Preprint of a manuscript submitted to Physical Review D (Regular Article). This deposit exists to timestamp the result and to carry the reproducibility scripts; it is not a journal publication. The result, in one line. The acceleration scale that organizes galactic rotation curves admits the exact closed form a0 = ½ c √(G ρΛ) = c2 √(Λ/32π) = cHΛ/√(32π/3) The three forms are algebraically identical, and the reduction cancels every factor of π and the 3. With the Planck 2018 ρΛ this evaluates to 9.43×10-11 m s-2, which is 0.70% from the value 9.36×10-11 m s-2 obtained by fitting the 175 SPARC rotation curves at a stellar mass-to-light ratio Υ3.6 = 0.70 (residual scatter 0.108 dex; Υ = 0.70 lies inside the 0.5-0.8 range expected for Spitzer 3.6 µm populations). The expression is parameter-free apart from the rational coefficient ½. What is NOT claimed, stated up front because it bounds the result. The coefficient ½ is fitted, not derived. An attempt to force it from ghost-freedom, unitarity and a holographic degree-of-freedom count is reported as having failed. The interpolating function used is identical to Eq. (9) of Milgrom, Phys. Lett. A 253, 273 (1999) — not a variant of it. That same paper derives a0 = 2cHΛ, a factor 11.58 larger, so the de Sitter-Unruh route does not leave the coefficient open: it predicts one and misses the measured scale by an order of magnitude. The present contribution is a renormalization of that coefficient to match data, not a new derivation. The coefficient here is not observationally distinguishable from the empirical cHΛ/2π of Milgrom (2020): √(32π/3) = 5.789 against 2π = 6.283 differ by 7.9%, i.e. 0.018 dex in gobs, only 16% of the fit scatter. No claim of improved fit is made. The contribution is one of form. √(32π/3) carries no geometric content; it is the unit conversion between HΛ and √(GρΛ) bookkeeping. Reading 32π/3 as a structure to be explained is an error the author has previously made and explicitly disavows here. No claim is made regarding particle physics, the Standard Model, or unification. The author publicly withdrew earlier statements of that kind and does not restate them. One falsifiable consequence. If a0 ∝ √ρΛ then a0(z)/a0(0) = (1+z)1.5(1+w0+wa) exp-1.5 wa z/(1+z), which is exactly constant for w = -1 and rises to a maximum before declining for the DESI-preferred w0 > -1, wa < 0 — in contrast to the monotonic rise predicted by a0 ∝ cH(z). A measured monotonic rise would exclude the relation. One reported determination of a rising a0(z) is recorded as a live tension, not as support. Known tensions, handed over rather than left to be found. (i) Clusters: the same closure underpredicts cluster-scale accelerations, with a median discrepancy factor 2.33 at R500 on uncorrected eRASS1 masses and a lensing-inferred cluster scale 21.6× the value above; because this a0 is lower than the conventional one, the discrepancy is 13% worse here. (ii) Solar system: applied exactly, the closure leaves a constant residual a0/2 three orders above Earth-Mars ranging bounds; an interpolation with a faster Newtonian approach leaves a02/2g instead and reduces this by five orders at a cost of 0.003 dex in the SPARC fit. The word “exact” should not be attached to the closure. (iii) a0 and Υ are degenerate in the fit; nothing here determines a0 to better than the 7.9% separating it from cHΛ/2π. Provenance. The relation was arrived at independently via three routes: an agentic automated-research harness (Karpathy, autoresearch) adapted by the author to search dimensional combinations of c, G, Λ and H0; symbolic regression in the spirit of AI Feynman (Udrescu and direct fitting to the SPARC compilation (Lelli, McGaugh route 3 fixed the coefficient. None constitutes a derivation, and a coefficient obtained by fitting remains fitted however it was found. AI-assistance disclosure. Portions of the analysis, numerical verification and drafting were carried out with the assistance of a large language model (Anthropic Claude). The author directed the work, specified and reviewed every load-bearing calculation, and takes full responsibility including for any errors. No AI system satisfies the criteria for authorship and none is listed as an author. Several intermediate claims produced during the work were found to be incorrect and were withdrawn before this deposit. Reproducibility. Every numerical claim is produced by a committed script that exits with a nonzero status if any internal consistency check fails. The verification script for this deposit is included.
Carl P. Zimmerman (Fri,) studied this question.