  Bethe/Gauge correspondence as it is usually stated is ill-defined in five
dimensions and needs a "non-perturbative" completion; a related problem also
appears in three dimensions. It has been suggested that this problem, probably
due to incompleteness of Omega background regularization in odd dimension, may
be solved if we consider gauge theory on compact $S^5$ and $S^3$ geometries. We
will develop this idea further by giving a full Bethe/Gauge correspondence
dictionary on $S^5$ and $S^3$ focussing mainly on the eigenfunctions of (open
and closed) relativistic 2-particle Toda chain and its quantized spectral
curve: these are most properly written in terms of non-perturbatively completed
NS open topological strings. A key ingredient is Faddeev's modular double
structure which is naturally implemented by the $S^5$ and $S^3$ geometries.
