A Tri-Scale Closure Interpretation of the Baryonic Mass-Halo Mass Relation

This preprint proposes a tri-scale closure interpretation of the baryonic mass-halo mass relation recently established by McGaugh et al. (2026, ApJ 1001: 65). McGaugh et al. report an empirical relation between the observed baryonic mass Mb and the enclosed dynamical mass M₂00 of extragalactic systems spanning nine orders of magnitude: Mb/M₂00 = fb tanh (Mb/M₀) ^ (1/4), with fb = 0. 157 and M₀ approximately 5 x 10¹3 solar masses. The present note argues that the main asymptotic structure of this relation follows naturally from a tri-scale closure within the Geometric Relay Programme (GRP). The galactic scale fixes the Baryonic Tully-Fisher scaling Mb proportional to Vf⁴ through the MOND/relay fixed point. The halo scale fixes the virial reference scaling M₂00 proportional to V₂00³. The cosmological scale fixes the high-mass saturation through the cosmic baryon fraction fb. The observed exponent 1/4 is therefore interpreted as the algebraic residue of closing these three scales simultaneously: 1/4 = (4 - 3) /4. Using the GRP-derived acceleration scale a₀ = m₀ c / 3 and a disk-geometry factor zeta = 0. 8, the BTFR coefficient is obtained as A = 49. 2 Mₛun km^-4 s⁴, close to the empirical value A = 50. The transition scale follows as M₀ = 6. 0 x 10¹3 Mₛun, corresponding to log M₀ = 13. 78, compared with the McGaugh et al. value log M₀ = 13. 7. The tanh interpolation is treated as structurally natural but not uniquely derived. The note distinguishes between derived, propagated, cosmological-input, and phenomenological ingredients. It also proposes a falsifiable residual prediction: if the tri-scale closure interpretation is correct, residuals around the McGaugh et al. relation should correlate with baryonic desynchronization, measured through the GRP Wronskian correction Phi (r) = ln (Vₛtar²/Vgas²). The upload includes the PDF preprint, the LaTeX source, the markdown source, and a reproduction script. The script reproduces the numerical chain a₀ -> A -> M₀ -> V₀ -> mb (Mb) without external data files and without fitting parameters to the McGaugh et al. relation.