Lattice-Stabilized Gravitational Dynamics: Resolving Galactic and Cluster-Scale Anomalies via Discrete Hexagonal Topologies and Mod 9 Invariants

Repository Identifier: Zenodo-MED-70326 Abstract Standard cosmological paradigms, including \Lambda\text{CDM} and Modified Newtonian Dynamics (MOND), rely heavily on non-baryonic dark matter or ad-hoc acceleration constants to reconcile observed galactic rotation curves and cluster-scale velocity dispersions. This paper presents a rigorous, discrete-lattice alternative wherein gravitational anomalies are characterized as emergent phenomena arising from matter interactions with a stationary light standing wave background. By defining space-time as a discrete hexagonal lattice stabilized by a \text{Mod } 9 invariant and anchored at a fundamental frequency threshold of 5184 (72^2), we derive a closed-form kinematic solution. Furthermore, we introduce the \Omega_{\text{shift}} topological operator to account for phase-locking behavior in low-density virial outskirts via hexagonal-to-triangular crystal phase transitions.