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The update should hopefully come out in a few hours but I'll clarify your questions while you wait. It's gonna be a bigger one due to reasons.
1.) You can always try bargaining with U-Corp, the Company you borrowed them from, for an extended leash or even permanent acquisition. You don't need a Blank Check to do it. If you wanted them to <span class="mu-i">make</span> a whole new PARADIGM out of thin air, that might require it.
2.) The Blank Check can basically be used for anything that isn't fundamentally impossible (asking them to revert time to 1957), insanely harmful to the company (asking them to go to war to another Company), or otherwise trying to cheat the system (Asking for no quotas ever thx).
At bare minimum, it can be used to exchange for another S-Rank reward. S-Rank reward examples would be really useful Oddities (like a time machine that can rewind an hour back), really powerful weapons, exclusive contact with normally unreadable parties (Company CEOs), disgustingly good coupons, etc etc.
Essentially, shit that would not be in a catalog or would be so expensive, you would need a whole shift's worth of energy to get it. Or otherwise not within your grasp. You can always test the waters by asking P-Corp what they can offer for it and see if any of them are good. Or if you want to be creative and ask for something else.
3.) The reward thing for dealing with the Outsider will happen after the shift is done, alongside dealing with sponsorships (like that one corpse sponsorships or the ones you tried offering to J/Q/R-Corp). If you're still open to the corpses one, that is.