify this value with a physical constant. The parameter audit then reduces the apparent adaptive freedom. The triangular retained-edge budget is fixed to B=3 by symmetric unit-edge normalization. The relaxation parameter rho controls convergence rate but cancels exactly from persistent fixed points. The leading local effects of eta and kappa_0 collapse into one logarithmic adaptive-response susceptibility, g_R = [d/ds log A(s)] at s=0 = eta/(1+kappa_0). For canonical ACRF-4, this equals 40/23, the same coefficient already present in the previously derived quadratic retained-history expansion because both descend from the same canonical adaptive law. No current Registration Theory theorem derives g_R from primitive registration principles. The surviving first-principles problem is therefore isolated sharply: derive, or prove underdetermined, the normalized adaptive-response susceptibility g_R without inserting a phenomenological response law A(s). This paper does not claim a derivation of electromagnetism, electric charge, Maxwell dynamics, quantum electrodynamics, or the physical fine-structure constant. Its timestamped contribution is the reduction of a diffuse coupling-selection problem to one named auditable response susceptibility within the RT/ACRF architecture.