The Relational Emergence Theory (RET): From Fundamental Matrices to Atomic Radii Prediction

This paper presents the Relational Emergence Theory (RET), a novel theoreticalframework that aims to derive macroscopic physical properties from the fundamen-tal structure of relational information. The theory is constructed in two cohesiveparts. First, we establish the fundamental mathematical objects: the RelationalMatrix Rij and the derived Relational Density ρR(x), which describe the web of re-lationships between basic relational units. Second, from this foundation, we extracta coarse-grained, computable parameter τ (tau), representing the "completeness"or closure strength of electron relationships within an atomic shell. A key emer-gent quantity, ∆0 = κ(1 − τ ), is defined, bridging the philosophical tension betweendiscrete quantum structure and continuous empirical trends. We implement thisτ -∆0 formalism into a predictive model (RET v6.7) for atomic radii. The model,without quantum mechanical wavefunctions, achieves a mean absolute percentageerror (MAPE) of 13.7% for elements Z=1-36 and successfully extrapolates to the 5thperiod (Z=37-54) with a MAPE of 19.4%, demonstrating its predictive power andphysical consistency. The theory provides a unified perspective where spacetime,matter fields, and chemical properties emerge from a primordial relational network.