Here f (Φ) is the scalar–coherence filter built from the phase gradient of the complex field Φ (with f (x) ≡ f ( ¯Φ (x))), G0 is the Newton coupling in a homogeneous reference background, ΛΦ  is the scalar–coherence scale, and S is the fixed “species sum” over the minimal bosonic gravitational subsector of the Scenario–B Axis EFT (complex scalar Φ, massless Xμ, massive Gμ with Stückelberg and ghost sectors) via their Seeley–DeWitt coefficients κs and weights ws. We prove that this map is invariant under trivial field reparametrizations that preserve the coherence filter f (Φ) and that it is gauge–parameter and BRST independent at one loop. For baseline content with S < 0 the resulting factor s is bounded and monotone, s ∈ [1/(1 + β), 1] and ds/df ≥ 0, which ensures a well–posed forward map from scalar coherence to Geff(x) in the EFT window (weak curvature, slowly varying f ). In the weak–field, single–factor regime this justifies the observational closure Geff = sG0 and yields concrete, falsifiable relations among lensing and standard–siren 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) corrections and through RG evolution of the EH coefficient; within the stated Axis EFT domain they do not remove the Einstein–Hilbert term. The paper thus places Geff(x) on a one–loop, first–principles footing and connects it directly to data through a single, testable parameter s(x).

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