Quantum field theory carries two unresolved structural problems: ultraviolet divergences requiring renormalisation, and a vacuum energy density 10¹²⁰ times larger than observed. We argue both problems have a common origin — continuous spacetime — and both are resolved when spacetime is treated as a discrete Planck-cell network. The Planck length provides a physical UV cutoff, rendering QFT automatically finite. The vacuum energy decreases continuously over cosmic time as the Planck length grows (ρ_vac ∝ c⁶), reframing the cosmological constant problem as cosmic evolution. QFT is entirely preserved as the correct effective field theory at all sub-Planckian energies. A new testable prediction: coupling constants should exhibit a small cosmological time evolution driven by the evolving Planck length, testable with the ELT/ANDES spectrograph at z < 1. This is Paper 5 in the Pixel Theory series. Companion papers: Paper 1: https://doi.org/10.5281/zenodo.20491684 Paper 2: https://doi.org/10.5281/zenodo.20529501 Paper 3: https://doi.org/10.5281/zenodo.20530178 Paper 4: https://doi.org/10.5281/zenodo.20525925 Paper 6: https://doi.org/10.5281/zenodo.20585009 Paper 7: https://doi.org/10.5281/zenodo.20540035