Emergent Logarithmic Potential from Classical Keplerian Phase-Space Counting

We show that a logarithmic gravitational potential emerges from the classical phase-space geometry of the Keplerian 1/r interaction together with a phenomenological weak-field statistical ansatz. The number of accessible phase-space cells in a 1/r potential scales as N(r)∝rη, where η is determined by the effective dimension of accessible action space (η=1 for rotationally constrained disks, η=3/2 for isotropic spherical systems), yielding a logarithmic Boltzmann entropy S(r)∼ηkBlnr for any η. The effective logarithmic potential then follows from the potential of mean force after integrating out internal Keplerian phase-space degrees of freedom, F=ΦN−ΘHσ, where the effective energy scale ΘH is identified with the depth of the Newtonian potential well at the cosmological transition radius, introducing no free parameters. The resulting framework naturally reproduces flat galactic rotation curves, the baryonic Tully–Fisher relation v_c⁴=GM_ba₀ where a₀∼cH₀ is the empirical BTFR acceleration scale, and the weak-field lensing signature of isothermal halos. Three parameter-free comparisons against 357 unique galaxies from the SPARC and ATLAS3D surveys confirm the predicted scalings, including a parameter-free prediction for the velocity dispersion of early-type galaxies (σ/e4=3/8GM_ba₀, R²=0.856, N=231), whose coefficient arises from an η=3/2 isotropic action-space geometry distinct from the η=1 disk relation. The framework is developed as a phenomenological statistical theory and does not claim to replace a complete microscopic description of dark matter at all scales.