Maybe the article neglects to repeat that the parameter t has to be positive; but it has little practical importance beyond culling sectors 1, 2 and 3 (because they don't contain the ray's starting point p and the direction vector points between up and right from somewhere in sector 4)
Sector 4, on the other hand, is guaranteed to contain an intersection because it contains p.

Don't expect this sort of paper to spell out everything in detail; some parts of the whole algorithm, in this case very important ones like locating a point (p) in a octree or discarding the sectors that intersect only the opposite ray (t<0) are obvious (to the authors) and already in common practice (and therefore outside the scope of a non-tutorial paper).

With few commendable exceptions, the purpose of a research paper that claims to give an algorithm to do something is illustrating some novel idea (in this case trickery with parametric line/ray equations and sector edges), not describing and demonstrating a complete and dependable solution for the stated problem; a practical implementation that has a good performance and doesn't choke on boring corner cases (e.g. 45° angles or rays through the center of a sector) is your job.

It is true research papers do not go into details, but this paper is a quite practical one, and going from paper to implementation is not complicated.

This is only because in this paper the combination of unusually detailed exposition and unusually simple math allows anybody to reconstruct the neglected details without leaving holes, but the authors actually skip I/O and special and symmetric cases like the best of them.