Which a0? The Coefficient of the Radial-Acceleration Relation Under Full Nuisances: A Population-Split Answer

The MOND acceleration scale is quoted as a0 ~ 1.2e-10 m/s^2; a Lambda-derived reading gives a0 = c^2 sqrt(Lambda/32pi) = 9.36e-11 -- a 22% fork degenerate with the stellar mass-to-light ratio. This note runs the named arbiter: a full hierarchical SPARC fit (Li et al. 2018 methodology; per-galaxy distance + inclination Gaussian priors from the SPARC table, lognormal Upsilon prior, profiled intrinsic scatter) on the Lambda-reading's OWN interpolation nu(y)=sqrt(1+1/y), with validation gates reproducing the published McGaugh-nu a0=1.20e-10 and the fixed-Upsilon 0.108-dex anchor, and full adversarial verification including one CAUGHT AND CORRECTED likelihood error (an inclination-transformation Jacobian) whose correction STRENGTHENED the adverse finding. Four results at equal weight: (1) THE FULL-SAMPLE TENSION IS REAL -- optimum a0 = 1.24-1.37e-10 on the Lambda-reading's own nu; 9.36e-11 excluded >3sigma-equivalent on every honest error treatment (naive Dchi2 161-270; galaxy-bootstrap ~3.3sigma; leave-out-most-influential holds; a global SPS zero-point cannot rescue). The missing-nuisance defense is closed. (2) NO SINGLE a0 FITS SPARC ON ANY INTERPOLATION: gas- and star-dominated populations prefer coefficients differing by x1.8 on the Lambda nu AND on McGaugh's -- every published single-number a0 is an average over a factor-two population disagreement. (3) THE ONLY MASS-TO-LIGHT-FREE MEASUREMENT LANDS ON THE LAMBDA VALUE: gas-dominated galaxies (Upsilon dynamically irrelevant), full nuisances, corrected likelihood -> a0 = 9.8e-11, statistically centered on c^2 sqrt(Lambda/32pi) (Dchi2 <= 1.5 both conventions; bootstrap P = 0.39). An earlier 7.7e-11 low-side hint is RETRACTED here as the Jacobian artifact. (4) THE HIGH-SIDE PREFERENCES COHERE AS ONE UPLIFT: the star-dominated SPARC preference (+0.146 dex in Upsilon) and the independent KiDS lensing preference (+0.20 dex in baryons) sit a COMMON x1.4-1.6 baryon uplift from the Lambda value -- one coherent systematic in the two regimes where baryon accounting is least certain, not independent misses. DIAGNOSIS rather than verdict: on current data the RAR coefficient is not a single measurable number; the deciding measurements are independent bulge mass-to-light ratios, absolute baryon budgets at the 0.05-dex level, and larger gas-dominated samples (BIG-SPARC) -- not more RAR statistics. Pre-committed symmetrically: population-synthesis confirmation of the baryon values kills the Lambda reading; confirmation of the coherent uplift vindicates it with its cleanest evidence already in hand. Credit: methodology Li+2018; data SPARC (Lelli-McGaugh-Schombert); gas-rich logic McGaugh; lensing Brouwer+2021/Mistele+2024. The Lambda coefficient is a modelling choice; earlier over-claims were publicly retracted (2026-06-23); the adverse and favorable findings are banked in the same public ledger.