Dark Matter as Local Geometry: A Derived Gravitational Enhancement from the Entropic Field Equation

We derive a gravitational enhancement function with zero free parameters from the entropic field equation ∇²S = −2gᴮ and Berger's theorem on the sectional curvature of CPⁿ. The result is G_eff(x) = (1 + 1/x²)^(1/K_eff), where K_eff(x) = 1 + 3/(1 + 8x²) and x = a_N/a₀. The factor of 8 is the product 2K: 2 from the field equation coefficient and K = 4 from the Bures manifold curvature, entering through the entropy-driven deviation equation J'' + 2KJ = 0. Tested against 175 SPARC galaxies (3,375 data points): χ²/N = 50.6, within 0.9% of MOND-simple. Weak lensing (KiDS) independently confirms the curve. The dark energy parameter α = π/16 matches DESI DR2 to 0.025%. The BKM identity d_BKM = arctan(x) is proven, and x = sinh(ΔE/(2k_BT)) is derived from the density matrix. Dark matter is the local geometry of state space responding to its content.