Decks with access to haste still work provided combat is the only way to deal a large amount of damage to the opponent. So if an unbounded computation is started during combat, we can't actually use it to deal unbounded damage to the opponent. (this also means we can't use first strike or double strike; as that would create an additional combat damage step).
Unfortunately, I've noticed another problem with this deck. True Polymorph allows for something with damage marked on it to become Ember fist Zubera and die, starting a computation without clearing our hand and board.
It can be fixed by using Goblin Boom Keg (since it can't become a creature); but then we need an extra card for the final damage output.
Gotcha. And hmm, in that case something with haste could take care of the final damage, but that does look like it would take a 17th card.
Edit: This is probably an issue with using Ember-Fist Zubera for iterated computations in general, since we can change its toughness with Artificial Evolution.
Could perhaps Solitude be replaced by something with lifelink and haste?
The problem is that we need a way to turn the output of the previous computation into the input of the next one. Solitude is our current way of doing that, by having us gain life based on the power of a creature. I couldn't find any cards with both lifelink and haste that would also let us do that with this setup.
For 17 cards, one thing that could help is Tempt with Vengeance. I assume the opponent would turn it down each time, I don't think the choice causes any issues.
Play Storm King's Thunder for however much we can get away with, flashback Echo of Eons that many times. Start a copy chain, then use it to set up the board for a computation. Verdant Sun's Avatar on the opponent's side can gain back life as they create tokens. Once we finish a computation, get an Avatar and a Necromancer back on our own side, then use Artificial Evolution to kill one or more damaged tokens on our side, triggering Necromancer and causing Avatar to gain life based on the computation's output. At the end, use Eiganjo Uprising to finish.
Yeah, one copy of Artificial evolution is enough between computations, though it requires a little bit more setup. First we set up the UTM as normal and add an output clock that stays constantly at 1 every step (IE 2 necromancers one 'heartbeat' that generates a death trigger every tick, and one that adds on to this output clock every tick) and a third necromancer that convers the output type to the input. These do not interfere with computation, and continue to behave as normal after the halt clock is tripped and computation winds down. At the end we have BB(X) of our output type, and each has a unique amount of damage marked on it from 0 to BB(X)-1
We then select Y such that BB(X-1) < Y < BB(X) as the next input and artificial evolution the token with Y+1 remaining toughness on it to another type, this causes all of them to shrink by 1 due to coat of arms, triggering a chain reaction where the first Y of them die and convert to the input type, ready for another computation, bounded by BB(Y) = O(BB(BB(X))))
Edit: a heartbeat clock like that is technically not allowed in TWM as multiple clocks zeroing simultaneously gives undefined behavior, however with our implementation we are forced into only having one way to resolve it: do both, in either order it doesn't matter.
It looks good! I guess one casting of Storm King's Thunder followed by Verdant Sun's Avatar is enough - VSA can substitute for Artificial Evolution for restarting the computation it looks like, but we can certainly cast Artificial Evolution as well after a copy of Echo of Eons is resolved. So we get one computation per Echo of Eons. We can cast Storm King's Thunder with X = 11,733, then cast Echo of Eons, and after resolving an Echo of Eons we can cast Eiganjo Uprising with X=1, gaining mana back again. We maybe need a couple of resolutions of Echo of Eons to build a big enough computation, but I imagine we get BB_1(X) with X not far from 11,733.
Hmm, I wonder if we can use any of this in the 60 card deck. It would be nice not to have March of the Machines so that Goblin Boom Keg can't be a creature, but we can't use True Polymorph, as we could use it to make more copies of our stage creature. (It could maybe work with Mimic Vat, but alas that is gone.)
It looks good! I guess one casting of Storm King's Thunder followed by Verdant Sun's Avatar is enough - VSA can substitute for Artificial Evolution for restarting the computation it looks like, but we can certainly cast Artificial Evolution as well after a copy of Echo of Eons is resolved. So we get one computation per Echo of Eons. We can cast Storm King's Thunder with X = 11,733, then cast Echo of Eons, and after resolving an Echo of Eons we can cast Eiganjo Uprising with X=1, gaining mana back again. We maybe need a couple of resolutions of Echo of Eons to build a big enough computation, but I imagine we get BB_{w+1}(X) with X not far from 11,733.
