Emergent Spacetime Geometry from Bounded Relational Relaxation

We investigate how spacetime geometry and non-linear field dynamics can arise as equilibrium descriptions of relational systems subject to a finite maximal propagation or relaxation flux. We show that bounded flux propagation 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 a 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 and stationary obstruction, the same mechanism yields the Schwarzschild geometry as the universal effective exterior solution. We further show that 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 a unified operator-based account of Born--Infeld electrodynamics, Lorentzian geometry, and horizon formation as consequences of bounded relational dynamics.