How sharp is the radial acceleration relation's transition? A calibration-limited measurement of the interpolating-function index, and the exclusion of a C0 onset to weak gravity

Summary. The radial acceleration relation (RAR) of disk galaxies turns from Newtonian to deep-MOND behaviour over a finite width, not at a point. This note measures that width and uses it to test the C⁰ ("kinked") onset to weak gravity predicted by van Putten (2017a, 2018) as the galaxy-scale signature of a causal threshold at the de Sitter acceleration a_dS = cH. Method. The one-parameter family μ_α(x) = x/(1+x^α)^(1/α) has the correct Newtonian and deep-MOND limits for every α, so a_0 keeps its meaning and α isolates the sharpness alone: α = 1 is the simple function, α = 2 the standard function, and α → ∞ is exactly the C⁰ kink. Fitting to 2696 SPARC points with a block-covariance likelihood that carries distance, inclination and mass-to-light errors as galaxy-level (rank-one) terms rather than per-point ones. Results. α = 1.25, 68% [1.20, 1.31], 95% [1.15, 1.37], with a global 3.6 µm mass-to-light calibration offset profiled out alongside a_0 and two intrinsic scatters; a_0 = 1.17 × 10^−10 m s^−2. At the standard normalisation Υ_disk = 0.5 the fit instead gives α = 1.10 [1.06, 1.14]: the sharpness is degenerate with the stellar mass-to-light calibration, and the shift between the two treatments exceeds either statistical interval. The strong C⁰ form is excluded (ΔlnL = 440; rms 0.1464 against 0.1346 dex with each model at its own calibration). A correction in van Putten's favour. His deep scaling a_N/α ≃ 2.1 ζ^(1/2) implies ν = (1/2.1) y^(−1/2), not 2.1. Fitted with the correct coefficient and the threshold free, the two-branch model places its threshold at 1.07 a_dS — essentially exactly where he predicts — while still fitting poorly (0.172 dex against 0.135). The threshold is in the right place; the shape around it is not. What limits this. The dominant systematic on the shape of the RAR, and on a_0, is stellar-population modelling rather than rotation curves. Reported negative and partly self-correcting results are included in full, including the withdrawal of a claim made in an earlier draft that the kink exclusion was insensitive to the mass-to-light calibration. All code, logs and intermediate outputs needed to reproduce every number are included. Rotation curves and the galaxy table are the public SPARC release (Lelli, McGaugh & Schombert 2016), https://astroweb.cwru.edu/SPARC/.