Quantum Field Theory as the Long-Wavelength Limit of Planck-Scale Discrete Spacetime: UV Completion, Vacuum Energy Drainage, and Cosmological Running of Coupling Constants

Quantum field theory (QFT) is the most precisely tested framework in physics, yet it carries two unresolved structural problems: ultraviolet divergences requiring renormalisation, and a vacuum energy density 10^120 times larger than the observed cosmological constant. We argue that both problems have a common origin — the assumption of a continuous spacetime manifold — 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 without fine-tuning. The vacuum energy decreases continuously over cosmic time as the Planck length grows (ρ_vac ∝ c^6), reframing the cosmological constant problem as a statement about cosmic evolution rather than fine-tuning. The renormalisation group acquires physical meaning as the description of coarse-graining over Planck cells at different scales.

Crucially, QFT is entirely preserved as the correct effective field theory at all sub-Planckian energies: Pixel Theory provides the UV completion QFT requires without replacing a single QFT prediction.

A specific testable prediction emerges: coupling constants should exhibit a small additional cosmological time evolution driven by the evolving Planck length, separable from the standard renormalisation group running. The accumulated drift is Δα/α ~ 1.8×10^-4 over 5 Gyr, with d(α)/α/dt ~ 3.6×10^-14 per year — testable with the ELT/ANDES spectrograph at z < 1 with precision ~10^-7 per measurement. Detection of this signal would distinguish Pixel Theory from standard cosmology; a null result at this precision would constrain the framework.

The hierarchy problem is reframed as epoch-dependent rather than fine-tuned: since the Planck mass m_P ∝ c^(1/2) decreases as c decreases while the Higgs mass is approximately c-independent, the ratio m_Higgs/m_P is a consequence of the current epoch, not an accidental cancellation.

This is the fifth paper in the Pixel Theory series. Companion papers are published at:
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