Edit: So, it looks like the progress for this has stalled a bit (in part because we've been working on the Standard deck). Of course, the priority is to get going on the writeups for both decks, but in case someone wants something to chew on in the meantime, here are several things we could talk about:
1. We could try to get the gigastage (Tier 4 stage) deck to actually work. Obviously, this is much less damage than a deck that can compute the Busy Beaver function, but this pretty much belongs in a different category from those decks - perhaps a category for decks with a finite game tree, or at least no infinite lines of play.
Hyp Cos has looked at these decks and mentioned some problems, so we have something to build off of.
2. We could shoot for the moon, and try to get higher order Busy Beaver functions. Hyp Cos also linked a relevant paper:
The idea is that doing roughly the same thing as our Busy Beaver decks, except over a higher order arithmetic formula rather than a recursive formula, will allow us to compute higher order Busy Beaver functions. However, the basic gist of the implementation is that the two players will represent different quantifiers in the arithmetic formula. So, the power of the construction seems to rely on the opposing nature of the two players. Therefore, for the more powerful formulas, I imagine what would happen when we consider what happens with opponent cooperation, is that we would go infinite.
So, for this challenge I believe we would need to amend the no-infinites rule: instead of considering opponent cooperation, a deck is only invalidated if we can deal arbitrarily high amounts of damage regardless of the opponent's actions.
3. There are of course many other formats that we haven't tried yet: Modern, Pioneer, Commander, Historic, Pauper, etc.
Of course we should keep plugging away at the writeups. (I haven't worked on them in a while due to irl busyness.)
Yeah I put some work into the standard writeup, will add more over the next week.
Making the gigastage deck work would be nice, but it is SO outclassed by BB decks that it is even more niche. Though the Chalice of the Void tricks were pretty impressive if we could ever get them to work properly.
Allowing the opponent to cooperate with us to get more damage seems wrong, even if we can convert their setup to work in one turn.
My thought for a higher order BB function was to use Shahrazad, and somehow use going infinite on later subgame turns as an oracle for the TM in the main game. But there doesn't seem to be a good way to cast Shahrazad enough during computation, nor a good way to communicate the answer up to the main game.
Other formats have the difficulty of going off turn 1, also their banlist is much more fluid. Tibalt's trickery barely missed the banhammer in standard (and I think is still banned in MTGA best of 1?).
Legacy: just a slightly worse version of the vintage deck, we have only had like 2 banned cards most iterations. (currently just Flash?)
Modern: Might be the home for the gigastage?
Pioneer: Not very familiar with this format, at a wild uninformed guess: something like turn 2 second order stage?
Commander: This is mostly legacy with bigger decks, commander color identity restrictions basically force us to play an uninteresting 5 color commander.
Historic: No clue, maybe like 4-7 stages?
Pauper: Would be slow and lose basically every engine we've ever used, probably at least one stage, but I'm not sure at all.
Oh, for the higher order Busy Beaver function we would still assume the opponent does their best to reduce our final score, so that part is the same as it is now. The difference would be that we don't invalidate the deck if we can do arbitrarily high damage, but only with opponent cooperation. I think this keeps the spirit of the challenge - of course it would make some things easier, but I think the challenge is interesting enough nevertheless.
A modern gigastage would be amazing! I don't know if we could get that far, but even getting to a hyperstage or better would be nice.
Most of the gigastage cards are legal in modern, or have close equivalents. In particular the spellweaver cards are both legal, along with worldfire and friends.
The whole thing probably doesn't port over cleanly, but there's gotta be at least an order 3 stage possible in modern.
A modern BB computation seems to be more than a few printings away, Artificial Evolution and Dralnu's Crusade in particular seem unlikely to be reprinted. Now Dralnu's Crusade might not be strictly necessary as we can implement flooding waterfall with it, but that seems unlikely to be TC.