Why Static Relational Optimization Cannot Generate Spatial Locality- A Systematic No-Go Analysis

Background: A recurring proposal in relational approaches to physical geometry holds that spatial locality can be obtained by optimizing a relational energy defined over pairs of internal node states, without presupposing a background metric. Methods: This paper examines a broad class of static (non-dynamical) optimization mechanisms proposed for this purpose: unconstrained pairwise optimization, fixed-budget threshold selection, degree-regularized optimization, algebraic-connectivity (spectral) optimization, correlation-based graph reconstruction, and sequential or predictive growth rules. Each mechanism is analyzed within a single unified mathematical framework based on quasi-linearity of the objective in the pairwise weights. Results: In every case, one of two outcomes occurs. Either the optimization collapses to a trivial or degenerate configuration that encodes no geometric information, or it produces a nontrivial graph only by importing an ordering, threshold, or objective that is fixed externally to the optimization problem rather than derived from it. A general theorem is stated and proved unifying these results: any static relational optimization problem whose objective is linear or quasi-linear in the pairwise weights, subject only to weight positivity and a finite family of linear or quadratic global constraints, either collapses or inherits its geometric content from the constraint data. The paper also proves, via an explicit closed-form four-node example, that degree regularization can select non-monotone support structures with respect to state mismatch, and shows, via a direct disconnection argument together with an eigenvalue-perturbation KKT analysis, that algebraic-connectivity optimization either collapses trivially to disconnection or, when constrained by a connectivity floor, introduces a bias toward long-range edges unrelated to state mismatch. Conclusions: Static relational optimization cannot generate spatial locality without presupposing it. These results motivate a companion investigation in which locality is sought as a stability property of relational dynamics rather than as a static optimum.