We formulate a limit framework for deriving a Harnack inequality for nonnegative viscosity supersolutions of the variational fractional infinity-Laplacian. The operator under consideration is the Hölder infinity-Laplacian arising as the \(p\rightarrow \infty \) limit of the fractional \(W^{s,p}\) -energy. The main point of the paper is to isolate and prove the nonlocal logarithmic estimate needed in this approach with constants uniform in p. The proof follows the logarithmic lemma mechanism of Di Castro–Kuusi–Palatucci, with the dependence of the constants on p tracked explicitly. Together with a fixed exterior datum, this yields a Harnack inequality with a nonlocal \(\infty \) -tail term for limits of approximating fractional p-supersolutions. The proof combines the uniform logarithmic estimate, a precise tail limit, an asymptotic use of fractional Morrey embedding, and an asymptotic stability result identifying the limiting function as a viscosity supersolution of the Hölder infinity-Laplacian.
