Torque-Mediated Cosmology (TMC) is presented: a cosmological theoretical framework proposing that the observed galactic dynamics, the baryonic asymmetry of the universe, and the critical acceleration of Modified Newtonian Dynamics (MOND, Milgrom 1983) emerge from a background torsion field with characteristic angular velocity ω₀ ≈ 10⁻¹⁵ s⁻¹, inherited from a containing structure called the Parent Stratum (PS). The TMC postulates that our universe originated through a Primigenial Cavitation Event (PCE): a low-pressure zone that forms when high-pressure energy from the PS crosses the transition membrane toward a lower-pressure stratum. The speed c is the normalization speed of that energy reaching equilibrium in the new stratum. The intrinsic angular momentum of the PS propagated to the resulting universe as a property of spacetime, not of specific objects. The entrainment and incorporation of baryonic matter that followed the PCE constitutes the Baryonic Consolidation Event (BCE). From these postulates, TMC derives without parameter fitting: (1) the Tully-Fisher relation v⁴ ∝ M, (2) the galactic radial scale Rd ∝ M^(1/4), and (3) the critical acceleration a₀ ≈ 1.68 × 10⁻¹⁰ m/s², a factor of 1.4 from the empirical MOND value. The model does not compete with MOND; it provides the physical mechanism that phenomenology lacks. The first quantitative analysis against the SPARC catalog (175 galaxies, Lelli, McGaugh & Schombert 2016) shows that all cases of significant deviation from TMC predictions correspond to dwarf galaxies with log M ≤ 8.6, without exception in 132 galaxies with usable data. The observational analysis of SDSS DR17 with 4,988 quasars at z = 4.0–7.0 shows a coherent dipole gradient with total rotation of 84.9° and an abrupt jump of 57.8° exactly at z = 5.5, statistically unambiguous (KS p < 0.0001). The Galaxy Zoo analysis of 97,176 spiral galaxies shows a 4% chirality deviation — 54% ACW vs 46% CW — with p < 0.000001, consistent with the primordial torque signature predicted by the model. Explicit falsification criteria with pre-declared numerical thresholds are stated.