The Mittermeier RG Attractor as an Essential Skeleton of the Renormalization Group: Closed two-threshold normal forms from γE, ρ and π as a conditioned candidate for UV completion in quantum gravity

Modern quantum-gravity programs based on the renormalization group (RG)—in particular the functional RG (FRG) and Asymptotic Safety—rely on a finite set of essential couplings and on robust, cross-scheme signatures (universality). This work isolates a candidate RG skeleton by compressing three deterministically reconstructed parameter clusters into closed normal forms with hard invariants. From the log-ansatz structures ln ξ(u), ln φ̂₀(u), and ln λ̂(u) in a monotone flow coordinate u ≥ 0, we obtain: (i) A geometric sector with an exact half-residue and π-threshold: ∂_u ln ξ(u) = -½ (1 + π/6 · u)⁻¹, together with a compact normalization key a₀ ≃ 1 - γ_E. (ii) An amplitude/matter sector with rational weights and the exact cross-scale constraint (Mittermeier-12 bridge): a₂ = 12b₂. (iii) A two-threshold beta-function normal form for β_λ(u) := -∂_u ln λ̂(u) with a single algebraic amplitude (plastic constant ρ, where ρ³ = ρ + 1) and a √π-coupled threshold pair: β_λ(u) = 15/16 - (ρ · p) / (1 + pu) + (ρ/7) / (1 + qu) with p = 15 / (28√π) and q = 15 / 28. Operational Normalization and Cosmology To normalize u without importing model-specific microphysics, we introduce a Misner–Sharp compactness anchor Φ_MS = 1/6. In Schwarzschild geometry, it selects the ISCO radius; in Λ-dominated regions, it implies r = 1/√Λ. With the canonically normalized curvature cutoff k = (3/8)¹/⁴ K¹/⁴, one obtains k = √Λ exactly in de Sitter space. Using the observed vacuum value λ̂_obs = Λ/M_Pl² ≃ 2.9 × 10⁻¹²² (Planck 2018), the full closed solution yields (under the adopted bridge) u_Cosmo ≃ 300.544 and δα ≃ 0.01455. These deviations quantify integrated threshold backreaction plus the Planck-boundary shift. Analytic Signatures and Scope Within the normal-form class, the construction yields hard, falsifiable analytic signatures: Strict monotonicity (no overshoot). A unique inflection point. A universal asymptotic-u approach law: Δ(u) = 15/16 - β_λ(u) ~ (11ρ/15) u⁻¹. Scope: This is not a derivation of quantum gravity from first principles, but a minimally parametric, reproducible normal-form diagnostic. It provides a rigid, constant-locked skeleton that can be cross-tested against independent FRG truncations and scheme variations as a candidate for UV completion.