Hmm, I wonder if we can use any of this in the 60 card deck. It would be nice not to have March of the Machines so that Goblin Boom Keg can't be a creature, but we can't use True Polymorph, as we could use it to make more copies of our stage creature. (It could maybe work with Mimic Vat, but alas that is gone.)
Problem is that between the Echo of Eons castings, we have to play at instant speed, so we can't replay Eiganjo Uprising or Verdant Sun's Avatar during that time. I think we'll end up needing to play Kaervek's Spite to refill on life, with whatever preparations that will require.
Oh, hmm... I think there may be a problem with setting things up then. We need to create a bunch of different Xathrid Necromancers on the opponent's side with various creature types. So it seems like we need a bunch of True Polymorphs and Artificial Evolutions. But, we only have one copy of Storm King's Thunder, and it will trigger on the next instant cast, so we can't get multiple copies of both TP and AE. We can't use multiple rounds of Echo of Eons, since Kaervek's Spite is not optional, with Echo of Eons requiring us to draw seven cards. Nor can we cast Storm King's Thunder in response to casting Kaervek's Spite, since Kaervek's Spite discards our hand as part of its cost...
Edit: Oh, I guess Kaervek's Spite is optional, since without it our hand gets shuffled into the library anyway. Scratch that.
Edit: I see this start gets the opponent 55 creatures, and we have to use those for everything the opponent needs for the first computation. So it needs to be checked what we can do with that.
It's also possible to use Bishop of Wings instead of Xathrid Necromancer, which lets the opponent survive the computation without an additional lifegain card. The Bishop can also gain us a bit of life with each computation start, though I don't think it can convert the output of the computation into life without additional help. Still, combining that with the 1 Artificial Evolution computation reset it looks like we are close to 15 cards. But I think we still need the 16th card to get started.
Finished getting caught up here, and wow these decks are small! After seeing the spoiler I did not expect Saw in Half to be such a good card for this challenge.
One thing to note about the BB decks is that if we can prove flooding waterfall to be TC, Dralnu's crusade can be entirely cut, saving a slot in all of those decks.
However flooding waterfall is very difficult to work with, so a proof seems unlikely.
It's also possible to use Bishop of Wings instead of Xathrid Necromancer, which lets the opponent survive the computation without an additional lifegain card. The Bishop can also gain us a bit of life with each computation start, though I don't think it can convert the output of the computation into life without additional help. Still, combining that with the 1 Artificial Evolution computation reset it looks like we are close to 15 cards. But I think we still need the 16th card to get started.
Are there some limitations with Bishop of Wings? I remember seeing some discussion about it vs Xathrid Necromancer in the 60-card thread, I forget what were the specifics of reasons for using one or the other.
I think we could get a decent start off it, but yeah, not being able to scale with the size of creatures limits how many roles it can fill.
Edit: If Bishop of Wings can keep the opponent alive, I think we can switch to Mana Echoes as our output card for a stronger start.
For the 60-card deck, the life gain of Bishop of Wings is a negative, since it keeps us from using life later on. But the distinct creature types would be an advantage, we don't have the problem with Xathrid Necromancer pumping up one of the associated creature types, so we wouldn't need a third creature type per waterclock.
Can any savings be made for a single Busy Beaver deck?
Here are the effects that a BB deck needs:
- The 6 card computational core [Coat of Arms, Xathrid Necromancer / Bishop of Wings / Rutlang Reanimator, Artificial Evolution, Arcbond, Comeuppance, Dralnu's Crusade]
- Starting computations [by dealing damage while also limiting our ability to interact until the stack resolves]
- Keeping the opponent alive [can be handled by bishop]
- Dealing the final damage blow
- Giving the opponent copies of things
- Starting card draw
- Starting mana
- Sufficient scaling to start a reasonably sized computation [probably by copying AE and other cards a large number of times]
- Profiting off of previous computations and setting up the next (for iterated BB decks)
Some of these effects involve multiple cards working together; and several cards also fulfil multiple roles.
