Bounded Relational Capacity and Its Conditional Continuum Dynamics

We organize the bounded-relaxation approach to emergent dynamics around a single structural axiom. The capacity axiom [A-cap] states that the per-node relaxation rate of an admissible relational configuration is uniformly bounded by a constant $c_{\mathrm{BI}}$. The axiom is motivated by non-injective projection together with finite local distinguishability; it is not a consequence of non-injectivity alone, which constrains information rather than rates. Which continuum quantity inherits the bound is an open problem: null field configurations carry arbitrarily large amplitudes at vanishing invariants, so no local Lagrangian of the field invariants bounds every constitutive direction. Born–Infeld theory is accordingly presented as a saturation candidate, distinguished by its known exceptional-propagation properties, not as a derived or unique completion. Under explicit hypotheses — an established continuum limit for the relational Laplacian and the existence of a second-order hyperbolic sector — the admissible signature is Lorentzian $(-+++)$, and stationary spherically symmetric flux conservation yields the Newtonian exterior profile $\Phi = \Phi_0 - C/r$. The Schwarzschild form requires additional dynamical input, and the reading of horizons as capacity saturation is relative to a preferred relaxation foliation; both are stated as conditional or open rather than as theorems. Interpretive status. The paper provides a structural scaffold connecting relational capacity to effective continuum dynamics. Its physical content becomes predictive only when the identified projection bridges — from capacity to constitutive response, from relaxation to hyperbolic dynamics, and from symmetric connectivity to gauge curvature — are closed. Keywords: relational dynamics, capacity bound, Born–Infeld saturation, non-injective projection, graph Laplacian, signature selection, emergent geometry