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 4. This version corrects twelve errors found over three rounds of adversarial pre-publication review; versions 1–3 are withdrawn. The measurement itself is unchanged and has been reproduced independently, from the printed equations and the public SPARC release, by three reviewers with no access to the author's code. Every correction was to the argument built around it. The paper's own "Note on versions" lists all twelve with the superseded values named. The largest: a comparison against Desmond, Hees & Famaey (2024) took n = 0.78 ± 0.03 from their Table 3, which uses a modified-gravity prescription rather than the algebraic MOND relation used here; the like-for-like value in their Table 1 is n = 1.02 ± 0.04, which agrees with this note. A claimed success for van Putten's threshold turned out to rest on an unequal mass-to-light treatment and on a kernel that was discontinuous rather than C⁰. A systematic quoted as ≈0.15 in α is 0.05. An "assumption-free" exclusion quoted as 4.4 is 3.5 once every parameter is fitted by the metric being compared. A conclusion attributed to Li et al. (2018) is one they explicitly reject. Summary. The radial acceleration relation 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 (2017b, 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) — Milgrom's n-family — 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. Fitted to 2696 SPARC points with a block-covariance likelihood carrying distance, inclination and mass-to-light errors as galaxy-level rather than per-point terms. Results. α = 1.25, 68% [1.20, 1.31], at Υ_disk = 0.61 and a_0 = 1.17 × 10^−10 m s^−2, with a global 3.6 µm calibration offset profiled out. At the standard Υ_disk = 0.5 the fit gives α = 1.10 [1.06, 1.14] — the sharpness is degenerate with the stellar calibration, and the shift between treatments exceeds either statistical interval. Desmond, Hees & Famaey (2024), marginalising ~900 per-galaxy parameters on the same points with the same family, obtain n = 1.02 ± 0.04 and a_0 = 1.08 ± 0.04: an independent confirmation at 1.4σ in the index and 0.3σ in the scale. The C⁰ limit is excluded by ΔlnL = 440; the rms exclusion depends on protocol and is +0.0080 ± 0.0023 dex (3.5) done consistently. On van Putten's threshold. His deep scaling a_N/α ≃ 2.1 ζ^(1/2) inverts to ν = (1/2.1) y^(−1/2), not 2.1. But the continuous two-branch relation that follows is algebraically the same kink already tested, with a_0 = a_t/4.41, and pinning its threshold at exactly cH_0 costs 0.0008 dex — because the 2.1 coefficient is the factor that makes those two statements equivalent. The threshold's apparent success is a property of the construction, not a measurement. This note constrains the shape, not the location. What limits this. The dominant systematic on the shape of the RAR, and on a_0, is stellar-population modelling rather than rotation curves. Taken with the external comparison, the index on this sample is determined to about ±0.15 within the algebraic MOND relation; a reader should carry 1.0–1.3 rather than any single quoted interval. Desmond, Bartlett & Ferreira (2023) reached a compatible conclusion by exhaustive symbolic regression on the same sample, and the novelty claim here is narrowed accordingly: what is new is the treatment of the C⁰ kink as the α → ∞ boundary member of a nested family with its threshold fitted, not the measurement of the index. All code, data, logs and intermediate outputs needed to reproduce every number are included, with a README giving run order and runtimes, and with the superseded scripts retained and labelled so each correction can be checked against what it replaced. Rotation curves and the galaxy table are the public SPARC release (Lelli, McGaugh & Schombert 2016), https://astroweb.cwru.edu/SPARC/.