For the 60-card deck, the life gain of Bishop of Wings is a negative, since it keeps us from using life later on. But the distinct creature types would be an advantage, we don't have the problem with Xathrid Necromancer pumping up one of the associated creature types, so we wouldn't need a third creature type per waterclock.
Can any savings be made for a single Busy Beaver deck?
That makes sense.
I took some tries at a smaller single Busy Beaver deck, but it seems difficult to make it work. The benefit to single Busy Beaver is being able to use Kaervek's Spite + Ember-Fist Zubera or Soulblast + Kaervek's Purge (or Traitor's Roar, or Phthisis) to condense the output, but those don't work with recursion, which makes it a lot harder to reach big numbers with a deck that small.
Edit: I think this one still doesn't scale fast enough to hit the target of 10^10,000, but it should do better than the previous one, and I don't think it has a window to set up an infinite.
I think the current best we have for 15 cards is four Saw stages + Finest Hour, I'm not sure what the threshold would be for a Busy Beaver deck to beat that.
Incidentally, I think I have some good cheap computations.
For exponentiation, we can use a heartbeat clock, a part-flooding clock, and a decrement clock. For the heartbeat clock, we will have one creature that keeps dying and creating another creature, and also creating another creature for the decrement clock, so the decrement clock will normally stay the same. The part-flooding clock will still have a "special" creature type, and we will have one creature of that special type that will create another one when it dies. But we will have two Bishop of Wingss for the non-special creature type, so that when they die, twice as many of the non-special ones get created. The special creature type dying will also create a creature for the heartbeat clock, delaying the heartbeat one turn, causing the decrement clock to go down 1. So each time the part-flooding group dies, the number of non-special creatures double, and the decrement clock goes down by 1. So when the decrement clock goes from N down to 0, the nonspecial creatures will have doubled N times, so this gives us exponentiation. (So the decrement clock is the input clock and the clock causing the computation to halt, and the part-flooding clock is the output clock.)
I think we just need one creature type for the decrement clock, which can be Samurai if we use Eiganjo Uprising for the opponent's creatures. For the heartbeat clock, I think we can also use one creature type, and have maybe 4 copies of Bishop of Wings, which, when a creature of the heartbeat type dies, will create a creature of the heartbeat type, a creature of the decrement type, and two creatures of the same type as Bishop of Wings, to keep those creatures from dying. (So I guess that type can be the output clock, since we will have the most of those.) If we make Angel the heartbeat creature type, we will need 4 copies of AE to hack the Bishops. For the part-flooding type, we will need two creature types, one copy of Dralnu's Crusade to make the special creature type to also be the non-special creature type, and 4 copies of Bishop of Wings, 1 to create a special creature when the special creature dies, 1 to create a heartbeat creature when the special creature dies, and 2 to create 2 non-special creatures whenever a non-special creature dies. So that looks like 10 copies of AE needed, but we can save 2 AE's by using Spirit for one of the creature types.
So, we will need 1 Dralnu's Crusade, 8 Bishop of Wings, and 12 Artifical Evolutions, not counting what we need for input. For input, we need 1 each of heartbeat/special/non-special, then the rest go into the decrement creature type. I guess we will need another 3 AEs, for 15 total.
For tetration, we can turn the decrement clock into a second part-flooding clock, and make another decrement clock and heartbeat clock. So the original heartbeat clock and part-flooding clock work exactly the same way, except the heartbeat clock now creates one of the second part-flooding clock rather than the new decrement clock. So the second part-flooding clock decrements once each time the first part-flooding clock hits 0, so if the second part-flooding clock starts at N, we will double the first part-flooding clock N times before the second hits 0. Then, we will create a special creature for each special creature that dies (so we only need one Bishop of Wings here rather than two), which will be more than 2^N, since we add a special creature to the second pf clock each time the first pf clock decrements without hitting 0. We also have a second heartbeat clock that will maintain the new decrement clock, and the second pf clock will increment the second heartbeat clock when it dies, causing the decrement clock to decrement. So the decrement clock will decrement each time the second pf clock hits 0, resulting in an exponentiation. So if the decrement clock starts at N, we will exponentiate the second pf clock N times, resulting in more than 2^^N output.
