Invariant Classes and Monotonicity Structures in Renormalized Tail Dynamics

We continue the renormalized-tail program for analytic power series by studying structural classes preserved by the nonlinear operator \[ S(F)(w)=F(w)-1/wF'(0). \] For a normalized germ \[ F(w)=1+∑n≥ 1 c_n w^n, c_n>0, \] its ratio sequence rₙ=cₙ₊₁/cₙ is shifted exactly by S. We show that every left-shift invariant class of bounded positive ratio sequences gives an S-invariant class of analytic germs. This yields a unified hierarchy of invariant cones, including bounded-ratio classes, log-convex and log-concave coefficient classes, and all higher finite-difference sign classes on the ratio sequence. The monotone-ratio classes have direct dynamical consequences. If (rₙ) is nondecreasing, then for every x≥ 0 in the disk of analyticity the orbit values SⁿF(x) are nondecreasing in n; if (rₙ) is nonincreasing, they are nonincreasing. In both cases we obtain sharp geometric bounds and monotone convergence to the universal geometric attractor. Our main structural result concerns the compact Stieltjes class \[ F_μ(w)=∫[0,τ]dμ(t)/1-tw, \] where μ is a probability measure. We prove that this class is exactly invariant under S, and that the induced dynamics on representing measures is the size-bias transform \[ dμ[n](t)=t^n/m_n(μ)\,dμ(t), m_n(μ)=∫ t^n\,dμ(t). \] Hence every iterate remains Pick/Stieltjes, the functions SⁿF_μ(-x) are completely monotone, the functions SⁿF_μ(x) are logarithmically convex on the positive axis, and the geometric attractor emerges from weak concentration μ[n]⇒δ_ρ at the edge ρ=μ. We conclude with explicit invariant simplices of rational germs, algebraic kernels (1-λ w)-α, and the exponential function.