Well, I'm constantly alluding to the Vaults, so you need to be a bit more precise. The first mention of the Vaults was in my second ever article, Shifty (also the archival section of Vault-SCORPIO). Then there's my Sandbox where I indeed alluded to infinite chess before (under TAURUS' tab) and finally, my "recent" post in the In-Universe subforum. There are also some other very tenuous allusions which are probably imperceptible for anyone but me, but that's fine. Whilst imposing bulwarks against a hostile universe, each Vault also has to fulfil some pragmatic roles, be it SCORPIO keeping backups of most records or PISCES literally doing laundry.
So you want to give the Unsinger both sentience and steal away with whatever it tries to hide? Then it will be clever enough to notice that the secret is already out and start chasing that instead. Base Onety-Nine is a diversion, not an entrapment.
The Ladle is, well, a Ladle, it's not a power source, but holding the fuel that is consumed upon conduction. I honestly haven't even thought about whether it is anomalous itself, you could (and probably would) replace it with an other industrial ladle. It's about the one-of-a-kind contents, not the container.
PE has… issues, as an article, and I tend to ignore it for all purposes of matter replications. I stick with House of All and its whacky interactions with other anomalies.
Escaping from the Unsinger without the City of Ruin isn't the problem. Most of Base Onety-Nine staff could probably just walk outside its area of effect if they were so inclined. The hard part is taking anyone who has been to the City of Ruin with you.
I doubt we went around intentionally replication the VMoD, so we probably have the original + a copy at Base Onety-Nine. As these tend to overlap only by a few dozen hours, that makes probably 2½ times the previous consumption. Even if we were vending 24/7, that would hardly be noticeable. The Foundation might have noticed, but won't be too concerned after the rate stabilized.
A semi-deterministic Büchi automaton is just a non-deterministic Büchi automaton that has all its accepting positions in a deterministic sub-set of states. This is very useful for chess, as when it comes to a mate, there are only finite amounts of threat overlaying around the king (obviously, you still have infinite different positions as rooks, bishops and queens could be arbitrarily far away, that's why you track threatened ranks and files instead of piece positions) and therefore has an deterministic end. (Sure, chess is a perfect information game and you could technically use dove-tailing to enumerate every single position on a plain unbounded TM, but where's the fun in that?)
Believe it or not, infinite chess has been studied quite extensively, but regrettably never got much main stream attention (mostly due to being deeply anchored in theory of computation), and since this is an article featuring SCHEMA, I won't shy away from getting very abstract.
When it comes to the transfinite ordinals, it's quite simple: if a player can stall their defeat by an arbitrary amount of turns, that means the game's length will be larger than any natural number, therefore 𝜔. If they can do that twice, it's 2𝜔, etc. up to being able to stall arbitrarily many times which would be 𝜔 times 𝜔 or simply 𝜔². Which is bad news, because whilst The mate-in-n problem of infinite chess is decidable, this is for finite n, so you may never arrive at a conclusion, no matter how much computation you throw at it.
Addendum is basically this plus some stuff about cellular automata encoded in piece movements and a reference to God's number. I was extremly afraid to make up some stuff just to have somebody prove me impossible, even though that won't happen.
I never bothered to get TTS to run on LibreWolf (does it run out-of-the-box on your *nix FF install? It does for FF on Win, but then MS's TTS is incredibly intrusive) and honestly, I can't be bothered because I'm still faster reading than listening to poor man's Vocaloid. Also, apparently there are assholes who keep using stuff like 𝜔 in their articles which is a challenge for reasonable TTS.