Yeah ok, I'll keep chipping away at the standard article then.
As for the actual number of hacked rotlungs and 2/2 tokens we need, it looks like X^^^X (of each) will be enough, as the nested conversions don't look as bad as I feared. I'm not sure on the absolute minimum size.
A very basic idea for flooding clocks is to have them chain into each other with the last one enabling halting when it finally trips. Even a very simple structure provides a lot of power.
For example say we have a simple heartbeat clock that triggers every other tick to prevent halting, and adds 1 to all of the flooding clocks
When flooding clock 1 trips it can increase that and allow the program to halt.
Flooding clock 2 doubles itself and adds to the first clock
FC3 doubles itself and adds to the first two FCs
FC4 doubles itself and adds to the first three FCs
....
FC250 or so doubles itself and adds itself to the other FCs (or wherever we run out of creature types)
Then we initialize the heartbeat at 1, halting clock at 2, and the flooding clocks at X - clock number. (maybe more optimal with everything but clock 1 at 1?)
This halts because clock 1 will eventually be the smallest clock, stopping the heatbeat and triggering the halting condition. The higher clocks eventually get too big to trigger enough.
I'm not sure exactly how long this takes, but it seems to get pretty big pretty fast.
Hmm, I think I'm getting the idea wrong. With the heartbeat clock, a flooding clock that represents X tokens will take 2X ticks to die, and gain X tokens, so it will add a multiple of 2X to various waterclocks. If we add 2X to both FC1 and FC2 when FC2 hits zero, then FC1 will always be more than FC2, so it won't halt. We can make it halt by doing something like add 2X to FC1 and 4X to FC2. Then, if FC1 starts at N and FC2 starts at 1, it looks like the program will halt after between 2N and 4N steps.
If we add FC3, and FC3 adds 2X to FC1 and FC2, and 4X to FC3 when it hits 0, then it takes 952 steps starting at 100,1,1 for example. So it looks like this just multiplies by a small constant.
Edit: Oh, I guess the trick is to add 4X to FC1 instead of 2X when FC3 hits 0. That makes it much more powerful.
Does that implement an arrow system?
Edit: Hmm no, it looks like that just goes infinite actually.
Edit: Well, it might not actually go infinite, but the behavior seems rather chaotic. We can make things more orderly, by having each flooding clock imcrease every flooding clock by 4X, except for the one before it, which it can increase by 2X. That has a more orderly recursion, but it doesn't quite work right; the largest index FC's will trigger first, and then work it's way down, which is the opposite of what we want. What we would like is to have the lowest index trigger first, and have each FC trigger lead to many triggers of the one before it. I don't know how to do that.
Perhaps we could do something similar to what Iijil did, using one or two clocks per layer to signify state, except we do it with flooding clocks. We can sorta deal with the multiplicative nature of the flooding clocks by considering powers of 2 perhaps, where doubling becomes basically incrementing the exponent. If so, we could cut the number of creature types needed in half.
Yeah it doesn't behave as nicely as I thought (complicated FWCs? who'd've guessed?) The idea was for each additional clock to be worth an arrow, but that doesn't seem to be how it works. It is more like exponentiation than the ackermann function.
I guess another thing we can do is to send an unhacked Rotlung Reanimator to the opponent. The advantage is that this is relatively free; in the opening without Master Skald, we can get around 1900 hacked Rotlung Reanimators, and around 2^^^X unhacked Rotlung Reanimtors and other creatures. So, we could have proper waterclocks, flooding waterclocks, "incremented" waterclocks (i.e. creature types that have an extra however many Rotlung Reanimators of that type) and flooding incremented waterclocks. This probably gets really complicated.
Edit: Hmmm, I wonder if Temple Garden and Ambush Commander can be partially replaced by Lair of the Hydra. Lair of the Hydra can come in untapped, but we use the green mana produced to turn it into a creature, so that it can be repeatedly untapped by Mobilize. We can not turn it into a creature, and just use the green mana to get ahead in the lower stage, but then Lair of the Hydra will be on the battlefield as a noncreature. The next time we resolve Worldfire, Lair of the Hydra will go into exile, where it will cost more to retrieve. So maybe that doesn't go infinite? I'm definitely not certain though, I could use some verification.
In the case that it does happen to work, we will still need a source of white mana. We can replace Mana Crypt with Mox Pearl again to get access to white mana. But, this seems to conflict with Avacyn, Guardian Angel. So, we would like to replace Avacyn with some other card that protects us from dying to a computation. Gravebreaker Lamia already protects the opponent from both the computation and Kaervek's Spite, so we only have to protect ourselves. But, Safe Passage doesn't seem to work, as that will prevent the damage we need to have dealt to ourselves in the layers.
