I added some references. If I now understand what Verdier specialization is, then by definition it differs from microlocalization just by a Fourier transform. I think the local system I am talking about will be present in the Fourier transforms of both the Verdier specialization and its vanishing part, possibly after shrinking our open set: if these two things differ by a relatively constant sheaf, then their Fourier transforms differ by something supported on the zero section.

Just that, right.

To give a sheaf on your example is equivalent to giving abelian groups A1 A2 and A3 and maps A3 --> A1 and A3 --> A2. (The group Ai is the stalk of the sheaf at i.) The constant sheaf Z is itself projective in your example, but the sheaf you call Z_{\ge 1} is not: the surjection from the constant sheaf to this does not split

In the first sentence of your proof you use assume that X is locally connected--is that necessary? I was just about to edit my answer to hedge a little more and say that I don't know what's going on with the Cantor set.

This answer is more helpful and embarrassing for me, but I think it means that Anton is dead right.