Yes obviously, but nowhere is it clear that you actually have a lattice of any kind. Specifically, I see no reason why pruning DAGs of unused computations would yield a lattice, and this is a result you seem to rely on. Program lattice does indeed sound nicer than program bounded poset, but I would be more wary about misusing terminology than having a nice name! :)
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.