We present two results concerning the MOND acceleration scale and the radial acceleration relation (RAR) within the framework of Verlinde's emergent gravity. First, we show that applying the Cai-Kim first law of thermodynamics on the ΛCDM apparent horizon introduces a matter-work correction (1 + Ωm/3) to Verlinde's prediction, yielding a₀ = cH₀ (1 + Ωm/3) /6 ≈ 1. 206 × 10⁻¹⁰ m/s². This matches the empirical SPARC value to 0. 5% with zero free parameters. Second, we demonstrate that the empirical RAR interpolation function ν (x) = 1 − exp (−√x) ⁻¹ is identically the Bose-Einstein occupation number: ν = 1 + ⟨nBE⟩ (√x), where ⟨nBE⟩ (y) = (eʸ − 1) ⁻¹. This provides a statistical-mechanical interpretation of the RAR as the thermal occupation of entanglement modes on the cosmological horizon, with the local Newtonian field setting an effective energy E ∼ √ (gN a₀) against a horizon temperature kT ∼ a₀. Together, these results yield a complete, zero-parameter framework for galactic dynamics. The Cai-Kim correction fixes the acceleration scale; Bose-Einstein statistics fix the transition shape. Testing against 175 SPARC galaxy rotation curves (3, 268 data points) confirms that the Bose-Einstein form matches the data (χ²/N = 33. 5), while a linear entropy-fraction model derived from Verlinde's entanglement strain is ruled out (χ²/N = 40. 7). The framework predicts a redshift-evolving acceleration scale a₀ (z) testable with JWST. Supplementary code: https: //doi. org/10. 5281/zenodo. 18881795
Christian W. Morris (Thu,) studied this question.