The Radial Acceleration Relation from Two-Horizon Entropy Sharing with Zero Free Parameters

We derive the MOND interpolation function from first principles using Jacobson’s thermodynamicgravity. Jacobson (1995) showed that the Einstein field equations emerge from δQ = T dS appliedto local Rindler horizons. We show that in a universe with a finite Hubble horizon, the entropyavailable to the local Rindler horizon is reduced by competition for vacuum entanglement. TheUnruh effect creates entanglement with planar symmetry (perpendicular to the acceleration axis);the Hubble horizon creates entanglement with spherical symmetry. These two structures competefor the same vacuum modes, and their different geometries make the competition angle-dependent.The sharing fraction is f = TR /(TR + TH,eff ), where TR is the Unruh temperature and TH,eff is theeffective Hubble temperature projected onto the acceleration axis. The mode overlap between planarand spherical entanglement goes as cos2 θ; integrated over the backward hemisphere, this yieldsTH,eff = TH /6. The result is f (a) = a/(a + cH0 /6), a modified-inertia interpolation function witha0 = cH0 /6 = 1.09 × 10−10 m/s2 —9% from Milgrom’s empirical value, with zero free parameters.Tested against the SPARC Radial Acceleration Relation (2,693 data points, 153 galaxies), thisprediction achieves σ = 0.133 dex scatter, identical to fitted MOND (σ = 0.133 dex with a0 fittedto 1.2 × 10−10 ).