The Cosmological Constant from an AdS-to-de Sitter f(R) Renormalisation-Group Trajectory with Standard Model Matter

We compute, to our knowledge for the first time, the renormalisation-group trajectory of polynomial f (R) gravity with Standard Model matter content from the ultraviolet non-Gaussian fixed point (NGFP) toward the infrared. The result is an AdS-to-de Sitter trajectory: the UV fixed point is AdS-like, with Λ̃* ≈ −1. 10 and G̃* ≈ 0. 90, while the physical trajectory crosses Λ̃ (k) = 0 at k ≈ 0. 21 Mₚₗ and enters the positive-Λ branch. The zero crossing is confirmed directly at n = 3–6, with Δtcross = 0. 008, and Newton's constant remains positive throughout. At n = 7 and n = 8, separatrix shooting identifies the unique trajectory threading the spin-2 singularity boundary layer, with separatrix amplitudes shifting by only ~7% between orders, supporting convergence of the physical trajectory. These results establish that the physical cosmological constant is determined by the RG trajectory, not by the fixed-point sign: a negative UV value does not preclude a positive infrared value. The three-dimensional UV critical surface supplies three relevant amplitudes. The separatrix/regularity condition removes one continuous freedom, and the two IR inputs Gₙ and Aₛ fix the remaining two. Within this truncation and boundary prescription, Λ is then an output, not a free parameter. For the effective cosmological constant, once the vacuum-energy/radiative-stability layer is protected by an independent mechanism, the smallness of Λ in Planck units, Λ/M²ₚₗ ~ H₀²/M²ₚₗ, reduces to the squared ratio of two boundary scales—because the framework provides derived reasons (the NGFP and the feedback attractor) for the dimensionless coupling to be O (1) at both ends, rather than requiring a cancellation. Combined with the Paper I feedback attractor and the Paper III two-boundary framework, the result supplies the UV trajectory side of a conditional determination programme for Λ. The Paper I attractor condition can be expressed as the vanishing of an effective infrared beta function, βIR_Λ = 0, making the UV and IR boundary conditions formally parallel: the dimensionless coupling λ (k) is pinned to O (1) at both ends by derived dynamics rather than assumption. Paper II proposes the Connected Singularity Hypothesis (CSH) as the series' candidate physical coupling mechanism; the present trajectory result does not require the CSH specifically. The trajectory does not yet bridge the full 140 e-foldings from the Planck scale to the Hubble scale. Applying the Friedmann equation to inputs that do not directly insert Λobs yields Λ = 1. 09 × 10⁻⁵² m⁻², within 1. 3% of the Planck 2018 value; some of these inputs retain model dependence through the distance-redshift relation. Paper V's spatial ODE, which navigates the spin-2 boundary layer directly, reaches Aₛ = 6. 6 × 10⁻⁹ at n = 6—within 0. 5 decades of the Planck target—and extending it to the converged n = 8 NGFP is the path to a more direct trajectory-endpoint extraction of Λ, reducing reliance on the Friedmann-equation reconstruction. --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) Asymptotic Safety with Standard Model Matter