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 M_b and the enclosed dynamical mass M_200 of extragalactic systems spanning nine orders of magnitude: M_b/M_200 = f_b tanh[(M_b/M_0)^(1/4)], with f_b = 0.157 and M_0 approximately 5 x 10^13 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 M_b proportional to V_f^4 through the MOND/relay fixed point. The halo scale fixes the virial reference scaling M_200 proportional to V_200^3. The cosmological scale fixes the high-mass saturation through the cosmic baryon fraction f_b. 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 a_0 = m_0 c / 3 and a disk-geometry factor zeta = 0.8, the BTFR coefficient is obtained as A = 49.2 M_sun km^-4 s^4, close to the empirical value A = 50. The transition scale follows as M_0 = 6.0 x 10^13 M_sun, corresponding to log M_0 = 13.78, compared with the McGaugh et al. value log M_0 = 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(V_star^2/V_gas^2). The upload includes the PDF preprint, the LaTeX source, the markdown source, and a reproduction script. The script reproduces the numerical chain a_0 -> A -> M_0 -> V_0 -> m_b(M_b) without external data files and without fitting parameters to the McGaugh et al. relation.