Zero Crossing of the Cosmological Constant in f(R) Asymptotic Safety with Standard Model Matter

Note: This has been amended with improved & clearer claims and re-titled as such. We compute the first renormalisation group trajectory of the cosmological constant in f(R) gravity with full Standard Model matter content (N_S = 4, N_D = 45/2, N_V = 12) at polynomial orders n = 3–6. The ultraviolet non-Gaussian fixed point established in Papers IV and V has Λ̃_* ≈ −1.10 (anti–de Sitter) and a three-dimensional 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. Using Radau IIA integration with real eigenvectors, 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. The crossing is confirmed at all truncation orders, with Δt_cross = 0.008 across n = 3–6, and Newton's constant remains positive throughout. This is the first demonstration that the Standard Model f(R) NGFP produces a de Sitter infrared through its RG flow. The trajectory confirms that Λ̃ is O(1) at the UV fixed point; combined with Paper III's observation that λ(H_0) = 3 Ω_Λ ≈ 2.06 is also O(1) at the Hubble scale, the 122-order hierarchy between Λ_obs and M_Pl^4 arises as the squared ratio of these two boundary scales. It is not a fine-tuning. Combined with Paper I's feedback attractor as the infrared boundary condition and Paper III's two-boundary framework, the trajectory addresses all three aspects of the cosmological constant problem: the old problem (why Λ ≪ M_Pl^4), the coincidence problem (why Ω_Λ ∼ Ω_m), and the determination problem (what fixes Λ). A consistency check against the Friedmann equation with Λ-independent inputs yields Λ = 1.09 × 10^−52 m^−2, within 1.3% of the Planck 2018 value. Paper V's spatial ODE method achieves A_s within 0.5 decades of the Planck target at n = 6; closing this gap at higher truncation order (Paper VII) would enable the direct extraction of Λ from the trajectory endpoint. --This is Paper VI of a six-part series, no more papers are planned. Paper I (Salmond 2026, DOI: 10.5281/zenodo.20156389): The Cosmological Constant as a Feedback AttractorPaper II (Salmond 2026, DOI: https://doi.org/10.5281/zenodo.20222173): Testing a Connected-Singularity Mechanism for Gravitational Feedback CosmologyPaper III (Salmond 2026, DOI: 10.5281/zenodo.20222351): Two-Boundary Determination of the Cosmological Constant from Asymptotic Safety and Gravitational FeedbackPaper IV (Salmond 2026, DOI: 10.5281/zenodo.20284172): The Cosmological Constant as a Zero-Parameter Prediction of Asymptotic Safety with Standard Model MatterPaper V (Salmond 2026, DOI: 10.5281/zenodo.20286625): Resolving the Spin-2 Boundary Layer in f(R) Asymptotic Safety Paper VI (Salmond 2026, DOI: 10.5281/zenodo.20286761): Zero Crossing of the Cosmological Constant in f(R)