Counting things up, the first heartbeat clock requires 4 copies of Bishop of Wings, the first part-flooding clock requires 4 copies of Bishops and 1 Dralnu's Crusade, the second heartbeat clock requires 2 copies of Bishops, the second part-flooding clock requires 3 copies of Bishops and 1 Dralnu's Crusade, and the decrement clock doesn't require anything. So it looks like we need 2 Dralnu's Crusades and 13 Bishop of Wings, along with one creature of each of the 6 creature types we need other than the decrement creature type. We will need 30 AE's if I've counted correctly.
More generally, it looks like each additional Knuth arrow will use up one more Dralnu's Crusades, 5 more Bishop of Wings, 3 more creature types used, and 3 more creatures of types besides the decrement type, along with 15 more copies of AE.
I'm not quite sure how to make things work with Xathrid Necromancers in the most efficient way; it looks like the numbers are going to be considerably higher. Maybe I'll look at it later.
So, looking at the 15-card deck: If we just cast Benevolent Offering and Storm King's Thunder after each of the first 6 copies of Timetwister, then after the sixth Benevolent Offering, each player will have more than 2.93 * 10^49 creature tokens, and we will have more than 5.86 * 10^49 life. Then, after the seventh Timewtister, we will be able to cast all our remaining copies, and use two SKTs to get enough copies of True Polymorph and Artificial Evolution.
2.93 * 10^49 probably isn't enough to implement that big of a Turing Machine; perhaps we can implement all 16-state Turing Machines, which is enough for Graham's Number, but not enough for 4 stages and change. So we probably want to look at what specific Waterfall machines we can implement.
The above constructions are pretty economical, but only go up to 87 Knuth arrows, no match for Ackermann function or above. So we have to figure out how to implement at least stages in the Waterfall model, to verify that this beats the four stage deck.
Edit: OH wait, I don't think we have quite enough cards at the end. We need to draw our 4 permanent cards Coat of Arms, Bishop of Wings, Dralnu's Crusade, and Ember-Fist Zubera. We also need to draw Kaervek's Spite to start the computation. But, that doesn't leave us enough cards to draw all 3 of True Polymorph, Artificial Evolution, and Storm King's Thunder. We need to cast True Polymorph and Artificial Evolution after we have those last permanents on the battlefield, and we need to have many copies of them. So I'm not seeing how to make it work out.
Edit: I guess we can cast Bishop of Wings from our opening hand. But that costs two additional mana, allowing us to only cast Storm King's Thunder for 4. That reduces the number of opponent's creatures from 2.93 * 10^49 to about 607 billion.
Hmm, that seems like it works, but yeah the scaling isn't quite quick enough
black lotus, GGG channel 19G, lattice 14, bishop 12, Offering 16, thunder x=6 7, twister 4, get 6 copies (7 total)
resolve first twister, Offering 14, thunder x=7 4,
resolve second twister, Offering 20, AE bishop of wings angel->spirit 19, resolve other offerings, 19+38+44+50+56+62+68+74=411 Thunder X>400
resolve third twister, Offering +80, polymorph spirit to bishop, AE spiritbishop to also gain 4 life per spirit. go to approximately 500000 life, thunder for half a million
Resolve 4th twister, do the polymorph trick again, get to about 10^10 life
5th twister 10^21
6th 10^43 now we need to stop and setup for computations, cast arcbond and comeuppance now, then a big Thunder as we need to copy both polymorph and AE
7th stack is finally clear, cast computation cards:
1 coat of arms
2 crusade
3 zubera
4 cast polymorph to give opponent lots of bishops and some crusades, give us some zuberas
5 thunder for the rest
6 AE to set up BB
7 Spite to start computation
10^43 is still probably enough to do some computation, but IDK if it is enough for a proper UTM...
