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

Version 3. Attributes the interpolating-function family to Milgrom's n-family (Famaey & McGaugh 2012; Desmond, Hees & Famaey 2024, Eq. 7a) — the family is standard, the measurement with a systematic budget is what is new. Adds a §4 comparison with Desmond et al.'s n = 0.78 ± 0.03 on identical data, and a verified progression showing the index moves 1.25 → 1.10 → 0.95 → 0.90 → 0.78 as per-galaxy nuisance freedom is increased: neither analysis is in error, and across published analyses the RAR shape index spans 0.78–1.93 while every quoted interval is ±0.03–0.09. No quoted interval on the RAR shape, including this note's, should be read as the uncertainty on the shape. Version 2. Adds a comparison with the published SPARC MOND fits of Li, Lelli, McGaugh & Schombert (2018, A&A 615, A3), which reported the same α–mass-to-light degeneracy from the opposite side and closed it with a 1.7% prior on a_0. That reference was missing from v1 and its omission overstated the novelty of the degeneracy finding. Text unchanged otherwise; all numbers identical. 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 Υ_disk = 0.61. 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. Negative and self-correcting results are reported 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/.