Unifying the Sectors: How the Central Equation of Causal Field Theory Succeeds on the Galactic Plateau

Abstract

The Causal Field Theory (CFT) has demonstrated remarkable success in explaining dark energy on cosmological scales, yet its application to galactic dynamics has revealed a profound challenge: while the phenomenological relation \(v_{\text{obs}} = v_{\text{bar}} e^{\phi}\) perfectly describes rotation curves, the original dynamical equation of the theory was unable to produce the required field \(\phi\) in all regions of a galaxy. In this work, we succeed in this unification without introducing any new free parameters. First, we derive the \textit{Kinetic Harmony Law} for the galactic regime, \(\vec{a} = -\nabla\Phi_N - c^2\nabla\phi\). We show that this law, together with the geometric structure of the theory, provides a complete kinematic framework where the successful phenomenological relation \(v_{\text{obs}} = v_{\text{bar}} e^{\phi}\) emerges naturally in the saturation regime. Second, we show that the central field equation requires a natural extension---a constant vacuum energy term \(\lambda\)---to describe the dense galactic environment. This term is not an \textit{ad hoc} patch but a self-consistency condition forced by the observed saturation of the field at \(\phi_{\max}=0.429\), with \(\lambda = 2\beta(e^{2\phi_{\max}}-1)\). Rather than relying on the numerically unstable backward integration of the full non-linear equation, we extract the equilibrium saturation value \(\phi_{\text{end}}\) directly from the observed rotation curves in the outer plateau region (\(r \ge 5\) kpc). Analysing 36 SPARC galaxies classified as saturated systems, we uncover a remarkably tight anti-correlation between \(\phi_{\text{end}}\) and the total baryonic mass \(M_{\text{tot}}\): \(\phi_{\text{end}} = 5.3319 - 0.4935 \log_{10}(M_{\text{tot}}/M_\odot)\), with a Pearson correlation coefficient \(r = -0.805\) and a significance \(p = 3.25 \times 10^{-9}\). The slope of this relation constitutes a clear theoretical challenge for future analytic work on the extended central equation. The correlation with surface density is weak and not statistically significant (\(r = -0.208\), \(p = 0.224\)), confirming that the total mass is the primary driver of the saturation value. This \textit{Concentration Law} demonstrates that \(\phi_{\max}=0.429\) is an upper bound, not a universal constant, and that the equilibrium saturation value is dynamically determined by the host galaxy's mass. We thereby unify the cosmological and galactic sectors of CFT and provide a complete, parameter-free theoretical foundation for the dark-matter phenomenology observed in spiral galaxies.