> So, I invite others to answer the second part of the question! I was very sloppy about the definition of \$$\mathcal{X}(x,y)\$$. If we do it carefully, the answer "\$$\infty\$$" should pop out in this case _automatically_, without any special extra rule.

We should take \$$\mathcal{X}(x, y)\$$ to be the meet -- the greatest lower bound -- over all paths in the graph from \$$x\$$ to \$$y\$$. When there are no such paths, every value is a lower bound of this set, so we just take the greatest -- in this case, that's \$$\infty\$$.

More generally:

\$\mathcal{X}(x, y) = \bigwedge \left\\{ \bigotimes p \mid p \text{ is a path from } x \text{ to } y \right\\} \$