This study presents a first-principles derivation of the critical acceleration aₔ 1. 2 10^-10 m/s² in galactic dynamics, using the Scale Gauge Theory (SGT) framework. By taking the cosmological constant S from Planck 2018 as the sole external input, we derive the fundamental constants of the theory without any free parameters. The derivation consists of three key conceptual steps: Effective Action from FRG: By integrating out the dilaton field in the SGT action, we derive an effective action for the Weyl gauge field W_. The couplings are determined by the functional renormalization group (FRG) analysis. The LPA' fixed point yields a Ginzburg–Landau parameter ₆₋² = 27. 2. Abrikosov Mapping: Based on the condensation-energy structure of the two-phase vacuum, we establish a precise physical mapping between superconducting critical fields and gravitational scales: the thermodynamic critical field Hc corresponds to the background acceleration a₋₎ₖ = c²S/3, while the lower critical field H₂₁ corresponds to the galactic scale aₔ = c²N/3. Quantitative Prediction: Applying the Abrikosov formula H₂₁/Hc = (₆₋) / (2 ₆₋), the theory predicts aₔ = 1. 21 10^-10 m/s². This result agrees with the SPARC database observational value within 1. 2%. Furthermore, as a significant parameter-free byproduct, the ratio of Dark Energy to the Dark Matter-equivalent vacuum energy (S/N) is predicted to be 19. 9, which matches the observed ratio of 20. 4 within 2. 4% accuracy. These findings suggest that the elusive "Dark" sector of the universe and galactic dynamics are both emergent properties of a single, unified vacuum structure governed by scale gauge symmetry.
Susumu Goto (Sun,) studied this question.