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) -1wF' (0). \ For a normalized germ \ F (w) =1+₍ ₁ cₙ wⁿ, cₙ>0, \ its ratio sequence rₙ=c₍+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_ (w) =[₀, d (t) 1-tw, \] where 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 \ d^[n (t) =tⁿmₙ () \, d (t), mₙ () = tⁿ\, d (t). \] Hence every iterate remains Pick/Stieltjes, the functions SⁿF_ (-x) are completely monotone, the functions SⁿF_ (x) are logarithmically convex on the positive axis, and the geometric attractor emerges from weak concentration ^n_ at the edge =. We conclude with explicit invariant simplices of rational germs, algebraic kernels (1- w) ^-, and the exponential function.