Two-Boundary Determination of the Cosmological Constant from Asymptotic Safety and Gravitational Feedback

The cosmological constant problem — why the observed vacuum energy is roughly 10¹²⁰ times smaller than the Planck scale — is recast as a boundary-value problem on the gravitational renormalisation group (RG) flow. One boundary comes from the ultraviolet: the Reuter fixed point of asymptotic safety, which restricts quantum gravity to a low-dimensional critical surface. The other comes from the infrared: the structure-formation feedback attractor established in Paper I (Salmond 2026, DOI: 10.5281/zenodo.20156389). Key observation. The dimensionless cosmological constant λ(k) = Λ/k² is O(1) at both boundaries — λ* ≈ 0.19 at the Planck scale (from Reuter 1998) and λ(H₀) ≈ 2.07 at the Hubble scale (from Friedmann’s equation with Planck 2018 parameters). The 10¹²⁰ between them is the squared ratio of the Planck and Hubble scales — a consequence of the scale hierarchy, not a fine-tuning. Inputs from published literature. UV fixed-point values are taken from 11 published truncations of the exact RG equation by 6 independent research groups (Reuter 1998; Souma 1999; Lauscher & Reuter 2002; Codello, Percacci & Rahmede 2009; Falls et al. 2013, 2016; Donà, Eichhorn & Percacci 2014; Eichhorn, Lumma, Pereira & Sber 2018). The predicted Λ turning point lies within a factor 3–4 of the Hubble scale in every truncation tested. Main results. Two conditions on a two-dimensional critical surface generically fix a unique RG trajectory, hence a unique Λ, with combined robustness E_total < 10⁻⁷⁰. Standard Model threshold corrections (EW at 246 GeV, QCD at ~200 MeV) are shown analytically to preserve the Jacobian of the critical-surface map exactly. The paper identifies the precise condition for the programme’s success: that continuous, trajectory-dependent matter-loop running breaks the rank-1 degeneracy of the pure-gravity flow. Scope and limitations. The paper is explicit that the result is conditional: the decisive numerical check — whether continuous, trajectory-dependent Standard Model matter-loop running breaks the rank-1 degeneracy of the pure-gravity flow (designated “Computation B”) — has not yet been performed. The phenomenological RG trajectory interpolating between the UV and IR boundaries is illustrative. If the rank condition holds, the 10¹²⁰ cosmological constant problem reduces to a concrete, computable rank-condition problem on the gravitational flow equations with Standard Model matter, rather than being resolved outright. This conditionality is stated in the abstract itself. Reproducibility. Full Python code for all three phases of computation (scale perspective and two-boundary intersection; parametric convergence across truncations; response to critique including measure problem, Jacobian degeneracy, and threshold survival) is included under CC BY 4.0. Series. This is Paper III of a three-part series: - Paper I (Salmond 2026, DOI: 10.5281/zenodo.20156389): mechanism-independent attractor theorem - Paper II (Salmond 2026, DOI: 10.5281/zenodo.20222173): CSH mechanism test against DESI DR2 - Paper III (this work): UV–IR two-boundary determination