Edit: oh deedlit has some better estimations, 3*10^49 is better but still seems small.
Hmm, why not use Echo of eons for the first twister so we can copy the flashback more?
black lotus, GGG channel 19G, lattice 14, bishop 12, AE angel->spirit 11, Offering 27, Echo 21
polymorph 15, ae 14, zubera 20 (is a spirit), Offering 60, Thunder X=50 7, Echo 51 times 4
resolve first echo, cast Offering 46, thunder X=39 4
resolve second echo, cast offering, polymorph trick, resolve other copies, thunder for about 7000?
... each echo is better than O(3x^2)
resolve fiftieth echo 10^(2^50)
Echo allows for creating multiple computations, as we can play out the permanents and then flashback echo; but that's incompatible with Ember-fist Zubera and True Polymorph.
Edit: I guess we can cast Bishop of Wings from our opening hand. But that costs two additional mana, allowing us to only cast Storm King's Thunder for 4. That reduces the number of opponent's creatures from 2.93 * 10^49 to about 607 billion.
It doesn't actually cost mana with Benevolent Offering involved. Lotus/Channel/Lattice puts us at 14, then Bishop/Offering puts us at 16. That's enough for X=6.
Echo allows for creating multiple computations, as we can play out the permanents and then flashback echo; but that's incompatible with Ember-fist Zubera and True Polymorph.
Oh, hmm yeah that is a problem, a damaged Zubera can stick around to die to AE and trigger a computation while we have Bishops.
Hmm, we are pretty close to being able to do multiple computations with that setup already, as we only need a zubera trigger to convert output to input and another (small one) to retrigger computation. Though we also need to cast arcbond again as that fell off when it halted. (and ensuring we don't fizzle the triggers...)
Oh, hmm yeah that is a problem, a damaged Zubera can stick around to die to AE and trigger a computation while we have Bishops.
Hmm, we are pretty close to being able to do multiple computations with that setup already, as we only need a zubera trigger to convert output to input and another (small one) to retrigger computation. Though we also need to cast arcbond again as that fell off when it halted. (and ensuring we don't fizzle the triggers...)
Hmm. If we want to keep Arcbond between computations, could protection work? We could put it on several creatures of a particular (non-red) color and also give them protection from that color, then point a Zubera trigger at each of them, so they avoid taking damage during each other's computations until the Zubera trigger hits them to start a new computation.
If there was a good way to incorporate that, could those computations be useful?
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Edit: This is probably an issue with using Ember-Fist Zubera for iterated computations in general, since we can change its toughness with Artificial Evolution.
For 17 cards, one thing that could help is Tempt with Vengeance. I assume the opponent would turn it down each time, I don't think the choice causes any issues.
Black Lotus, Channel, Mycosynth Lattice, Tempt with Vengeance for X=9, Benevolent Offering for 24 mana. Then we can play Storm King's Thunder for X=11 before playing Echo of Eons the first time, and build to higher numbers from there.
2 Channel
3 Mycosynth Lattice
4 Storm King's Thunder
5 Verdant Sun's Avatar
6 Echo of Eons
7 Eiganjo Uprising
8 True Polymorph
10 Kaervek's Spite
11 Coat of Arms
12 Xathrid Necromancer
13 Artificial Evolution
14 Arcbond
15 Comeuppance
16 Dralnu's Crusade
Play Black Lotus, Channel, Mycosynth Lattice. 14 mana.
Verdant Sun's Avatar, Coat of Arms, Eiganjo Uprising for X=5. We create 5 Samurai, opponent creates 4, Coat of Arms gives them all +8/+8, Avatar gains 10 life for each of ours. 50 mana.
Play Echo of Eons (44 mana). Play True Polymorph, turning a Samurai into a second Avatar (38 mana). Play Xathrid Necromancer, while it's on the stack, play Artificial Evolution to turn it into a 10/10 Samurai (54 mana). Play Eiganjo Uprising for X=52, making a total of 103 Samurai, each Samurai is 113/113 and gains us 226 life (11,752 mana). Play Goblin Boom Keg and Dralnu's Crusade (11,745 mana).
