Information-Geometric Matter Dynamics and a Dimension-Matched Gravitational Response

We develop a physical realization of controlled-completion entropy geometry and derive its action, field equations and observable response relations. This results in analytic realizations to General Relativity resulting in corrections to galaxy rotation curves which are tested against SPARC, resulting in statistical significance of \( Z \geq23 \). Relative entropy fixes a positive kinetic metric and, once a symmetric finite comparison is prescribed, a restoring potential. We state the foundation's conditional local-horizon closure to Einstein gravity and the additional spacetime and scale inputs. A two-coordinate oscillator then yields an Einstein-coupled matter model whose canonical matching links four contact interactions. We prove an invariant nonrelativistic polarization, an exact relation among response coefficients, and connected predictions for core support, gravitational instability, coherent pressure and cosmological dilution. A stress decomposition identifies how internal gradients and thermodynamic preparation affect gravitational dynamics beyond total density. We also develop a conditional infrared extension: a capital-\(\Gamma\) capacity-clock argument supplies a proper-time interpretation, horizon thermodynamics motivates an acceleration scale, and spatial Weyl invariance selects \( p=d \) within a local homogeneous gradient class. For \( d=3 \), the deep AQUAL equation yields a logarithmic spherical exterior, flat rotation curves and \( v_c^4=GM_ba_* \). Contracted relative entropy identifies the required differential response. An explicit Gaussian variance constraint yields a cubic entropy cost at criticality and a quadratic correction at finite correlation length. For a sufficiently small positive response floor, conditional gravitational matching permits a nearly flat interval followed by a Keplerian decline; the observable, state and local gravitational coupling remain to be derived. Prepared-state calculations verify the toy model, while SPARC comparisons illustrate conditional state and infrared responses. The adopted inner-calibrated approximation, evaluated on all 3389 defined rows, gives a conservative fixed baryon-only Gaussian-marginal null bound \( p_G\leq5.90\times10^{-123} \), with two-sided normal equivalent \( Z_G\geq23.58 \). This rejects that specified null, without identifying the microscopic mechanism.