PRE-THRESHOLD FRACTIONAL SUSCEPTIBILITY FUNCTION: HOLOMORPHY AND RESPONSE FORMULA

The paper proves both holomorphy (radius > 1 for 0 ≤ η < 1/2) and the fractional response identity by constructing a parameter set Ω with quantitative density, building uniform towers, and deriving uniform mixing/renormalization-period control (Theorems 1, 4, 5) before executing a clean proof of Theorem 5 via an H^s_p estimate and Fubini’s theorem . By contrast, the candidate solution assumes a uniform spectral decomposition in a full neighborhood (“CE is open”) and then informally disposes of root-of-unity peripheral spectrum by “shrinking to mixing,” steps that are not justified in general and contradict known subtleties addressed by the paper’s construction of Ω and the constancy of the renormalization period over Ω(t0) . The paper’s proof of the response identity uses the identity (I − L_{t0+t})(ρ_{t0+t} − ρ_{t0}) = (L_{t0+t} − L_{t0})ρ_{t0} together with justified interchanges of sums and integrals, exactly producing M_{η,Ω(t0)}(R_φ)(t0) = Ψ^{φ,t0}_{Ω(t0)}(η,1) . The candidate’s reasoning would be correct only under stronger assumptions (uniform spectral gap and mixing on a full neighborhood), which the paper shows cannot be taken for granted without the delicate Ω-construction and tower machinery.