I don’t think there’s much difference between our concept of empire and what Polybius called “ἀρχή” in his Histories (“dominion”, perhaps). He compares Rome to Persia and yes, some smaller states like Sparta, but it’s clearly in the context of acquiring and controlling large territories. That the Roman war machine had a particularly efficient and (for the most part) inclusive manner of absorbing territories doesn’t really change the nature of the resulting entity even if we use different words.

Any abstraction on concrete concepts of public administration across millennia (especially involving translations from ancient/dead languages) is very lossy.

> I don’t think there’s much difference between our concept of empire and what Polybius called “ἀρχή” in his Histories (“dominion”, perhaps)

Yes there is. You could write a doctorate thesis on the differences.

> That the Roman war machine had a particularly efficient and (for the most part) inclusive manner of absorbing territories doesn’t really change the nature of the resulting entity even if we use different words.

What is 'the Roman war machine'? How did it absorb territories? Which territory? Are we talking italic peninsula, client kingdoms, consular provinces or imperial dominions? From which time specifically? Efficiency is an entire topic in itself. The levels of corruption, the predatory nature of the tax collection enterprises (it was typically farmed out to private investors), were all very inefficient.

Saying Roman Empire and Persian (Seleucid?) Empire is a didactic oversimplification and I don't think there's any grounds to equating ἀρχή (which is also the root in archaic, archaeology, archetype, among others) to the modern concept of Empire or the the ancient concepts used in ancient polities to handle the intricacies of moving around troops, collecting taxes and dispensing justice across millions of square miles in Roman, Persian, or any Hellenistic state contemporary to Polybius for that matter.

Instead of getting into the weeds, let's rewind a bit. Do you believe the Roman Republic was ever in possession of something _we_ should recognise as an empire? You seemed to imply that you don't, so I'm interested in where our definition might fall short (and how would _you_ translate Polybius?) And then beyond that, what do you think is most interesting about how the Romans (or their subjects or enemies) perceived their many conquests and the vast territories that they acquired? What's the difference between their conception and ours? Genuine questions, hoping to learn!