Play Storm King's Thunder for however much we can get away with, flashback Echo of Eons that many times. Start a copy chain, then use it to set up the board for a computation. Verdant Sun's Avatar on the opponent's side can gain back life as they create tokens. Once we finish a computation, get an Avatar and a Necromancer back on our own side, then use Artificial Evolution to kill one or more damaged tokens on our side, triggering Necromancer and causing Avatar to gain life based on the computation's output. At the end, use Eiganjo Uprising to finish.
Does that check out?
We then select Y such that BB(X-1) < Y < BB(X) as the next input and artificial evolution the token with Y+1 remaining toughness on it to another type, this causes all of them to shrink by 1 due to coat of arms, triggering a chain reaction where the first Y of them die and convert to the input type, ready for another computation, bounded by BB(Y) = O(BB(BB(X))))
Precursor golem is safe as it doesn't change ownership of the spell (Got blown out by Slice in twain in limited...)
Edit: a heartbeat clock like that is technically not allowed in TWM as multiple clocks zeroing simultaneously gives undefined behavior, however with our implementation we are forced into only having one way to resolve it: do both, in either order it doesn't matter.
Hmm, I wonder if we can use any of this in the 60 card deck. It would be nice not to have March of the Machines so that Goblin Boom Keg can't be a creature, but we can't use True Polymorph, as we could use it to make more copies of our stage creature. (It could maybe work with Mimic Vat, but alas that is gone.)
Problem is that between the Echo of Eons castings, we have to play at instant speed, so we can't replay Eiganjo Uprising or Verdant Sun's Avatar during that time. I think we'll end up needing to play Kaervek's Spite to refill on life, with whatever preparations that will require.
Edit: Oh, I guess Kaervek's Spite is optional, since without it our hand gets shuffled into the library anyway. Scratch that.
Edit: I see this start gets the opponent 55 creatures, and we have to use those for everything the opponent needs for the first computation. So it needs to be checked what we can do with that.
One thing to note about the BB decks is that if we can prove flooding waterfall to be TC, Dralnu's crusade can be entirely cut, saving a slot in all of those decks.
However flooding waterfall is very difficult to work with, so a proof seems unlikely.
I think we could get a decent start off it, but yeah, not being able to scale with the size of creatures limits how many roles it can fill.
Edit: If Bishop of Wings can keep the opponent alive, I think we can switch to Mana Echoes as our output card for a stronger start.
Can any savings be made for a single Busy Beaver deck?
- The 6 card computational core [Coat of Arms, Xathrid Necromancer / Bishop of Wings / Rutlang Reanimator, Artificial Evolution, Arcbond, Comeuppance, Dralnu's Crusade]
- Starting computations [by dealing damage while also limiting our ability to interact until the stack resolves]
- Keeping the opponent alive [can be handled by bishop]
- Dealing the final damage blow
- Giving the opponent copies of things
- Starting card draw
- Starting mana
- Sufficient scaling to start a reasonably sized computation [probably by copying AE and other cards a large number of times]
- Profiting off of previous computations and setting up the next (for iterated BB decks)
Some of these effects involve multiple cards working together; and several cards also fulfil multiple roles.
I took some tries at a smaller single Busy Beaver deck, but it seems difficult to make it work. The benefit to single Busy Beaver is being able to use Kaervek's Spite + Ember-Fist Zubera or Soulblast + Kaervek's Purge (or Traitor's Roar, or Phthisis) to condense the output, but those don't work with recursion, which makes it a lot harder to reach big numbers with a deck that small.
Edit: Best I've come up with for 15-card Busy Beaver is Chromatic Orrery, Thousand-Year Storm, Replication Technique, Scrambleverse, and Wizard's Spellbook, but it doesn't scale fast enough to hit the numbers we want.
Edit: I think this one still doesn't scale fast enough to hit the target of 10^10,000, but it should do better than the previous one, and I don't think it has a window to set up an infinite.
