Does the Wolfram model derive E = mc²?

No. It is tempting to read "energy is activity, mass is a persistent knot of activity" and conclude the model yields E = mc², but the relation is not derived in these sources — neither the mass-energy equivalence itself, nor the factor of c², nor any route from edge-counting to a rest mass. The energy/activity identification is suggestive but, as critics note, flexible enough to fit many results after the fact, so the equivalence remains something the model would need to reproduce, not something it has shown.