Notes on the cosmological constant in E₈ group field theory

We record the current status of the cosmological-constant programme attached to the E₈ group field theory of the companion foundations paper [doi:10.5281/zenodo.19843240] and to the long-wavelength sigma model on the Wolf orbit EIX = E₈ / (E₇ × SU(2)) of the companion sigma-model paper [doi:10.5281/zenodo.19843827]. The candidate two-step mechanism of Hypothesis CC.1 splits the dimensionless cosmological constant into a perturbative pair L_bg + L_{1-loop} and a non-perturbative residue, and ties the latter to a Euclidean saddle of the sigma model on EIX with action (D − 4) ln π = 244 ln π. Two of the four operative gates have closed numerically: (C1) The Skyrme-type Bogomol'nyi reference saddle on EIX has action S^EIX_Skyrme = 244.06, strictly inside the band (12π², 244 ln π) — the "R2-path" verdict. (C2) The perturbative cancellation gate (G1) closes at the naturalness boundary ξ = 0 at leading + sub-leading Camporesi–Higuchi order, with conservative residue |Δ c_H''^(CH)| / |2π²| ≤ 3.70 %. The non-perturbative residue gate (G2) — separate-saddle closure — splits into four sub-deliverables (alternative 4-cycle representatives; non-Skyrme L-BFGS saddle search; one-loop Coleman pre-factor; multi-saddle dominance verdict) and is open. We document the conditional candidate value Λ ℓ_P² = π^(−244) ≈ 4.96 × 10^(−122) together with its 0.23-dex offset from the PDG 2024 reference, and we record explicitly the four-scenario falsification window that the closure of (G2) will eventually decide. These are working notes, not a finished paper. Verification scripts: https://github.com/lukasbednarik/E8-GFT