Characterize the second saddle term in the open TS/ST correspondence

Determine the precise characterization of the second “saddle” contribution J_- in the background-independent open topological string wavefunction for the Y^{N,0} mirror curves, namely the second term in the sum representation given by equation (eq:sigmasum), so that its definition holds in full generality across moduli and parameters.

Background

Within the open topological string/spectral theory correspondence, the non-perturbative open partition function solving the quantized mirror curve is written as a sum over two contributions (or saddles). The first contribution J_+ is explicitly constructed from the full grand potential J, but the second contribution J_- is only specified implicitly.

For local cases (e.g., in prior work on specific geometries) simple prescriptions exist, while for the general Y^{N,0} family the authors propose shifts of x and the closed moduli as a candidate relation and provide preliminary tests. Nevertheless, a complete, general, and geometrically clear characterization of the second term remains to be established.