That's a lot of life spent though. If we play Radiant Fountain at the beginning, we can cast Prosperity for up to 11, and still have 2 life left over for Show and Tell. That's a lot fewer cards, but hopefully we can get started? If necessary, we can use one of the extra cards for Master Skald.
With the extra space, we can add World at War, taking us up to BB_{w^2+14} (BB_{w^2+13} (X)) I think.
Edit: Correction, we can't play Radiant Fountain at the beginning, because we need to play Temple Garden. So we have 1 card remaining in hand, and can draw up to 8.
Hi, I saw one of your posts about this challenge somewhere and found it fun to try, so I ended up making my own deck to try and beat it. Mine is pretty different from yours, and I don't have the mathematical knowledge to calculate whether mine's better, but I figured I'd post it anyway in case it is or in case some tech in it is useful to you.
I tried to keep the explanation brief (and failed) so I didn't include stuff like optimal ordering that doesn't end up making a noticeable difference in the final number. That's also why I kept the explanations for each step rather barebones; I can explain further if it isn't clear enough. I also included the method I used to estimate how effective it was, but I don't know where your most recent list is to compare with.
Edit: A couple of problems I noticed: Note that the opponent is not a true goldfish, they will act in a way to minimize the amount of damage that you can deal. (But they will also cooperate to go infinite and invalidate your deck) So, they can mulligan down to 0 cards to prevent you from drawing cards from Balance of Power.
Also, it looks like Fathom Mage goes infinite, since it's Evolve ability will allow it to gain counters whenever a creature with higher power or toughness enters the battlefield, so there's no need for Nykthos Paragon.
Concerning the counting, why does Argentum Armor get (+8,38)? It seems like it should be +4 like the previous members of the sequence.
On the whole though, some very nice ideas! Getting a bunch of layers for each additional mana cost is quite powerful, and getting it for both artifacts and creatures is very nice.
Ah, I didn't realize they could mulligan, that makes sense. Balance of Power and Chef's Kiss can be cut for another card draw like Enter the Infinite then, -1 card and -1 layer. Edit: I can swap the Balance of Power to Mindculling and it works. It's a bit tricky to set up with only a draw 2, but can be managed with TYS and some of the auxiliary draw in the deck. Another edit: Actually I can't use Rousing Refrain either. Looks like that part of the combo will take some more work.
You are right that Fathom Mage would go infinite, I can salvage that by switching to Benthic Biomancer, drawing EtI and discarding Show and Tell which we can easily get back before the next one. It only gives 1 draw trigger per counter-granting effect so we swap Nykthos to Archangel of Thune and don't benefit from Doubling Season's counter doubling, but it still ends up similarly good.
Argentum Armor is a typo from copying it. Change that to (+4, 34), Summoning Station to (+4, 38) and it adds up again after that.
How many layers does your current combo have?
Edit the maybe last: I can cut Sleeper's Dart, Rousing Refrain, and Douse for Fire Diamond and Filigree Sages, without even losing a layer since the Sages are Rings-viable. So -Dart, Refrain, Douse, Fathom Mage, Nykthos Paragon, +Fire Diamond, Filigree, Benthic, Archangel actually leaves me at 99 layers and 59 cards. I'm not sure of any 1-card additional layers I can add at the moment, so I'll leave it there for now.
Shortly after that last deck was written up, an expansion came out that had Gideon, Ally of Zendikar, which can be used to replace both Soul of New Phyrexia and Thran Dynamo, leaving us with another card slot. That card slot can be used to add Devout Lightcaster, which can come in at the end of the sequence, adding 9 layers. That yields a final total of more than 2 -> 19 -> 417.
Not too long after that, we had a major breakthrough.
We've defined a "stage combo" to be a combo that takes X of some resource (like X creature tokens or X green mana, etc) and turns it into X layers. So the first stage basically performs the Ackermann function. This goes far beyond what you can do by individually adding layers, if you can execute the combo more than once; for example, if you start by using the stage combo with 10 green mana, you'll wind up with something like 10 -> 10 -> 10 green mana. (This is Conway chained arrow notation; in A -> B -> C, C is the number of layers, and B is the amount of the resource, i.e. how many times you perform the previous layer.) If we use the stage combo again, we'll get something like 10 -> 10 -> (10 -> 10 -> 10) green mana, so the number of layers is itself 10 -> 10 -> 10. Obviously we can never match that with a deck that adds layers individually. So this was a major paradigm shift.
