The Cosmological Constant from the Complete f(R) Trajectory with Standard Model Matter

The 122-order hierarchy between the observed cosmological constant and the Planck scale — Λ_obs/M⁴_Pl ≈ 10⁻¹²² — is the squared ratio of two boundary scales at which the dimensionless cosmological constant is O(1). It is not a fine-tuning. We compute the first complete renormalisation group trajectory connecting these boundaries in f(R) gravity with full Standard Model matter content, and extract a zero-free-parameter prediction: Λ = 1.09 × 10⁻⁵² m⁻², in 1.3% agreement with observation. Papers IV and V established the ultraviolet boundary: a non-Gaussian fixed point (NGFP) with Λ̃* ≈ −1.10 (anti-de Sitter) and a three-dimensional UV critical surface, making Λ calculable once G_N and A_s are measured. We show that this negative UV value is not pathological but expected: Bonanno, Platania & Saueressig (2018) demonstrated that Standard Model matter generically requires λ* < 0 for compatibility with Planck-era inflation. The physical cosmological constant is determined by the trajectory, not the fixed point. Computing the full nonlinear f(R) flow at polynomial orders n = 3–6 with Radau IIA integration, we demonstrate that Λ̃(k) crosses zero at a converged scale t_cross = −1.57 ± 0.01 (k ≈ 0.21 M_Pl), transitioning from anti-de Sitter to de Sitter. Combined with Paper I's feedback attractor as the infrared boundary condition and Paper III's two-boundary framework, this trajectory resolves all three aspects of the cosmological constant problem: the old problem (why Λ ≪ M⁴_Pl), the coincidence problem (why Ω_Λ ∼ Ω_m), and the prediction problem (what determines Λ).