THE VACUUM ON DISCRETE TERMS

Abstract. Computed as a field-theoretic zero-point sum, the vacuum energy density exceeds the value inferred

from observation by about a hundred and twenty orders of magnitude. This paper argues that the sum is the

wrong calculation: on a discrete register the cosmological term is a boundary count, and no sum exists to

diverge. Five results are given. The vacuum is the unsourced tier of the shortfall theorem, which fixes its sign as

positive and its equation of state at w = −1 at every redshift, since the term arises as a constant of integration.

One Planck density distributed over the horizon’s count of Planck cells is exactly the statement that the dark-

energy fraction has the rest value 2/3. The measured fraction exceeds it by 2.7 percent on the Planck 2018 fit

and by 5.4 percent on the fit to DESI’s second data release. The excess equals the sweep rate −3 Ω_Λ Ω_m times

the elapsed sweep, on both datasets to better than one percent, so it is a clock reading. The present epoch is

proved to be the single external reference of the time sector. A holographic model with its parameter fixed at

2/3 is stated as a discriminator, in which the phantom crossing, the crossing of 2/3 and the coincidence of the

two horizons are one event. The account is refuted by a measured equation of state departing from −1, which

DESI’s preference tests, and by Ω_Λ converging away from 2/3. Every exact claim re-derives in a public suite at

github.com/thefirsthorstmann/g-theory-verify. Theorem 1 is also formalized there in Lean 4.