First-Principles Derivation of the Critical Acceleration in Scale Gauge Theory : Functional Renormalization Group Fixed Point and Abrikosov Mapping

This study presents a first-principles derivation of the critical acceleration aᵤₚ 1.2 × 10⁻¹⁰ m/s² in galactic dynamics, using the Scale Gauge Theory (SGT) framework. By taking the cosmological constant ΛS 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 φ in the SGT action, we derive an effective action for the Weyl gauge field W_μ. The couplings are determined by the functional renormalization group (FRG) analysis. The LPA' fixed point yields a Ginzburg–Landau parameter κGL² = 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 alow = c²√ΛS/3, while the lower critical field Hc1 corresponds to the galactic scale aᵤₚ = c²√ΛN/3. Quantitative Prediction: Applying the Abrikosov formula Hc1/Hc = ln(κGL) / (√2 κGL), the theory predicts aᵤₚ = 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 (ΛS/ΛN) 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.