(Note: This glitch appears to be already patched in the JP release of the game, though it should still work on at least the current US version.)
So apparently, when pausing the game (with the + button) while a cutscene is being skipped, the cutscene continues being fast-forwarded but with certain elements (like player/NPC animations and movement) being frozen. Shoutouts to @Setonkeys (and probably also @RepentMF, indirectly) for noticing this interaction (and bringing it to my attention).
While messing around with this glitch, I discovered a way to abuse it to warp to the normal ending. Basically, you can use this cutscene pause glitch to hold the big Core shutter open. There's an invisible wall or something that spawns there to prevent you from leaving during the fight, but it luckily disappears after the fight. By ending the fight close to the exit (and above the water), there's just enough time to leave the room before your air runs out. If you then backtrack to Labyrinth W and drown in any of the puddles there, you'll trigger the normal ending "breakdown" scene, which ends with Kazuma saving you and the credits rolling.
Here's a (potato quality) video where I demonstrate this setup:
And here's a technical explanation of why this works: The game stores its progression state in a bunch of numbered "flags", which can be either set or unset, and are usually manipulated by "events" which are coded in the *.tsc script file for each area and triggered by various in-game actions. Upon beating the Core, the post-fight cutscene event sets flag 4000. Flag 4000 is a special flag: Normally, when Quote runs out of air while underwater, the game runs event #0041 ("You have drowned. Want to retry?"), but if flag 4000 is set when this happens, the game instead runs event #1100. What event #1100 actually does depends on which map you're in (most maps don't even have an event #1100), so it's intended that the only place you can drown while flag 4000 is set is in the Core, whose event #1100 is the usual "Losing... consciousness..." cutscene. (That cutscene also happens to unset flag 4000, though you also get an air tank so you couldn't drown afterwards anyways...). But by using the cutscene pause glitch to escape the Core room after the fight, you can keep flag 4000 set permanently (or at least until you go back to the Core and drown there). This means that, from there on out, whenever you drown, the game will run the current's map's event #1100 instead of killing you. Anybody who's worked on routing Cave Story 3D glitched categories will know all too well where this is headed: Drowning in Labyrinth W runs that area's event #1100...which doesn't exist, but that's okay because the TSC parser lands on the next highest event instead, which is #1200. That event is part of the normal ending "breakdown" sequence, and so that's how you end up at the credits. (And if you're wondering how you're able to re-enter the Labyrinth to begin with...the flag that locks you out of the Labyrinth is set by the Core drowning cutscene, not by the post-fight cutscene.)
If you know anything about CS3D routing (and even if you don't), you may be starting to realize that this "core escape glitch" (so to speak) has enormous potential—it's like a baby version of CS3D's inventory glitch! Although it's more restrictive than what you can do in CS3D, there are still a lot of interesting things that are possible. In particular, the high event number of #1100 is useful for TSC "overflow" glitching (which CS3D routers will certainly have experience with). (For those unfamiliar with what any of this means, I have an in-depth CS3D inventory glitch explanation here: https://pastebin.com/gpsHt7aE )
For example, loading in Labyrinth I and then exiting to Labyrinth H and drowning triggers Labyrinth I's life capsule event, which you can run infinitely (even if you've already collected that life capsule!) to obtain as much health as you want:
If, instead of escaping to Labyrinth M, you instead save in Dark Place, you can exit the Labyrinth entirely with the help of a second save file, by entering Mimiga Village in that second save file, then exiting, reloading back into the first Dark Place save, and then drowning (this triggers the door to First Cave). Professor Booster won't appear in Arthur's House to give you the Booster v2.0 (his trigger only spawns upon beating Ironhead), but you can obtain the Booster v2.0 anyways (and enable the Labyrinth teleporter location, as a bonus) by entering Arthur's House, then exiting to Mimiga Village and drowning there:
Unfortunately, I haven't as of yet found a way to progress the game from there (the teleporter sends you to normal Egg Corridor, just for the record), or otherwise warp to the Best Ending credits or even find any kind of usable skip for the Best Ending route. There are several significant hurdles involved with achieving the best ending, so routing it (if it's even possible) will take a lot more effort. This is the closest I've gotten so far:
(it takes you to the normal ending credits afterwards BibleThump )
All of this brings up another important discussion point: Since a credits warp (at least for normal ending) now exists, we'll need to decide what happens with the current Normal Ending speedrun category. Currently, the rules say that timing ends "when the HUD disappears when jumping off the island after Undead Core", but obviously this new route doesn't do that. So that means either the timing rule for the existing Normal Ending category will need to be adjusted to accomodate the new credits warp, or the credits warp will need to become a separate category leaving the current Normal Ending category as a sort of "no wrong warp" (or "no major glitches"?) category. (I'm slightly partial to the latter option, but I'd like to hear other peoples' thoughts.) And if the core escape glitch ever leads to a major skip in the Best Ending route, we may need to have this discussion again for that category :) .
As of the v1.3 patch, the cutscene skip pause glitch is now patched, which means the credits warp is no longer possible (unless you kept a save file from after performing Core Escape).
Since there isn't a proper way to revert to an older version of the game after updating, I've chosen to delete the credits warp category. Here are the two runs that were on the leaderboard before it was deleted:
23:58 on normal difficulty by me:
38:15 on easy difficulty by @Lor:
For anybody that happens to still be on v1.2 of the game and wants to do some credits warp runs, feel free to share them here! Otherwise let's all mourn the passing of a cool glitch that had a lot of wacky potential.