Parameter-Free Derivation of Vacuum Energy Density from Six-Dimensional Discrete Spacetime Lattice

The origin and magnitude of the cosmic vacuum energy density represent one of the central problems in modern cosmology. Based on the self-consistent microscopic assumption of a six-dimensional discrete spacetime Planck-scale lattice (3 spatial + 3 temporal dimensions), this paper proposes a parameter-free derivation of vacuum energy density, where the two additional temporal dimensions are compactified within the Planck scale and Planck time. Gravity emerges as the statistical average of phase-density bias in the lattice wave function, described by a relativistic nonlinear wave equation. To verify the correctness of this equation, we calculate the regime where quantum gravitational effects become significantly measurable. Under the weak-field approximation, we obtain the condition g c², demonstrating that cosmological scales constitute the regime where quantum gravitational effects are significant, requiring the solution of vacuum energy density using a nonlinear wave equation that incorporates quantum gravity. Under the quasi-static approximation at cosmological scales, the wave equation reduces to an algebraic relation, yielding the probability density of lattice excitations as topological defects with vortex structure: ||² = / (4 mP). The spherical symmetry of time contributes a geometric factor, while the degeneracy between left-handed and right-handed defects (both with zero-point energy E₀ = 12mP c²) doubles the total production probability, giving the lattice defect excitation probability as p = 2/P. Utilizing the quantum gravity significance condition, we obtain the effective mass density of ₕ₀₂ c²/ (G Rᵤ²), which is consistent with the observed dark energy value. This theory contains no free parameters and makes testable predictions: the dark energy density evolves with cosmic scale as ₕ₀₂ Rᵤ^-2, which can be tested by near-future surveys such as DESI and Euclid.