Lower bound for Kaneko's val function

Establish the conjectural strict lower bound < \operatorname{val}(\gamma) for every hyperbolic element \gamma\in PSL_2(\mathbb{Z}), complementing the proven upper bound \operatorname{val}(\gamma)\leq 744.

Background

Kaneko's val function is the normalized cycle integral of the elliptic modular j-function along the closed geodesic associated with a hyperbolic element of PSL_2(\mathbb{Z}). The paper explains that earlier work established the optimal upper bound 744 and, subsequently, the complementary optimal lower bound given by the value at the golden-ratio class.

The introduction identifies the conjectural condition -1<\operatorname{val}(\gamma)<1 as an unresolved part of Kaneko's third problem. The paper resolves the associated distribution question for values ordered by geodesic length, but does not resolve this pointwise bound.