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

The cosmological constant problem — why the observed vacuum energy is roughly 10^120 times smaller than the Planck scale — has resisted solution for decades and driven much of theoretical physics toward anthropic reasoning. We argue that Λ is not the outcome of an improbable cancellation, nor a selection effect in a multiverse, but the unique solution of a boundary-value problem (BVP) on the gravitational renormalisation group (RG) flow. One boundary condition 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, "The Cosmological Constant as a Feedback Attractor"; hereafter Paper I), which fixes Λ through gravitational self-consistency with bounded sensitivity. The dimensionless cosmological constant λ(k) = Λ/k² is known to be O(1) at the UV fixed point (λ* ≈ 0.19, from the asymptotic safety literature since Reuter 1998) and is trivially O(1) at the Hubble scale (λ(H₀) ≈ 2.07, from the Friedmann equation). Taken alone, this restates the hierarchy in different variables. The present work supplies the missing ingredient: an independently determined IR boundary condition that, combined with the UV fixed point, imposes two conditions on the two-dimensional critical surface, generically fixing a unique RG trajectory and rendering the 10^120 ratio a consequence of the dynamical range (k_Pl/k_H₀)², not a fine-tuning. Two conditions on a two-dimensional surface generically fix a unique RG trajectory, hence a unique Λ, with combined robustness E_total < 10⁻⁷⁰: perturbations to Planck-scale physics are exponentially irrelevant for the late-universe vacuum energy. Using published fixed-point data from eleven truncations of the exact renormalisation group equation, we show that the predicted Λ turning point lies within a factor 3–4 of the Hubble scale in every case, providing a candidate resolution of the coincidence problem that does not require anthropic reasoning. We demonstrate analytically that Standard Model threshold corrections preserve the Jacobian of the critical-surface map exactly, and originally posed the programme's success as a binary rank question (rank 1 vs. rank 2). A computation reported in this revision recasts that question. Using the Donà–Eichhorn–Percacci (DEP14) Einstein–Hilbert truncation with Standard Model matter content, we find that the endpoint map Φ: S_UV → (G_phys, Λ_phys) is formally rank 2 in both pure gravity and the SM case, with a finite but large condition number cond(DΦ) ≈ 3 × 10⁵ and 5 × 10⁵ respectively. The decisive observation, reported in this revision, is that the cond-number measures inverse-amplification of generic UV perturbations into IR-observable shifts, not a hidden tuning of the physical trajectory: backward integration from the observed IR endpoint (G_N, Λ_obs > 0) converges to the SM NGFP (Route 1, Theorem 9.2), and the trajectory's NGFP-side tangent is the universal slow eigenvector v₂ for any IR endpoint with O(1) dimensionless data — confirmed by an Option 3 scan across 47 orders of magnitude in G_N·Λ_obs. The v1.2 "BVP-selectivity" assumption is therefore demoted from a postulate to a derived statement in this truncation: the leading tangent at the NGFP is automatic by saddle-point dynamics, and the framework's surviving assumption set is {A1, A2, A3, A4} (NGFP existence, finite critical surface, smooth IR limit, IR attractor). The 10^120 cosmological constant problem is thus reduced, in DEP14 1L-II SM, to a fully derivable consequence of the boundary-value structure: 120 digits from the dynamical range (k_Pl/k_H₀)², an O(1) infrared boundary value Λ̃(H₀) ≈ 2, and Paper I's attractor. Higher truncations (Computation B, Step 2, §11) remain to be performed. A partial Step 2 computation is reported in this revision. Using the Codello–Percacci–Rahmede f(R) truncation at polynomial order n = 3, we confirm the eigenvalue hierarchy Re(θ₁,₂) = 2.71 > θ₃ = 2.07 that makes the R² coupling the slow relevant direction on the three-dimensional critical surface. The one-parameter family left by two IR conditions is identified with the Starobinsky scalaron: different departures along v₃ produce different physical R² couplings, hence different CMB scalar amplitudes A_s. If dim(S_UV) = 3, the framework therefore correlates the cosmological constant with the inflationary amplitude — a zero-free-parameter prediction testable with existing CMB data. The quantitative determination of A_s(Λ_obs) requires Standard Model matter content in the f(R) truncation and is deferred to Paper IV. Series. This is Paper III of a six-part series: 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)