Matthew, can you elaborate on why reducing Petri net reachability to SAT would imply \$$\text{EXPSPACE} \subseteq \text{NP}\$$? Is Petri net reachability known to be EXPSPACE-complete? I don't think you're necessarily wrong, but the critical step is eluding me.

Approaching this similarly: We have an exponential lower bound on space for Petri net reachability. As you said, this necessarily imposes an exponential lower bound on time, since you can only write one cell per unit time (per tape). Suppose a reduction to SAT existed. If SAT had a subexponential algorithm, then we could defeat the exponential lower bound; so SAT, and by extension every NP-Complete problem, must not be solvable in subexponential time. Therefore, \$$\text{P} \ne \text{NP}\$$.