Geometric Tension Equilibrium (GTE): A Covariant Action Principle Framework Unifying Galactic Dynamics and Cosmological Redshift Evolution

Abstract Contemporary astrophysics faces a profound theoretical bottleneck: the elusive physical nature of dark matter and the "cosmological coincidence problem" whereby local galactic kinematics numerically parallel global cosmological scales. This paper introduces the Geometric Tension Equilibrium (GTE) framework, which models the spacetime continuum as a compact 3-sphere superfluid constrained by an invariant hardware limit—the Planck tension T0. By evaluating a modified Einstein-Hilbert action, we demonstrate that Newton's gravitational constant G is not fundamental, but an emergent geometric projection (G = c4/T0). Under the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, the covariant geodesic equation strictly requires the minimal background geometric acceleration a0 to be the inverse radial projection of the cosmic horizon: a0(t) = c2/R(t). This geometric coupling yields the exact differential identity H(t) = −(da0/dt)/a0(t), leading to a rigid redshift evolution law: a0(z) = a0(0)(1+z). Furthermore, we model the baryonic galactic disk as a viscous fluid using the Navier-Stokes stress tensor, establishing a quantifiable coupling efficiency between the geometric topological torque and the baryonic velocity dispersion. We derive an exact analytic equation for local galactic rotation velocities that naturally reduces to the Baryonic Tully-Fisher Relation (BTFR) at low redshifts while predicting the anomalous Keplerian decline observed in high-redshift (z ≈ 1.5) turbulent galaxies. This provides a complete, falsifiable paradigm shift devoid of unobserved mass parameters. This work builds upon the initial geometric framework presented in: https://doi.org/10.5281/zenodo.20263271