Bounded Relational Capacity and Its Conditional Continuum Dynamics

We investigate how spacetime geometry and non-linear field dynamics can arise as equilibrium descriptions of relational systems subject to a finite maximal relaxation flux. We show that the existence of a universal bound on admissible relational gradients excludes purely quadratic effective actions and uniquely enforces a Born--Infeld-type structure as the minimal local representation compatible with saturation.Starting from a weighted relational Laplacian with irreversible relaxation, we derive an effective continuum description in which the metric tensor emerges from the principal symbol of the operator, while antisymmetric perturbations enter as an effective gauge field strength. In homogeneous regimes, the bounded-relaxation constraint dynamically selects flat spacetime with pseudo-Riemannian signature $(-+++)$ as a stable equilibrium. When homogeneity is broken by a localized stationary obstruction, the same mechanism yields the Schwarzschild geometry as the universal effective exterior solution. Horizon formation corresponds to saturation of admissible relational flux and to a loss of projectability of the continuum description rather than to a physical singularity. These results provide an operator-based unification of Lorentzian geometry, Born--Infeld electrodynamics, and horizon formation as consequences of bounded relational dynamics.