Effective Closure Theorem for the Galactic Kernel and Structural–Cosmological Bridge

This note presents an effective closure theorem within the framework of the Log-Harmonic Field Theory (LHFT). The manuscript develops a mathematically explicit infrared description of the galactic structural sector and extends it toward a first structural–cosmological bridge. The theorem starts from the logarithmic structural coordinate u = ln (r/r0), a projected structural potential PhiS, and a nonlinear baryon-sourced matching law. An explicit interpolating response function mu (y) and its parent action density F (X) are introduced, yielding a closed effective description of the structural acceleration gS (r). In the strong regime, the construction recovers Newtonian/Poisson-like behavior. In the weak exterior regime, it generates a logarithmic potential and a halo-like 1/r law. This leads directly to a baryonic Tully–Fisher-type relation: vᵢnf⁴ = G kappa Mb a_*. The same projected structural mode is then shown to contribute to the weak-field Weyl sector, providing a corresponding halo-like lensing density. On larger scales, the note formulates a minimal structural–cosmological bridge based on a coarse-grained split of the remainder field phi = Df - 3 into smooth and clumping sectors, together with an effective background potential Uₑff (phi), linear growth equations, and scale-dependent response functions muₑff (a, k) and Sigma (a, k). A further appendix demonstrates, at the level of an effective low-energy / coarse-grained derivation, how ΛCDM can emerge as a macroscopic limit of LHFT. In this reading, dark-matter-like and dark-energy-like sectors are not treated as fundamentally separate ontologies, but as effective manifestations of one and the same underlying structural field. This document does not claim the final microscopic closure of the full cosmological sector. Rather, it provides a publication-ready theorem note with technical appendices, establishing a clean and variationally explicit effective core from which further mathematical and empirical developments can proceed.