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

This paper is part of a research program proposing that the cosmological constant arises not from total vacuum energy but from the finite-domain residual to which gravity actually couples. This paper develops the third component of a research program examining the consequences of finite-domain vacuum energy 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 ρ_grav(L) ~ ħc / (ℓ_p² 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 (ℓ_p / L)², reflecting the natural disparity between ultraviolet and cosmological scales as expressed through the gravitationally constrained survival weighting. Series Context This paper forms the third component of the Finite-Domain Vacuum Energy research program addressing the cosmological constant problem. The series investigates how the observed cosmological constant may arise from the residual vacuum energy that survives symmetry cancellation within the finite causal domain of the universe. Papers 1 and 2 establish the observational constraints and conceptual framework distinguishing vacuum state-energy from the gravitationally operative residual. The present paper develops the quantitative structure of the framework by deriving the scaling of the finite-domain residual from gravitational consistency bounds and horizon thermodynamics.