Effective-Resistance Flexibility and Its Limits on a Fixed Relational Substrate: Geometric Capacity, Uniform Rigidity, and the Failure of Spontaneous Geometric Selection on K_N

A fixed combinatorial substrate does not imply a fixed effective geometry when distance is defined through the weighted Laplacian rather than by hop count. This paper separates three logically distinct questions: combinatorial rigidity, effective-metric flexibility, and genuine dynamical emergence. For a connected graph sequence G_N with strictly positive edge weights, ordinary graph distance is unchanged as long as the support is fixed. By contrast, effective-resistance distance can vary strongly under positive reweighting. We first derive the exact resistance between the endpoints of a single weak edge in a weighted cycle and show that proximity rankings can reverse without any edge deletion. We then prove a general approximation principle: if H_N is any connected spanning subgraph of G_N, assigning unit weight to H_N and weight epsilon_N > 0 to the complementary edges drives the effective-resistance metric of the weighted G_N toward that of H_N as epsilon_N tends to zero. Consequently, even a fixed complete substrate K_N can realize radically different effective-resistance scaling classes while every formal edge remains present. Explicit constructions reproduce cycle-like Theta(N) resistance diameter and two-dimensional torus-like logarithmic resistance growth on the same fixed K_N support. We next show the complementary rigidity result: if all weights remain uniformly bounded above and below independently of N, then the weighted and unweighted resistance metrics are uniformly bi-Lipschitz equivalent, so asymptotic resistance scaling cannot change arbitrarily. Finally, we test whether the relational weight dynamics used in the programme spontaneously exploits this geometric capacity. On K_N, starting from generic random data and without manually selecting strong edges, the dynamics remains close to the homogeneous-weight sector and generates only modest effective-resistance deformation over the tested interval. The combined result is therefore two-sided: fixed support permits substantial effective-geometric capacity, but the tested relational dynamics does not spontaneously select the required weight hierarchy. This distinguishes representational capacity from geometric emergence and sharpens the question of whether emergent locality requires a dynamical edge set or can instead arise from a dynamically generated effective support on a formally complete substrate.