Abstract

The Axis Model gives an environment-dependent gravitational coupling of the form Geff (x) = s(x)G0, where the scalar-coherence factor s(x) is fixed by the coherence filter and by the one-loop curvature response of the projected scalar-vector sector. The remaining problem is the homogeneous coherent-domain anchor G0. This paper closes that normalization on the signed-z field-excursion branch used here.

The central point is dimensional. Gauge couplings are dimensionless and can be normalized by projector-restricted stiffnesses, whereas the Einstein–Hilbert operator requires a microscopic area scale. We obtain that scale from the same scalar-filtered parent construction used in the electroweak Appendix-J closure. The parent reduction selects a stable coherent scale and stationary displacement vector without using measured Newton gravity, the Planck length, the Planck mass, black-hole entropy normalization, or an equivalent gravitational anchor.

The new step is the identification of the curvature-normalizing displacement. In the parent convention used here, the stationary z-coordinate is a one-sided signed displacement from the scalar-coherent reference configuration. The curvature-normalizing length is identified with the full branch-to-branch field excursion of the z-sector coordinate, rather than with the origin-to-branch amplitude. This signed-field excursion fixes the gravitational displacement without introducing an additional normalization parameter and without modifying the Einstein–Hilbert heat-kernel coefficient or the ultraviolet trace invariant.

On the minimal metric-normalization branch, the resulting homogeneous Newton anchor is 

GAxis 0 = 6.65413218 × 10−11 m3 kg−1 s−2. 

Compared with the reference value used as a held-out diagnostic, this gives GAxis 0 /GCODATA =0.99697829, a fractional offset of −3.0217 × 10−3. The reference value is used only after the derivation for comparison. The environmental extension then follows by inserting this homogeneous anchor into the previously derived scalar-coherence map Geff (x) = s(x)G0.

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Keywords: Newton’s constant, emergent gravity, Einstein–Hilbert normalization, heat-kernel methods, scalar coherence, scale selection, field excursion, effective field theory, pre-geometric models, Axis Model.