Complementary Relational Dynamics and Three-Dimensional Closure: From the CRP to Pair Exchange, Phase Flux, and Same-Type Hodge Return

Abstract

This preprint develops the Complementary Relational Principle (CRP) and its mathematical realization as Complementary Relational Dynamics (CRD). The foundational proposal treats a standalone physical zero not as a primitive state but as a possible cancellation, balance, projection, or limiting readout of nonzero relational structure. Continuous oriented norm-preserving exchange on a real complementary pair yields a complex structure. With additional Hilbert-space, symmetry, and measurement assumptions, the framework is connected conditionally to Schrödinger evolution, Born weights, tensor composition, generalized measurements, decoherence, pointer records, Bell–CHSH correlations, and canonical uncertainty. A relation-preserving compression hierarchy distinguishes reversible global-phase compression from irreversible diagonal coarse graining. The central spatial result is independent of these quantum extensions. Every bilinear alternating relation of two one-forms factors through their wedge product. Requiring a nonzero, orientation-preserving-isometry-natural return to the same one-form type, without an additional primitive tensor, selects three dimensions, where the return is a scalar multiple of the Hodge star. A gauge-covariant pair-to-space bridge and a candidate feedback equation yield a falsifiable self-generated holonomy signature. Stereopsis and two-path interference illustrate the shared input–relation–readout architecture, while Fermat arithmetic is retained only as a separate boundary comparison. The conclusions are explicitly conditional: the paper does not claim to derive quantum mechanics or the dimensionality of physical space from the CRP alone.