Relational Architecture of Emergent Dimensionality via Reduction of Relational Radius R: Conduction, Optical Guiding, and Magnetism as Constraints on Degrees of Freedom

We present a strict exploratory framework in which observable material properties—conduction, optical guiding, and magnetismare interpreted as architectures of constraint that reduce a system’s relational radius R, an operational measure of phase mismatch scaled by a fixed reference frequency ω. The framework is built on explicit level separation (postulates, internal definitions, tentative correspondences, falsifiable predictions, and programmatic speculation). A minimal effective-field implementation is developed for two-dimensional phase coherence (XY/BKT on a honeycomb lattice) with a single independently estimated control parameter κκ encoding activation of out-of-plane dephasing channels. To block the critique that “κκ only renormalizes JJ”, we pre‑register two minimal non‑degenerate extensions (vortex‑core energy modification and κκ-induced next‑nearest‑neighbor coupling). The model generates sharp, testable predictions: (i) universal scaling collapse of order and roughness under x=T/[J0(1−κ)]; (ii) the Nelson–Kosterlitz jump of the helicity modulus; (iii) the correlation exponent η=1/4ηat the BKT transition; and (iv) a disciplined transport ansatz σ=σ0 Ψ/(1+αR) to be compared against null models via objective criteria (AIC/BIC). Experimental protocols are outlined, linkingκ to independent proxies (Raman, AFM, weak‑localization dephasing) and referencing existing open datasets. Success or failure of these pre‑registered tests will either support or refute the relational hypothesis in a scientifically useful way.