Spacetime Densification Theory: the nonlinear AQUAL bridge and the galactic transition scale

We formulate the quasi-static galactic-sector bridge between the Newtonian limit of Spacetime Densification Theory and its SPARC/RAR phenomenology: an AQUAL-type k-essence completion of the densification kinetic sector. We derive, in closed form, the kinetic function F(ξ) = √(ξ(1+ξ)) − sinh−1√ξ, whose quasi-static field equation is the elliptic Bekenstein–Milgrom equation with the standard interpolating function, recovering the Newtonian limit at high gradient and the deep-MOND limit at low gradient. The associated propagating mode is ghost-free; its longitudinal characteristic speed on the static galactic background, v∥2/c2 = (2+ξ)/(1+ξ) ∈ [1, 2], is superluminal in the deep-MOND regime — generic to AQUAL-type theories — yet the theory remains causal because the substrate selects a preferred frame furnishing a global time function. These results fix the form of the radial acceleration relation and the slope-4 baryonic Tully–Fisher relation vc4 = GMba†, and are independent of the absolute scale. We then show that the transition scale reduces to a single dimensionless closure constant τ, through a†(τ) = (√τ/2π) cH0√(2ΩΛ), with the phase-space factor 6π2, the kinetic stiffness, and the inheritance of the cohesive energy density all derived. We do not derive τ: six independent microscopic routes are tested, and none, within the present coarse-grained substrate framework, yields the value τ = 1 required to place a† at the observed MOND scale (within 1.6% of a0, and bracketed by the SPARC estimators τ ∈ [0.89, 1.08]). We adopt τ = 1 as a calibrated statistical closure, and advance its universality across galaxies — required by the tightness of the radial acceleration relation — as a falsifiable prediction tested by the intrinsic scatter. The closure constant τ is found to share the infrared-versus-ultraviolet fine-tuning structure of the cosmological-constant problem: the substrate carries two vacuum-level calibration constants — the cohesive coupling χ, fixed to ΩΛ, and τ, fixed to a0 — that protect the effective description from ultraviolet overshoots. In summary, within the quasi-static galactic sector the densification programme fixes the form of the galactic dynamics and anchors the scale to a single universal calibrated constant whose universality is falsifiable; the result is the galactic AQUAL bridge of SDT, not a complete cosmological replacement for dark matter. It does not derive a0 from first principles, but reduces it to one named constant and identifies the linkage of χ and τ — which, if realized by a common ultraviolet completion, would predict a0 = cH0√(2ΩΛ)/2π — as the controlling open problem.