Fractional Susceptibility Functions for the Quadratic Family: Misiurewicz–Thurston Parameters

The paper’s Theorem C rigorously proves the frozen 1/2–susceptibility decomposition Ψ_fr,φ(1/2,z)=U1/2Σφ+W+Vsing+Vreg with the stated analyticity radii and the explicit residue formula at z=1 when sgn(Df^p(cL))=+1, using Ruelle’s density expansion, a key identity (Proposition 2.5) converting the Marchaud derivative in the parameter to one in the space variable plus a regular correction, and an operator-theoretic analysis of the Hilbert-transform contribution and tower projectors . The candidate’s outline reaches the same formal decomposition and residue but crucially (and incorrectly) passes the two-sided Marchaud derivative in the parameter through compositions/transfer operators as if a simple chain rule held, a step the paper explicitly warns is not valid without Proposition 2.5; the model also asserts key error bounds and rank-one/rational structure without the paper’s spectral-projector construction or decay estimates . Hence the paper is correct and complete for Theorem C, while the model’s proof is not rigorous.