Empirical Calibration of the Dimensional Transition Constant (Λ) in Scale-Variant Entropic Gravity Using Wide Binary Kinematics

Standard cosmological and quantum models rely on the foundational assumption that spacetime operates as a perfectly smooth, three-dimensional continuous manifold (D=3). This assumption demonstrably breaks down at extreme scales, requiring the insertion of non-physical mathematical placeholders such as singularities and dark matter. The Thermodynamic Cascade framework models spacetime as a scale-variant fractal geometry where the effective spatial dimension D(S) fractures toward D=2 in low-entropy density regimes. This paper establishes the mathematical chain of custody for scale-variant entropic gravity, redefining it not as a fundamental force, but as an emergent pressure gradient driven by the downward flow of energy through an infinite fractal funnel. Utilizing a non-linear regression model applied to wide binary orbital kinematics simulated after Gaia DR3 parameters, we isolate an optimal transition constant of Λ = 0.29715 ± 0.00172. This calibration provides the precise mathematical infrastructure required to map orbital velocity excesses at low accelerations without invoking dark matter fields.