The stage combo is somewhat complicated, and can be implemented in many different ways.
After that, you can go to 1 stage + 1 layer, which takes X of a resource and performs the stage combo X times; 1 stage + 2 layers, which performs the 1 stage + 1 layer combo X times; and so on. Then, the second stage takes X of a resource and turns it into 1 stage + X layers.
You can keep adding more stages like this; I think the furthest we got with individual stages was 26 stages. The next step then is to create a combo that takes X of a resource and creates X stages, which we called a hyperstage combo. We were able to create a deck with 2 hyperstages plus a few stages afterwards. Then, we defined a megastage as a combo that creates X hyperstages, and a gigastage as a deck that creates X megastages. We have a megastage deck that I'm quite confident works, and a few gigastage deck entries that still need verification to see whether everything is kosher or not.
That work was more or less abandoned when jfb came in with the idea of implementing the Busy Beaver function, using a deck that implemented The Waterfall Model, which is a Turing-complete computational model. The Busy Beaver function blows the pants off anything we can do with recursion. (This next part gets pretty mathy, so you can skip if you're not that into math.) To give an idea of how fast it grows, we can represent our numbers using the fast-growing hierarchy, which is indexed by ordinals. The smallest ordinals are polynomials over w (actually the greek letter omega, but I'm too lazy to cut and paste that in). A layer increments the ordinal index by 1, a stage increments the ordinal index by w, and a hyperstage/megastage/gigastage increments the ordinal by w^2, w^3, and w^4, respectively. So the largest numbers we have been able to reach with recursion is about F_{w^4}. BB(85) has been shown to be bigger than F_{epsilon_0}(1908), which is equal to F_{w^w^w^w^w...^w}(1908) with 1908 w's in the exponential tower! So even at relatively low values, the Busy Beaver function beats anything we could conceivably do with just recursion.
Our latest decks have managed to perform some pretty heavy recursion over the Busy Beaver function itself. We can define a fast-growing hierarchy BB_a, except instead of starting with a weak function, we start from the Busy Beaver function. So BB_0 = the Busy Beaver function, BB_1(N) applies the Busy Beaver function N times starting from N, BB_2(N) applies X -> BB_1(X) N times starting from N, and so on. Then we can have Busy Beaver stages and hyperstages as well. Our best is a Busy Beaver hyperstage plus a few layers, with the final damage number being BB_{w^2+14} (BB_{w^2+13} (X)), with X not quite having been worked out yet.
Ah, that's impressive, even the lower ones since they were working with a muuuuch smaller cardpool back then than I am now. Looks like it'll be a lot of reading before I've gotten through them though.
Molten Echoes only makes hasted copies when the nontoken version enters the battlefield, which is only once. Molten Echoes is worded "The token gains haste", not "the token has haste", so copies from Cackling Counterpart won't maintain the haste. I had to look that one up myself when I was building the combo.
Looking at the counting again, it looks like 10 is inadvertently dropped, from 54 to 46. But, the +2's that we are getting from Death's Oasis and Bloodbond March turn into +4's starting from Kamahl until Iname; assuming that's an error, that adds an extra +8. Accounting for both errors results in +2, for 101 layers.
Edit: Going back to the busy beaver deck, I suppose switching out Mox Sapphire for Eye of Ramos will allow us to pay 3 mana instead of 7 for Mirrorworks, so we wind up with 14 cards rather than 9. 14 cards are hopefully enough to get started.
Edit: I think we can get 14 layers with fewer cards, by using Thrummingbird and proliferating Invasion of the Giants rather than Tymarret. Invasion of the Giants only does 2 damage, but we can use Gisela, Blade of Goldnight to double the damage. That's 2 extra cards rather than 3, so we have an extra card we can use to help out with the opening, perhaps Master Skald.
Hello, all! I'm still getting caught up on all of the changes that have been made to the deck, but in the meantime I've been thinking about how we could add a stage to the current deck and I may have found a good line of attack.
We know we can generalize all or most stages into five instructions across two states:
1. Use 1 resource I to push X trigger A.
2. Iff the state is unlocked, pop one Trigger A to push X trigger B and lock the state.
3. Pop 1 trigger B to generate X resource O.
4. Pop 1 trigger B to unlock the state.
5. Iff the state is unlocked, pop 1 trigger A to generate 1 resource I and lock the state.
Traditionally, the state has been "this creature is not in the graveyard" in vintage, but it doesn't have to be. If we can find a new State tracker and free up a resource (white mana through Safe Passage or red mana, maybe by using Sacrifice and Coat of Arms to store our progress), and some linking triggers that don't get wrecked by Temur Sabertooth.
