If determinism is true, then any possible world in which at t0 I act differently from how I actually act at t0 is a world where either (a) the laws are different from the actual world's laws or (b) the laws are the same but the past is different at every time prior to t0. In case (b), it follows that my own causal history will be different in that world. But if essentiality of origins is true, my own causal history could not have been different. Therefore, if essentiality of origins and determinism are both true, then any possible world in which at t0 I act differently from how I actually act at t0 is a world where the laws are different from how they are.
On the plausible assumption that I cannot do something the doing of which entails that the laws are other than they are at @ (where "@" names the actual world), it follows that if determinism and essentiality of origins are true, then I cannot act otherwise than I do.
Or, to put it differently, if essentiality of origins holds, the compatibilist's "Had I wanted to, I would have acted differently" conditionals are counter-nomic. But it is most implausible that a counter-nomic conditional would suffice to capture our "could have done otherwise" conditionals.
Therefore, if essentiality of origins is true, either determinism is false or we are not free.