Radial Structure in the Baryonic Mass Discrepancy of SPARC Spirals: Beyond the Tully–Fisher and Mass–Radius Relations

We investigate the logarithmic baryonic mass discrepancy, φ(r) = ln[v_obs(r)/v_bar(r)], in the saturated kinematic regime (r ≥ 5 kpc) of N = 83 SPARC disc galaxies with M_bar < 10¹¹ M_⊙. The median discrepancy is well described by a linear relation with baryonic mass, φ_end = 4.04 − 0.351 log₁₀(M_bar/M_⊙), with Pearson r = −0.797 and R² = 0.635. We show that the slope of the endpoint value φ(R_max) with respect to log M_bar is fully accounted for by the combination of the Baryonic Tully–Fisher Relation (BTFR; slope α) and the empirical mass–radius relation R_max ∝ M_bar^p: the predicted slope, ln(10)(α − 1/2 + p/2) = −0.271 (from the same-sample values α = 0.230 and p = 0.304), reproduces the observed slope to numerical precision—a direct consequence of the algebraic identity. The BTFR+R(M) identity accounts for 77% of the observed slope of the median discrepancy–mass relation; the remaining 23% is attributable to a distinct observable—the radial structure Δφ ≡ φ_end − φ(R_max)—which correlates significantly with galaxy size R_max at fixed baryonic mass (ρ_p = −0.42, p < 10⁻⁴). This correlation shows the same sign in all three mass tertiles (r = −0.574, −0.375, −0.455), though it is marginal in the middle tertile, and shows no significant residual correlation with gas fraction (ρ_p = +0.18, p = 0.10 after controlling for M_bar). The radial structure constitutes a quantitative constraint on galaxy-dynamics theories. We present this as a purely empirical decomposition, keeping the theoretical interpretation open.