The Universal Scaling Law of Topological Stiffness: A UV-Complete Proof of the Complexity Ratchet

We provide a scale-invariant proof for the Stiffness Functional (\Xi) within the Coherent Field Model (CFM). This work reinterprets the "Dark Matter" effect as a physical manifestation of vacuum lattice tension sourced by the Soft Breaking of Continuous Time-Translation Symmetry. By identifying a discrete temporal update cycle, we derive the Modified Newtonian Dynamics (MOND) acceleration constant, a_0, as a direct consequence of the Hubble expansion rate modified by the universal azimuthal flow harmonic (v_2 = 0.12). Using the Sim-Omega simulation suite, we demonstrate that galactic rotation curves and large-scale filamentary structures emerge naturally from a 0.9994 recovery-efficient medium, removing the requirement for undetected particulate matter. Key results include: Symmetry-Derived Acceleration Floor: The MOND constant a_0 is derived as a_0 = \frac{cH_0}{2\pi} \cdot (1 + v_2) \approx 1.21 \times 10^{-10} \text{ m/s}^2, matching internationally observed scales. Metric Tensor Perturbation: We define the lattice potential as a perturbation of the metric (g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}), where h_{\mu\nu} is scaled by the energy floor (1-R), identifying the 0.06% non-recovered energy fraction as the source of gravitational "stiffness". Quasar Dipole Prediction: The model accounts for the observed spatial dipole variation in the Fine-Structure Constant (\alpha) by identifying vacuum pressure differentials (v_2) across Gpc scales. Cosmic Web Integrity: Sim-Omega validates that galactic velocity dispersions remain asymptotic (flat) due to the topological resistance of the lattice, aligning with observed Tully-Fisher relations. This work provides a UV-complete, scale-invariant framework that unifies the subatomic energy floor with cosmological binding, reclassifying "Dark Matter" as the energetic cost of a discrete spacetime manifold.