As long as this isn't another attempt at trying to legitimise Core into some kind of major power in the Omicrons, it looks solid.
While I don't think the motto references anything particularly offensive, I think it's just a pretty lousy motto for a corporation, it sounds like something for a labour union or a trade association.
Oh yeah, and let me give some actual feedback too:
The formatting looks really good.
I like the concept overall, too. I'm hardly the only one who considered the old Core to be absurdly overinflated in terms of proclaimed strength, influence, and ambition, and it's good to see a more realistic take on the faction. Other Omicron factions will likely benefit from having a good Core faction as an antagonist.
The Diplomacy section is uh, pretty minimalist. Surely there's more than three factions the Core has an opinion on? "highly values cooperation with all organizations within the Sirius Sector and Gallia and engages only on rare occasions against hostile forces" sounds a bit too passive and pacifist for a PMC.