Mass of the AGL-EFT Condensate from Galactic Dynamics: α ∝ V −1 c and the Emergence of a Universal Scale

The Adiabatic Geometric Leveling eective eld theory (AGL-EFT) introduces a con-densate eld with a characteristic length scale α that has been treated as a free parameterin Papers IIIV of this programme. Here we derive α from the equilibrium of the conden-sate in the galactic gravitational potential, following the GrossPitaevskii equation for anon-relativistic scalar eld. The equilibrium condition yields α = ℏ/(√2 maVc), where mais the condensate mass and Vc is the asymptotic rotation velocity. Using the 12 galaxies ofPaper IV for which the metrological scale αinf was independently measured, we infer:ma = (5.69 ± 0.45) × 10−25 eV/c2,with a galaxy-to-galaxy scatter of 8.3% and no correlation with Vc (Spearman ρ = 0.12,p = 0.71), consistent with a universal mass. The predicted scale αpred(Vc) = ℏ/(√2maVc)reproduces the observed αinf values with a scatter of 0.037 dex (8.5%) across galaxiesspanning Vc ∈ [25, 225] km/s. Re-tting 171 SPARC galaxies with α xed by the pre-diction and only Λobs free, we obtain a median χ2ν = 1.052 versus 1.541 for the base model(Wilcoxon p < 10−4), with 67% of galaxies showing improvement. The AIC comparison gives∆AIC ≈ 0, meaning the AGL model with one free parameter plus a theoretical predictionfor α is statistically equivalent to the two-parameter base model. This converts the Paper IIsignal from a two-parameter t into a one-parameter model with a testable prediction for α.