As far as I know the only case that can't be closed is mindcrash, since they don't give you a CHIME law to report with. (I'm pretty sure I've tried reporting the mindcrash toggle to no effect. Claims to the contrary are welcome). Things move on pretty promptly upon solving that case anyways. Samhammering does totally go to your case log, allowing them to be closed. Hyppi even proudly reminds you that this is some janky hack when you do so. Somehow didn't even occur to me that all cases would mean hunting down the ones that aren't actually handed to you, but totally makes sense. Good variety in types of ways to quickly access each of them, too, honestly.
I am not a speedrunner by any means, but I got it in my head to maybe give this a go, and an 'All Cases' category is what I started trying to work out before realizing that it didn't exist. Not sure if it would just mean having them all in your case log or if it would mean needing to smash the close button on (almost) all of them. Like, it's not required in the game even in its own text but im here for thematic speedrunning apparently. all i know is dylan is a pal who spots you all the coin you need for hypnocure since there's a lot of time waiting for emails elsewhere to handle other important business.
then of course there's the question of who's gonna be the first to go for a 100% archive run