Correcting the Verlinde Acceleration Scale and Identifying Bose-Einstein Statistics in the Radial Acceleration Relation

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