A Bandlimited Framework for Quantum Gravity: Second-Order Low-Pass Filtering, Dual Horizon Singularity Avoidance, and the Vacuum Residual as Dark Energy

A novel, non-dissipative framework for quantum gravity is proposed, modeling the fabric of spacetime as a covariant, bandlimited Shannon-sampling system. By modifying the standard position-momentum commutation relations with a quadratic momentum regulator characteristic of the Generalized Uncertainty Principle (GUP), it is shown that spacetime behaves fundamentally as a critically damped, second-order low-pass filter. This geometric cutoff establishes a universal ultraviolet (UV) threshold at the Planck length (lP), natively eliminating point-like singularities. Gravitational collapse is resolved through a dual horizon framework that reorients the GUP deformation parameter as two dynamic scaling metrics: βM, quantifying the inward-facing mass-energy metric strain, and β_Λ, quantifying the outward-facing vacuum metric strain, governed by the conservation of spacetime bandwidth (βM + β_Λ = 1). During black hole collapse, β_Λ redshifts and stabilizes the diverging Hawking flux into a constant power output on the order of ≈ 0.05324 MP for a distant observer, halting collapse into a stable, sub-atomic Planck remnant that serves as a non-interacting Cold Dark Matter (CDM) candidate. Finally, applying the invariant momentum measure to vacuum energy loops reveals that the smooth second-order roll-off leaves an irreducible infrared (IR) residual phase leak. Bounded by the physical Nyquist limit, this vacuum energy is dictated by the absolute dynamic range of the cosmic horizon (ΛIR/MP), which uniquely fixes the neutrino mass scale (M_ν ~ 10⁻³~eV) and predicts the observed Dark Energy density (~ 10⁻⁴⁷~GeV⁴) without fine-tuning. In the primordial universe (β_Λ → 0), the scaling of vacuum energy density (ρvac = ρbaseline / β_Λ²) drives rapid cosmic expansion, naturally relaxing the manifold toward its contemporary macroscopic baseline (β_Λ → 1).