So considering that this is the Imperium dealing with what seems to be a human Successor State, and their likely assumption that they found a lost fragment of Humanity from the Dark Age of Technology, they are likely to identify themselves as belonging to the Imperium of Man, the premier successor of the Human domains before the Long Night. The terminology might be different than what the Hegemony would be using, but it ticks enough boxes for them to consider that the Domain of Man collapsed and that this Imperium has risen to take it's place. The shock for them to learn that around 15k years have passed since the start of the long Night would come as a shock to them, considering it'd have been less than 300 years for the Sector. The explanation of what a Warp storm is, what the Warp is, and how time flows oddly in relation to the Warp might lead them to make the conclusion that the reason why the Gates collapsed was because of these Warp Storms.
The Imperium is likely to appear in a favourable light to the Hegemony: martial state that doesn't enforce too many obligations on its constituent parts, only a tithe for mutual protection and to accept this God-Emperor, and his Ecclesiarchy. With the exception of the enforcement of religious homogeneity, it does more or less the same as what the Domain and the Hegemony rule over their worlds. They'd be likely asked what the other local powers are, but if I remember right the Hegemony holds more worlds than the other factions in the Perseus Sector, so they'd be the likely go-to choice for the Imperium to designate as the ruling authority of the Sector. The fact that they were the faction that met the Imperial fleet likely only helps their position to be considered as the primary authority of the sector.
I'm sure the techpriests attached to the vessels would immediately call for an Ark Mechanicus to arrive in the Sector, especially in order to analyse the "STC"s that the Hegemony possesses and to analyse their vessels, weapons, armour, and shields. The phase-capable craft will have them in a frenzy, as will the local power armour which I'm sure the Imperial Guard and Navy will fight over the jurisdiction on the primacy of these suits. Tri-Tachyon will likely see the Admech swarm their holdings with mass boarding parties of skitarii as soon as they can, both in order to halt their operations as well as secure any and all results of their R&D department. The presence of the [REDACTED] would likely see the Basilikon Astra scour the entire Sector to destroy and analyse every form of them they encounter.
Much like Tri-Tachyon, the Luddic Path will likely see itself condemned by the Imperium as dissidents and conventional terrorists. Nothing that condemns them too harshly or would warrant Exterminatus, but the Hegemony would be tasked with their elimination. The Luddic Church would be approached by the Ecclesiarchy and enter into talks and theological debates in order to see if they can be integrated as the local version of the God-Emperor's worship. I'm sure some leeway will be allowed, the Ecclesiarchy can be quite flexible with its tenets, especially with well established and non-Chaos influenced cults. Depending on the urgency, the separatist desire of the factions, and the resources available to be spared, there would either be a diplomatic delegation, likely there to support the Hegemony elements in their unification efforts, or a small task force, be it from the Basilikon Astra or the Imperial Navy depending on which Imperial institution is able to acquire primacy, or a full scale minor Crusade force with Adeptus Astartes support. I'm sure the Imperium would prefer to have them peacefully integrate within the greater Imperium, depending on how Andrada is willing to put aside his differences and ego with the Hegemony, or if he's still alive by the time the Imperium arrives, it could be a peaceful reintegration within the Hegemony, likely as a special system organisation under the Hegemony, or it could be a fierce defiance. The Persean League would also depend on how its leadership is willing to set aside their differences with the Hegemony, but also how their constituent worlds would react to the Imperial support of the Hegemony. Pirates are really just going to be considered the same as pirates are across the rest of the Imperium: a pest. At least this one doesn't interact with Xenos.
There'd be a good amount of exchange between the ascending Hegemony and the Imperium. Hypermatter Drives would likely be quite intriguing for the Admech and the Imperial Navy. Yes, it's much slower than the Warp Drive, but hey, you don't have to worry about Gellar Field failure or being thrown across time! It'd make travel throughout Subsectors in the Imperium much more reliable. The Admech and the Imperium would also immediately examine the Comm Relays. Those would be extremely useful to the Imperium, allowing for rapid communications across increasing distances throughout the Imperium without interference from the Warp, and not needing an Astropathic Choir to function. The vessels of the Persean Sector would also be likely to be proliferated across the Imperium as escort vessels, since I don't think that the vessels would be big enough to be considered anything else (going off their crew numbers, the Onslaught, one of the most crew intensive vessels in Starsector has a maximum crew of 1,5k, compared to the Imperium's Light Cruisers which seem to compose of at least 60k crew. Now I'm pretty sure that the ships in Starsector contain a lot more automation and require a lot less crew as a base, but even if we were to be quite generous and say that they can have 10x less crew compared to their Imperial equivalents, it still leaves them as quite small compared to Imperial designs. As a result, I'm pretty sure the larger vessels of the Sector would fit in quite well with the Frigates and Destroyers of the Imperial Navy). The various weapons would draw a lot of interest for the Admech to disassemble and understand their workings, and the carriers, or rather their hangars, would see a lot of interest. Since the hangars built fighters during the battles as well as the munitions, the integration of such systems into Imperial vessels would see a sharp improvement in the development of carrier-based designs for the Imperial Navy.
Naturally, the Persean Sector would also receive a lot of Imperium tech: Terraforming technology would be able to revitalise the Core Worlds and heal the scars of the AI Wars. Their construction and materials tech would allow for tougher constructs, better infrastructure, and larger vessels for the Sector. Access to the greater Imperial Market would likely see a revitalisation of their economy through an export focus, while they import various goods in order to catch up to the rest of the galaxy technologically and in their colonisation efforts. The introduction of rejuvenats, hive constructs, and the introduction of the STCs from the Imperium would see the productivity and security of the Sector skyrocket. It'd be a revival of the Sector the likes of which would have never come about without their introduction. Perhaps most importantly, it's likely that the Persean Sector is unreachable through conventional (by 40k standards) means, meaning that they'd have all the benefits of the Imperium without the risk of Chaos, Orks, Eldar, Necrons, Tyranids, or Tau. That means that it would grow into an economic and industrial powerhouse with barely any hindrances to it.
Or at least, that's how I imagine it could go.
