This work presents a non-relativistic, weak-field framework for emergent entropic gravity derived from total entropy conservation. Using a minimal variational principle, the conserved total entropy (S_tot) generates an additive gravitational acceleration field g_tot = g_N + g_ent, where g_N is the Newtonian acceleration and g_ent is an emergent entropic contribution sourced by vacuum quantum information density (rho_Q). The framework reproduces galactic rotation curves and gravitational lensing phenomenology without invoking dark matter or modifying General Relativity in the weak-field regime. Fractional contributions of g_ent are calculated for representative galaxies, including the Milky Way, bulge galaxies, Andromeda, and dwarf galaxies such as Fornax and Sculptor. The derived equations naturally recover the Poisson equation in the weak-field limit, with the entropic term providing a dark-density contribution similar to Lambda-CDM. Included in this repository is the PDF Phase2_EntropicGravity.pdf containing the full paper, including derivation, table of g_ent fractions, and figure showing fractional entropic acceleration versus galactic radius. Optional source files can also be included for reproducibility. This work demonstrates a self-consistent, Noether-conserved entropic gravity framework that connects fundamental principles (entropy conservation, holography) to observable galactic phenomena, providing a testable alternative to dark matter models. Keywords: Entropic Gravity, Emergent Gravity, Galactic Dynamics, Dark Matter Alternative, Holographic Principle, Weak-Field Limit