We prove that this map is invariant under trivial field reparametrizations that preserve the coherence filter and is gauge/BRST - independent at one loop. We establish boundedness and continuity—s ∈ [1/(1 + β), 1], ds/df ≥ 0—ensuring well- posed inference in the EFT window (weak curvature, slow variation of Geff). In the weak - field, single - factor regime the result justifies the observational closure Geff = sG0 and yields concrete, falsifiable relations among observables (linear scaling of weak- lensing fields; predictable rescalings of Einstein angles, time- delay differences, and standard- siren amplitudes), including the joint invariant R∆t = (Rθ )2. We also specify pre- registered tests and deterministic artifacts (figures, checksums, code hash) that fix the empirical surface for validation. Higher- order effects enter as O(R2, ∇R ) and RG evolution of the EH coefficient; they do not remove the EH term within the stated domain. The paper thus places Geff on a first - principles footing and connects it directly to data via a simple, testable parameter s.