The Cosmological Constant as a Feedback Attractor

Three independent research programmes — Lombriser (2019, PLB), Farrah et al. (2023, ApJL), and Bonanno & Reuter (2002, PRD) — have arrived at structurally similar results suggesting the cosmological constant Λ may be dynamically determined rather than fine-tuned. This paper formalises the common mathematical structure behind these proposals and proves that any cosmology with negative feedback between Λ and gravitational structure formation possesses a unique, stable fixed point. Observational inputs. The analysis uses Planck 2018 cosmological parameters (Ωm = 0.315, ΩΛ = 0.685, σ₈ = 0.811, H₀ = 67.4 km/s/Mpc), the Sorini, Peacock & Lombriser (2024, MNRAS 535, 1449) stellar-fraction dataset f*(α) for Λ/Λobs ∈ [0.001, 10⁴], and four published halo mass functions: Press-Schechter (1974), Sheth-Tormen (1999), Tinker et al. (2008, ApJ 688, 709), and Watson et al. (2013, MNRAS 433, 1230). Main results. An exact elasticity–slope identity, E = 1/(1 + |σ(Λ*)|), bounds the fixed point’s sensitivity to the coupling amplitude. Five physically distinct coupling mechanisms and four mass functions all converge to the same attractor. A 100× overestimate of the coupling constant shifts the predicted Λ by only 1.3–5× (depending on baseline convention), compared to the 10¹²⁰ range of the original problem. The feedback-attractor posterior is 332× more concentrated at the observed Λ than the SPL anthropic posterior; this density ratio is reported as an illustrative comparison, not a full Bayesian model selection. A stochastic-attractor demonstration shows convergence from four orders of magnitude of initial spread to 0.087 dex scatter in fewer than ten iterations. Scope and limitations. The framework proves existence and stability of the attractor but does not specify the physical coupling mechanism. The specific mechanism determines the precise elasticity but not the attractor’s existence. A companion paper (Paper II) proposes one candidate mechanism and derives testable predictions. Reproducibility. Full Python code reproducing all figures, numerical results, and sensitivity analyses is included in the bundle under CC BY 4.0. All code uses only standard scientific Python (NumPy, SciPy, Matplotlib). No fitted or calibrated parameters are introduced — all inputs are taken directly from the published sources cited above. Series. This is Paper I of a three-part series: Paper I (this work): mechanism-independent attractor theorem.Paper II (Salmond 2026, DOI: 10.5281/zenodo.20222173): test of a specific coupling mechanism (Connected Singularity Hypothesis) against DESI DR2 BAO data .Paper III (Salmond 2026, DOI: 10.5281/zenodo.20222352): UV–IR two-boundary determination connecting asymptotic safety to the IR attractor