Metric Refraction Across the Cosmic Web: A Dilatonic Interpretation of Local Hubble Anisotropy

Localized low-redshift anisotropy in the Hubble residual field appears to evade compression into a scalar local expansion rate plus a single externally reconstructed flow correction. This paper develops a phenomenological bridge between that empirical situation and a dilatonic complex-tension framework. We start from a constrained complex tension field $\Lambda=\Lambda^P+i\Lambda^I=\Lambda_0 e^{i\Theta}$, where the angular variable $\Theta$ controls the partition between a rigid structural channel and a relaxed coherence channel. The associated Einstein-frame coupling is logarithmic, $\Delta\ln A(\Theta)=\tfrac12\ln\!\left(\cos\Theta_{\rm struct}/\cos\Theta\right)$. We then show that the natural thermodynamic order parameter is $\mathcal U=\ln(\Lambda^I/\Lambda^P)=\ln(\tan\Theta)$, and that a sector-resolved logarithmic free energy reduces at leading order to a Ginzburg--Landau effective field theory in $\mathcal U$. The corresponding domain-wall solution is a kink profile, which yields the transfer law $\tan\Theta(z)=\tan\Theta_{\rm struct}\exp[\gamma\tanh((z-z_0)/\xi)]$. This provides a principled phenomenological foundation for the saturating \texttt{tanexp\_tanh} closure used in the observational pipeline. The present manuscript is organized to connect an inherited empirical target to a logarithmic scalar framework through a bridge-specific reduction from local geometry to fixed support, a derivation of the effective-field closure, and a fixed-support confrontation between closure families.