Self-Consistent Vacuum Energy and the Particle Horizon: A Fixed-Point Solution to the Cosmological Constant and Coincidence Problems

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 fourth component of a research program examining the consequences of finite-domain vacuum energy for the cosmological constant problem. The preceding papers in this series established that gravitational consistency constraints uniquely select a vacuum residual energy density scaling as ρ_Λ ∝ ℏc/ (lₚ² L²) within a finite causal domain of size L, with an O (1) prefactor α (Paper 3). The present paper demonstrates that this scaling, combined with standard ΛCDM cosmology, defines a self-consistency condition when L is identified with the particle horizon. The present analysis does not treat the particle horizon as a dynamical infrared cutoff governing the equation of state w (a), as in standard holographic dark energy models; it treats the particle horizon as the finite causal domain entering a static self-consistency condition for the residual vacuum energy. The central finding is that the self-consistency equation reduces to a constraint on the pure number R ≡ ρ_Λ/ρₘ: the matter density cancels completely, yielding R = 8πα (r + 1 + R) / (3 I (R) ²), where I (R) is a dimensionless integral over the expansion history. This equation has a unique fixed point under the self-consistency map, and the map is a strong contraction near that fixed point. For α ≈ 0. 83, the predicted composition matches Planck 2018: Ω_Λ = 0. 685, Ωₘ = 0. 315. It is shown that the de Sitter radius yields no constraint on ρ_Λ, and that pure de Sitter space admits no self-consistent solution — matter is essential for breaking the single-scale degeneracy. The result reformulates the coincidence problem: the ratio ρ_Λ/ρₘ at the epoch of saturation emerges from a structural self-consistency condition rather than from cosmic timing, and is not a free parameter once α is specified. Series Context This paper forms the fourth component of the Finite-Domain Vacuum Energy research program addressing the cosmological constant problem. Papers 1–3 establish the observational constraints, conceptual framework, and gravitational consistency conditions underlying finite-domain vacuum energy. The present paper extends the framework to cosmological evolution by demonstrating that the ratio of vacuum energy to matter density emerges as a self-consistency condition determined by the particle horizon.