Relational Interaction Networks and Conditional Constraints on Spatial Dimensionality

Abstract We examine when a noncrossing relational network and local two-body binding can coexist, and how that conditional dimensional constraint changes when the force law is perturbed. Distinguishable units carry response variables, reciprocal weights, and unitary comparison maps. An additional scalar mediator connects this structure to a Gauss-law attraction under specified boundary and continuum assumptions. A nonplanar graph realized by disjoint channels in one local patch requires at least three spatial dimensions. For an unmodified Gauss force, a nondegenerate frozen-epoch radial minimum with nonnegative background acceleration then permits only three. We analyze 120 prescribed orbit conditions in thermal-reference, static, and accelerating backgrounds, including controls that do not select three. An analytical force-slope criterion and a fixed-angular-momentum persistence bound are tested alongside 217 force-comparison conditions with matched attraction at a reference separation. In the marginal four-dimensional Gauss case, a weak constant-tension contribution or a softened source can produce a nondegenerate minimum, whereas a massive Helmholtz mediator steepens the force and does not restore such binding in four or more dimensions on static or accelerating backgrounds. The dimensional restriction therefore lacks universal robustness to physically motivated force corrections. Unlike winding-annihilation scenarios, the topological input concerns channel preservation; its joint use with binding restrictions is not claimed as an unprecedented principle. The result is a reproducible compatibility and robustness analysis of an effective scalar pair model, not a solution of full Einstein dynamics or a simulation of spatial-dimension emergence. Supplementary Materials and Reproducibility The study provides numerical simulation code, datasets, analytical and numerical verification procedures, and figure-generation resources. The numerical results are based on synthetic model conditions rather than direct cosmological observations. The accompanying supplementary materials document the model parameters, numerical methods, and reproducibility checks.