Omnidirectional Complementary Relational Dynamics: Sphere-Indexed Kernels, Gauge Transport, Memory, and Two-Turn Return

This paper develops a constructive local-to-global extension of the Complementary Relational Principle (CRP) and Complementary Relational Dynamics (CRD). It investigates how local two-input relations can be assembled into omnidirectional networks with consistent comparison, observation, reduction, and dynamical evolution. Directional states are indexed by the circle S¹, the sphere S², or the frame-rotation group SO(3). These angular parameter spaces are distinguished from physical spatial dimensions. Positive two-direction kernels provide a common relational representation, while cocycle consistency and connection-based transport specify how locally represented states are compared. The mathematical constructions include positivity-preserving finite directional sampling, gauge-invariant transported relations, reference-routed kernels in the presence of curvature, spherical-harmonic reduction, and quantitative memory corrections under weak anisotropic coupling. Exact block elimination produces a joint direction–time memory kernel, and a contraction argument constructs a causal nonlinear memory functional under stated hypotheses. Two physically distinct applications are developed: visibility-weighted multiview reconstruction with error and covariance bounds, and coherent interference through quadratic detector readouts. An observation-fiber criterion characterizes autonomous reduced dynamics, while a two-state error bound quantifies the predictive consequences of discarded information. A specified kernel–connection feedback model supplies a globally well-posed, positivity- and trace-preserving dynamical core. The paper also distinguishes ordinary 360-degree complex-phase return from spinorial 720-degree frame return. The latter is constructed through an associated SU(2) spinor bundle and becomes observable as a relative sign against a coherent reference; it is not inferred from the imaginary unit alone. These results establish a modular relation–observation–promotion architecture without identifying stereopsis, interference, gauge transport, and fluid-mode reduction as the same physical law. The paper is a companion to “Complementary Relational Dynamics and Three-Dimensional Closure” and preserves its conditional Hodge-degree argument.