Relational Realization: Emergent Geometry from Latent Possibility

We present Relational Realization Theory (RRT), a mathematical framework – developed here through three research iterations, unified into a single account – for investigating a specific hypothesis about physical existence: that reality is not fundamentally a collection of objects located in a pre-existing space, but a network of latent possibilities that becomes physical through local, relational realization. The mathematical core is a continuous realization variable qi∈[0,1] on a pre-geometric substrate, crossing a threshold qc under coupled node–edge gradient-flow dynamics derived from a single free-energy functional F[q,w]. We report what this specific mathematical object actually does, not what the informal narrative suggests it might do. On the positive side: an exact fixed point and closed-form Hessian yield an analytic dispersion relation (machine-precision-verified on regular graphs) and a derived bifurcation condition that explains a previously unexplained partial-saturation plateau seen throughout this research; a growing-substrate mechanism gives sustained exponential expansion with an analytically predicted rate confirmed numerically to four significant figures; curvature, spectral entropy, and an area-law test give a clean, internally consistent geometric picture (logarithmic, not area- or volume-law, entropy scaling). On the negative side, reported with equal weight: a structural argument, not merely an empirical one, shows the model’s first-order dynamics cannot support a finite propagation speed or Lorentzian causal structure; a direct nonlinear-dynamics test shows that localized linear eigenmodes – initially proposed as candidate matter-like defects – delocalize and grow within a few relaxation times rather than persisting, undermining that interpretation; and a targeted search for emergent locality (effective distance organizing local realization) returned a statistically insignificant signal. Every claim in this paper is tagged with an explicit evidence status (derived, numerically supported, consistent with, conjectured, or not supported), collected in a final status table, so that the strength of the underlying hypothesis can be judged from the mathematics and experiments actually performed rather than from the narrative surrounding them.