A Discrete Lattice Framework for Spacetime Geometry and Energy Density Constraints at the Planck Scale

We propose a discrete, phenomenological framework for spacetime geometry whereinthe cosmic manifold is modeled as a rigid, close-packed lattice of immutable Planck-scalespheres. Space is treated as non-continuous, characterized by localized voxels of a fixeddiameter equal to the Planck length (lp). By evaluating a localized quantum harmonicfluctuation baseline within this spherical geometry, we assert a linear scaling relation betweenthe localized energy density (ρE) and the effective angular frequency (ω), yielding ρE = γω.This coupling constant γ is formulated entirely from fundamental physical invariants. At theabsolute frequency boundary, the expression naturally resolves to the order of the Planckenergy density. We extend this static boundary condition into a discrete wave equation onthe lattice, analyzing how a characteristic propagation velocity matching the speed of lightcan emerge as a structural property. Furthermore, the geometric packing leaves an inherent25.95% interstitial void volume; we analyze the severe sub-Planckian quantum constraintsof this boundary, proposing a non-local field interpretation to reconcile the framework withastronomical dark matter observations and current Lorentz invariance violation limits.