GR, MOND, and the Cosmological Constant from One Entropy Law: Completing the Jacobson Derivation in de Sitter Space

Jacobson (1995) derived the Einstein equations from the Bekenstein-Hawking entropy law S=A/4G applied to a flat Rindler horizon via the Clausius relation. The derivation is incomplete in de Sitter space. Including the exact Narnhofer-Peter-Thirring temperature T_eff = sqrt(a²+H₀²)/(2π) and two new lemmas — (i) the Wald entropy for the ERG action is A/4G exactly, since F(Q) contains no Riemann tensor; (ii) the Bunch-Davies vacuum contributes zero energy flux through any null surface by the null-vector identity — the Clausius relation uniquely gives F'(Q) = T_U/T_eff = sqrt(Q)/sqrt(1+Q). Integration yields F(Q) = (Q)·sqrt(1+Q) − arcsinh(sqrt(Q)), the ERG kinetic function, with zero free parameters. This is the unique prediction of S=A/4G in de Sitter space: any other interpolating function requires S(A) ≠ A/4G. Standard MOND [μ(x)=x/(1+x)] requires an entropy that violates the Generalized Second Law. The 15 independent observational tests of Papers I–V constitute multi-scale tests of the Bekenstein-Hawking law at Rindler scales 10⁷–10¹⁰ kpc; eRASS1 and ACCEPT confirm S=A/4G to ~3% precision. GR, MOND, and Λ all emerge from S=A/4G. The full Jacobson derivation beyond the quasi-static limit gives viscous Friedmann corrections (0.024%, unobservable) and a thermodynamic prediction of the H₀ tension amplitude: different epochs sample different T_GH/T_eff ratios, predicting ΔH₀/H₀ = μ(sqrt(4π²−1)) × 13% = 12.8% at z~0, consistent with the observed 13.0% tension. The pre-registered Euclid η(k,z)=1.000 prediction is reframed as a test of S=A/4G at cosmological scales. Loop quantum gravity provides microscopic justification for S=A/4G via γ_BI = ln2/(π√3). The wide binary prediction is corrected to γ_v = 0.91–0.99 (ERG+EFE). A 2.4σ first signal of a₀(z) ∝ H(z) is presented.