Finite-Domain Vacuum Energy and the Cosmological Constant: Gravitational Consistency and the Quadratic Survival Function

This paper is part of a research program examining finite-domain vacuum energy and its implications for the cosmological constant problem. The observed dark energy density is smaller than the naive quantum field theoretic estimate of vacuum energy by roughly 120 orders of magnitude. This discrepancy can be clarified by distinguishing bulk vacuum energy defined over an infinite-domain idealization from the residual vacuum energy gravitationally accessible within a finite causal region. The present work determines the scaling of this finite-domain residual. Two independent gravitational consistency conditions—the Cohen–Kaplan–Nelson gravitational collapse bound and the holographic entropy bound—both imply that the gravitationally admissible vacuum energy density in a causal domain of size L must scale as ρgrav (L) ~ ħc / (ℓₚ² L²) Within the quantum field theoretic mode sum, this constraint determines the spectral weighting of vacuum modes. Specifically, any survival function S (kL) whose ultraviolet behaviour follows a power law must have suppression exponent n = 2. This is not a choice within the power-law class but a necessary condition imposed by the independently derived gravitational scaling. The resulting energy density is consistent with the observed magnitude of the cosmological constant without fine-tuning. The 120-order-of-magnitude suppression relative to the naive Planck-scale estimate is consistent with the ratio (ℓₚ / L) ², reflecting the natural disparity between ultraviolet and cosmological scales as expressed through the gravitationally constrained survival weighting.