2 Channel
3 Mycosynth Lattice
4 Benevolent Offering
5 Storm King's Thunder
6 Timetwister
7 True Polymorph
9 Kaervek's Spite
10 Coat of Arms
11 Bishop of Wings
12 Artificial Evolution
13 Arcbond
14 Comeuppance
15 Dralnu's Crusade
By starting with Black Lotus, Channel, Mycosynth Lattice, Benevolent Offering, Storm King's Thunder, and Timetwister (and Bishop of Wings), we we should be able to copy Timetwister 6 times, with no way to start a computation before the last one resolves.
I think the current best we have for 15 cards is four Saw stages + Finest Hour, I'm not sure what the threshold would be for a Busy Beaver deck to beat that.
For exponentiation, we can use a heartbeat clock, a part-flooding clock, and a decrement clock. For the heartbeat clock, we will have one creature that keeps dying and creating another creature, and also creating another creature for the decrement clock, so the decrement clock will normally stay the same. The part-flooding clock will still have a "special" creature type, and we will have one creature of that special type that will create another one when it dies. But we will have two Bishop of Wingss for the non-special creature type, so that when they die, twice as many of the non-special ones get created. The special creature type dying will also create a creature for the heartbeat clock, delaying the heartbeat one turn, causing the decrement clock to go down 1. So each time the part-flooding group dies, the number of non-special creatures double, and the decrement clock goes down by 1. So when the decrement clock goes from N down to 0, the nonspecial creatures will have doubled N times, so this gives us exponentiation. (So the decrement clock is the input clock and the clock causing the computation to halt, and the part-flooding clock is the output clock.)
I think we just need one creature type for the decrement clock, which can be Samurai if we use Eiganjo Uprising for the opponent's creatures. For the heartbeat clock, I think we can also use one creature type, and have maybe 4 copies of Bishop of Wings, which, when a creature of the heartbeat type dies, will create a creature of the heartbeat type, a creature of the decrement type, and two creatures of the same type as Bishop of Wings, to keep those creatures from dying. (So I guess that type can be the output clock, since we will have the most of those.) If we make Angel the heartbeat creature type, we will need 4 copies of AE to hack the Bishops. For the part-flooding type, we will need two creature types, one copy of Dralnu's Crusade to make the special creature type to also be the non-special creature type, and 4 copies of Bishop of Wings, 1 to create a special creature when the special creature dies, 1 to create a heartbeat creature when the special creature dies, and 2 to create 2 non-special creatures whenever a non-special creature dies. So that looks like 10 copies of AE needed, but we can save 2 AE's by using Spirit for one of the creature types.
So, we will need 1 Dralnu's Crusade, 8 Bishop of Wings, and 12 Artifical Evolutions, not counting what we need for input. For input, we need 1 each of heartbeat/special/non-special, then the rest go into the decrement creature type. I guess we will need another 3 AEs, for 15 total.
For tetration, we can turn the decrement clock into a second part-flooding clock, and make another decrement clock and heartbeat clock. So the original heartbeat clock and part-flooding clock work exactly the same way, except the heartbeat clock now creates one of the second part-flooding clock rather than the new decrement clock. So the second part-flooding clock decrements once each time the first part-flooding clock hits 0, so if the second part-flooding clock starts at N, we will double the first part-flooding clock N times before the second hits 0. Then, we will create a special creature for each special creature that dies (so we only need one Bishop of Wings here rather than two), which will be more than 2^N, since we add a special creature to the second pf clock each time the first pf clock decrements without hitting 0. We also have a second heartbeat clock that will maintain the new decrement clock, and the second pf clock will increment the second heartbeat clock when it dies, causing the decrement clock to decrement. So the decrement clock will decrement each time the second pf clock hits 0, resulting in an exponentiation. So if the decrement clock starts at N, we will exponentiate the second pf clock N times, resulting in more than 2^^N output.
