Gravitational Wave Dispersion and Emergence of Negative Energy Density at Topological Boundaries in a Discrete Spacetime Framework

We investigate the macroscopic physical consequences of a discrete topological spacetime framework based on the 3-dimensional hypercube graph Q3. When the energy of gravitational waves approaches the Planck scale, the topological spectral gap leads to the dispersion of high-frequency components. We derive a modified group velocity v_g(E) ≈ c(1 - 2/9 * E/E_P) from the discrete wave equation. Furthermore, using the heat kernel method for boundary conditions governed by 5th roots of unity, we demonstrate the emergence of local negative energy density in highly symmetric phase-transition boundaries: ρ_neg ≈ -0.242 * (ħc/L^4). This model strictly satisfies the Ford-Roman quantum inequalities and provides clear falsifiable signatures for future high-frequency gravitational wave observatories and precision quantum measurements. (Note: The attached PDF is written in Chinese).