The ΛCDM paradigm models galactic rotation curves via collisionless particulate dark matter halos, yet it faces persistent structural failures, most notably the “cusp-core” discrep ancy in dwarf galaxies. This paper tests an alternative continuous-media framework, “The Geometric Thaw,” which reinterprets dark matter phenomenologies as the macroscopic vis coelastic frame dragging of a 14-dimensional spacetime manifold. Treating the vacuum as an Oldroyd-B viscoelastic fluid, we derive a two-parameter kinematic equation dependent on Geometric Stiffness (k) and a Viscoelastic Relaxation Length (λ). Using Markov Chain Monte Carlo (MCMC) regression, we evaluate this model against the standard Navarro Frenk-White (NFW) profile utilizing strictly filtered, high-fidelity kinematic and 3.6 µm photometric data from 175 galaxies in the SPARC database. Point-by-point deconstructions of representative morphologies—NGC 5055 (massive HSB), UGC 06614 (giant LSB), and IC 2574 (gas-dominated dwarf)—demonstrate that the viscoelastic formulation natively reproduces linear inner cores, extended flat profiles, and declining outer edges without ad hoc baryonic feedback tuning. The viscoelastic manifold achieves a vastly superior global Reduced Chi-Square (χ2 ν) and Bayesian Information Criterion (BIC) score, suggesting that supplementary galactic rotation velocities are the direct dynamic geometric response of the vacuum to baryonic shear stress. Keywords: Dark Matter, Galactic Kinematics, SPARC Database, Viscoelasticity, NFW Profile, Modified Gravity