Unoriented Phase Connection in Registration Theory: Exact Compact-Phase Descent, a Bilateral Circle Invariant, and Holonomy Constraints in Canonical ACRF-4

This paper develops the next mathematical bridge in Registration Theory (Physical Foundations) from the canonically derived local complex structure of persistent ACRF-4 closure toward a possible phase-connection framework. Prior work established that a nonzero persistent closed registration carries a normalized local complex structure J satisfying J^2 = -I, while a subsequent quotient theorem showed that the internal-label gauge does not canonically distinguish J from -J. Only the unoriented class [J] = {J, -J} descends. The present work first proves that the associated norm-one compact phase subgroup nevertheless descends exactly: T_[J] = {aI + bJ : a^2 + b^2 = 1} ≅ SO(2) ≅ U(1). Thus the compact phase circle is canonical even though its orientation is not. The paper then proves that the already established participant-exchange lifts transport this circle only by identity or inversion. Consequently, those lifts alone can generate at most a Z2 orientation holonomy and cannot produce a continuous U(1) phase translation. Independently, the exact bilateral ACRF-4 quotient contains the gauge-invariant relational matrix M_AB = c U V^T. From its trace and antisymmetric part the paper derives a new normalized circle-valued invariant, zeta_AB = (X_AB + i Y_AB) / sqrt(X_AB^2 + Y_AB^2), where X_AB = tr(M_AB) and Y_AB = (M_AB)_12 - (M_AB)_21. The exact identity X_AB^2 + Y_AB^2 = c^2 ||U||^2 ||V||^2 places zeta_AB on U(1) whenever M_AB is nonzero. The invariant survives all internal S2 x S2 relabelings and satisfies zeta_BA = zeta_AB^(-1) under participant exchange. The bilateral phase has no unique continuous extension to the persistent closed stratum, where M_AB = 0. This sharpens an earlier orientation no-go result: the closed registration possesses a canonical compact phase group, but an actual relative phase element arises only on a nondegenerate relational stratum. The paper next derives the general orientation-compatible edge transport P_AB(z) = zeta_AB z^(epsilon_AB), with epsilon_AB in {+1, -1}, showing that the natural transport structure is U(1) semidirect product Z2 ≅ O(2). An explicit triangle-holonomy formula is derived for three participant-relative registrations. A further no-go theorem shows that if the pairwise phases factor through participant-independent local phase frames, the continuous phase cancels around every triangle. Therefore nontrivial continuous holonomy requires genuinely participant-relative edge structure or another nonfactorizing transport mechanism. These results establish a rigorous mathematical foundation for a future Registration Theory phase-connection and curvature program. They do not derive electromagnetism, electric charge, spin, quantum mechanics, Maxwell dynamics, or a physical wavefunction. The remaining bridge requires a canonical dynamical map from the bilateral circle invariant to the closed-sector phase fiber, a multi-registration realization with nontrivial holonomy, and a controlled continuum recovery of connection, curvature, matter coupling, conservation laws, photon polarization, and Maxwell dynamics. A deterministic Python audit and frozen JSON output accompany the paper. Canonical theory name: Registration Theory (Physical Foundations)