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 empirically fitted parameter clusters into closed, analytically tractable normal forms with hard invariants. Starting from the log-ansatz structures ln xi (u), ln phi0hat (u), and ln lambdahat (u) in a monotone flow variable u >= 0, we derive: A geometric sector with an exact half-residue and a pi-threshold: d/du ln xi (u) = -1/2 * (1 + (pi/6) * u) ^ (-1), together with a compact normalization key a0 approx 1 - gammaE (Euler-Mascheroni constant). An amplitude/matter sector with rational weights and an exact cross-scale constraint, the "Mittermeier-12 bridge": a2 = 12 * b2. A two-threshold beta-function normal form with a single irrational amplitude (the plastic constant rho defined by rho³ = rho + 1) and a sqrt (pi) -coupled threshold pair: betaₗambda (u) = - d/du ln lambdahat (u) = 15/16 - (rho * p) / (1 + pu) + (rho/7) / (1 + qu), with p = 15/ (28*sqrt (pi) ) and q = 15/28. The combined structure yields hard, falsifiable analytic signatures: strict monotonicity (no overshoot), a unique inflection point, and a universal UV approach law: Delta (u) = 15/16 - betaₗambda (u) ~ (11*rho/15) * u^ (-1), while the best-fit parameters lock to the closed constants gammaE, rho, pi and the rational "15-family" at sub-percent level. Scope: this is not a derivation of quantum gravity, but a minimally parametric and reproducible normal-form diagnostic that can be cross-tested against independent FRG truncations and scheme variations as a candidate universality-class skeleton for UV completion. Keywords: RG Attractor, UV completion, quantum gravity, asymptotic safety, functional renormalization group, FRG, threshold functions, universality, beta function, critical exponents, effective field theory, Euler-Mascheroni constant, plastic constant.