Bridge Construction: From Lawful Continuation to Einstein-Form Representation

This work proves the structural uniqueness of the Einstein-form field equation G⏛⏜ + Λ g⏛⏜ = κ B⏛⏜ under five explicit bridge constraints arising from an upstream representation layer. Using Lovelock’s theorem in four dimensions, we show that when the following conditions hold: - second order locality, - no additional dynamical fields, - diffeomorphism invariance, - variational origin, - symmetric divergence-free rank 2 tensor constructed solely from the metric, the Einstein form is uniquely forced. No higher-curvature corrections, additional fields, or modified terms are admissible without relaxing at least one constraint. We further connect this uniqueness to the Law of Minimal Identional Disruption (LMID) and its finite approximation hierarchy. In LMID, admissible continuations minimize rupture vectors lexicographically (κR ≫ κB ≫ κV). The Einstein form emerges as the large-n shadow of this lexicographic descent when scalar representation functionals approximate strict priority ordering. Modified gravity theories are interpreted as finite-nₑff relaxations of bridge constraints, permitting channel trade-offs forbidden in the strict lexicographic regime. The result provides a conditional structural explanation for why the Einstein equation appears inevitable under the stated premises, while preserving falsifiability and explicitly forbidding backflow from the physical representation layer to the upstream law layer.