For that construction to work, these 'virtual' nodes need to exist in the poset. Your poset is just the vertex set of your DAG ordered by reachability, so they don't exist. I think your intuition is good, but the mathematics just doesn't add up. Perhaps you should look into things like trace monoids, pomsets and petri nets?
Please understand that I'm not trying to be mean here. You mention you've been applying for grants to work on this, but if you don't explain the mathematics adequately or demonstrate how it differs from existing formalisms then I imagine people are just going to reject you instantly. I would suggest seriously trying to formalise some of this stuff in a proof assistant like Coq. That'll quickly tell you if your theory works or not.