The Nomadic Principle V2: A Lagrangian Explanation for Coherence Kernels and Invariants

This paper presents a revised theoretical formulation of the Nomadic Principle, explicitly subordinated to recent empirical results. Prior work has established (i) a fixed-shape, two-scale gravitational kernel that reproduces galaxy rotation curves without per-galaxy tuning, (ii) a coherence–topology invariant that constrains any viable modification of gravitational dynamics, and (iii) a single sharp, falsifiable prediction for wide binaries in the Solar neighbourhood. These empirical results are treated here as logically prior. The present manuscript introduces no new empirical fits. Instead, it derives the observed kernels and invariants from a minimal fourth-order coherence Lagrangian. Variation of this action yields a bi-Helmholtz field equation whose Green’s function naturally produces two intrinsic coherence scales and a weak-acceleration threshold. When projected into different regimes, the same operator reproduces both the fixed-shape galaxy kernel and the wide-binary kernel, while automatically respecting the coherence–topology invariant. The Lagrangian is not claimed to be unique; it is presented as a sufficient, non-ad-hoc explanation of patterns already measured in galaxies, clusters, and wide binaries. The theory introduces no adjustable functions beyond parameters fixed by the empirical kernel. Its predictions—most notably a sharp break in wide-binary dynamics near 7.5–8.5 kAU—are decisively falsifiable with forthcoming observational data.