A First-Principles Derivation of the Recognition Physics Formula

We present a derivation of the Recognition Physics formula entirely from first principles. Starting from the core postulates of dual recognition and minimal informational overhead, we define a coverage function in R3, construct a cost functional that incorporates both boundary constraints and synergy (or reflection) penalties, and show that its minimization yields a unique optimal parameter Xopt = ϕ/π ≈ 0.5149. We then derive both a modified Poisson equation (non-field form) and a recognition-modified Einstein field equation (tensor form) in which every term is justified from these first principles.