Counting things up, the first heartbeat clock requires 4 copies of Bishop of Wings, the first part-flooding clock requires 4 copies of Bishops and 1 Dralnu's Crusade, the second heartbeat clock requires 2 copies of Bishops, the second part-flooding clock requires 3 copies of Bishops and 1 Dralnu's Crusade, and the decrement clock doesn't require anything. So it looks like we need 2 Dralnu's Crusades and 13 Bishop of Wings, along with one creature of each of the 6 creature types we need other than the decrement creature type. We will need 30 AE's if I've counted correctly.
More generally, it looks like each additional Knuth arrow will use up one more Dralnu's Crusades, 5 more Bishop of Wings, 3 more creature types used, and 3 more creatures of types besides the decrement type, along with 15 more copies of AE.
I'm not quite sure how to make things work with Xathrid Necromancers in the most efficient way; it looks like the numbers are going to be considerably higher. Maybe I'll look at it later.
2.93 * 10^49 probably isn't enough to implement that big of a Turing Machine; perhaps we can implement all 16-state Turing Machines, which is enough for Graham's Number, but not enough for 4 stages and change. So we probably want to look at what specific Waterfall machines we can implement.
The above constructions are pretty economical, but only go up to 87 Knuth arrows, no match for Ackermann function or above. So we have to figure out how to implement at least stages in the Waterfall model, to verify that this beats the four stage deck.
Edit: OH wait, I don't think we have quite enough cards at the end. We need to draw our 4 permanent cards Coat of Arms, Bishop of Wings, Dralnu's Crusade, and Ember-Fist Zubera. We also need to draw Kaervek's Spite to start the computation. But, that doesn't leave us enough cards to draw all 3 of True Polymorph, Artificial Evolution, and Storm King's Thunder. We need to cast True Polymorph and Artificial Evolution after we have those last permanents on the battlefield, and we need to have many copies of them. So I'm not seeing how to make it work out.
Edit: I guess we can cast Bishop of Wings from our opening hand. But that costs two additional mana, allowing us to only cast Storm King's Thunder for 4. That reduces the number of opponent's creatures from 2.93 * 10^49 to about 607 billion.
black lotus, GGG channel 19G, lattice 14, bishop 12, Offering 16, thunder x=6 7, twister 4, get 6 copies (7 total)
resolve first twister, Offering 14, thunder x=7 4,
resolve second twister, Offering 20, AE bishop of wings angel->spirit 19, resolve other offerings, 19+38+44+50+56+62+68+74=411 Thunder X>400
resolve third twister, Offering +80, polymorph spirit to bishop, AE spiritbishop to also gain 4 life per spirit. go to approximately 500000 life, thunder for half a million
Resolve 4th twister, do the polymorph trick again, get to about 10^10 life
5th twister 10^21
6th 10^43 now we need to stop and setup for computations, cast arcbond and comeuppance now, then a big Thunder as we need to copy both polymorph and AE
7th stack is finally clear, cast computation cards:
1 coat of arms
2 crusade
3 zubera
4 cast polymorph to give opponent lots of bishops and some crusades, give us some zuberas
5 thunder for the rest
6 AE to set up BB
7 Spite to start computation
10^43 is still probably enough to do some computation, but IDK if it is enough for a proper UTM...
Edit: oh deedlit has some better estimations, 3*10^49 is better but still seems small.
Hmm, why not use Echo of eons for the first twister so we can copy the flashback more?
black lotus, GGG channel 19G, lattice 14, bishop 12, AE angel->spirit 11, Offering 27, Echo 21
polymorph 15, ae 14, zubera 20 (is a spirit), Offering 60, Thunder X=50 7, Echo 51 times 4
resolve first echo, cast Offering 46, thunder X=39 4
resolve second echo, cast offering, polymorph trick, resolve other copies, thunder for about 7000?
... each echo is better than O(3x^2)
resolve fiftieth echo 10^(2^50)
seems like enough for BB computation
Yeah, I avoided Echo and Emergency Powers for that reason.
Hmm, we are pretty close to being able to do multiple computations with that setup already, as we only need a zubera trigger to convert output to input and another (small one) to retrigger computation. Though we also need to cast arcbond again as that fell off when it halted. (and ensuring we don't fizzle the triggers...)
If there was a good way to incorporate that, could those computations be useful?