Huh. One of the things that drew me into following this timeline was the idea of an Empire that was not centered on Persia but somewhat shifted west. A fascinating prospect was that it might shift north, to develop the Black Sea region and be centered there the way the Roman Empire was essentially the Mediterranean shore with appendages.
Also I've always been full of doubts and alarm whenever the Pontic kings contemplated vast new conquests, fearing that they would overstretch and the basis of their rule would be broken.
So, here we have Arcathius proving to be capable of retaking most of the Achaemenid Empire's territory. I quite understand he is pleased because from his point of view, he is merely returning to his ancestral status as a great prince of that Empire, whose family had been exiled. Well and good for him, if he and his successors can keep it a while, as talk of a "later" phase of the realm, one more properly called "Persian," suggests they shall.
Nevertheless, even though this phase is still called "Pontic" and not "Persian" yet, it looks to me like another Persian Empire has been founded. Its role will be determined by geography and demographics; future Pontic/NeoPersian Padishahs will inherit the same perspectives and preoccupations the many Iranian-centered empires of OTL had.
To be sure there is some grounds to expect some innovative extensions--Pontus did already hold land north of the Black Sea, so perhaps an expansion into southeastern Europe; Mithridates did form an alliance with the Greeks so the new Empire picks up where Darius proposed to leave off as it were, with full possession of Greece and an implication of yet more westward expansion; at any rate consolidation of the eastern Med--but of course the Achaemenids already ruled most of that except the Greek lands, so what is new there is Greece itself plus the possibility of expanding up the Balkan peninsula.
But any such expansions would take place on an essentially Persian basis; in conquering the Iranian plateau Arcathius may have won glory and riches and a firm basis for his dynasty all well enough, but he's walked away without a glance back from the different prospects of an Anatolian-centered empire.