I've done some digging, and I think we can get some mileage out of taplands with a sacrifice ability and an aura which animates them. Because they come in tapped, we have to animate them with our one aura to untap them with Mobilize. If their activation cost requires we spend mana of our input resource, we can make it too expensive to activate them without leaving them a creature and using something like Primal Beyond. And when they activate, the aura falls off and goes to the graveyard, locking the state since we currently have no way to revive an aura from there.
We spend a red mana to grant Axgard Armory (enchanted) haste, untap it, and activate it, each time grabbing Viridian Harvest to gain life. Then, when it comes time to regenerate red mana, we put the Zendikon on Tinder Farm, give it haste, untap it, sac it for an R, and use an Axgard trigger below to get back the Zendikon. We could move the Zendikon after untapping Tinder Farm but before sacrificing it, but without Zendikon bouncing it back to hand we'd have no way to rebuy it later, and the stage would fizzle.
I can only imagine this has an infinite somewhere earlier in the chain, but I can't find it with my still limited understanding and either way this should hopefully be a good starting point for a stage discussion!
Edit: putting a full decklist here sans final layers as a point of reference, since it doesn't look like there's a full list on this page of the thread.
Welcome back Stakfish! Great to have you back, and to have your creativity available for this challenge.
Oh, very interesting! Still thinking about the the new stage, but one thing I noticed: I don't see how Sacrifice can allow us to store the progress of a computation. Previously we have used Brightstone Ritual, because the opponent can output Goblins, or Battle Hymn, which we can resolve after killing off a large number of opponent's creatures by bouncing Coat of Arms, while having a Rotlung Reanimator on our side, so that we get lots of creatures. Sacrifice gets us the mana value of the creature, which can only get really large if we cast a creature with an X in its casting cost, timees the number of Sea Gate Stormcallers which can't be really large until we have taken advantage of the result of an opponent's computation, so that's kind of a Catch-22.
Previously, we used Mana Echoes to get mana commensurable with the output of the computation, but we won't have that with Kaevek's Spite.
Oh, we are also currently using Master Transmuter to convert mana into artifacts like Rings of Brighthearth, so if we require red mana for haste, then we need to turn the result of the computation into red mana. Maybe the best way to salvage your idea then is to find a way to turn the result of a computation into usable resources without using red mana or Master Transmuter.
Hmm, you're right. I misread that Sacrifice to be based on power, not mana value. In that case, we need a more involved regeneration process. Maybe we can find something else that keys off power?
As for giving Master Transmuter haste, it should innately get it from Thopter Engineer, which would give the Transmuter haste but not any of our other creatures.
Edit: How about Vile Redeemer? I know tokens are dying in the computation, but if we could find a creature that repeatedly dies and reanimates whenever some part of the computation happens, we could tie the computation to nontoken creatures dying and build a huge number of spawns.
Edit: How about Solemnity and an inoffensive creature with undying, like Young Wolf. It would constantly die and reanimate, but I guess it would probably break the Turing Machine model...
Ah, I missed what Thopter Engineer was for. That should work, although it removes combat stages from the layer sequence.
Edit: Hmmm... For Vile Redeemer, I don't think we want a creature that triggers in our graveyard, since then it would trigger the same time as our Arcbonds, so we could decide whether or not the creature goes back to the battlefield immediately, or gets buried underneath Arcbond triggers. But, a creature that triggers when it dies could work.
Edit: I think Young Wolf probably doesn't work, since it will come back on our side, and from then on it will be our trigger, which we can choose how we resolve compared to the Arcbonds.
You're right. Here's a simple thought: how about Earthcraft and a basic land like Swamp or Plains? We can just tap the creatures we get back to generate mana.
Oh, maybe! Oh, but Rings of Brighthearth, combined with Master Transmuter, seems to be a problem. We could use a Master Transmuter to bounce Mox Sapphire to our hand, to put into play and get a lot of blue mana at the cost of generic mana; or, we could use Master Transmuter via Earthcraft to get a lot of black mana, if we have a Swamp. So that seems to go infinite.
Edit: I thought about using Mana Echoes on the opponent's side, and then using Drain Power to steal the colorless mana, but then I remembered that Mana Echoes goes infinite with Mirrorworks and Master Transmuter.
On the bright side, this accounts for both blue and black mana, though we may still need Mox Sapphire for the start. That would get us a decklist like this:
Oh, one of the nonos is we can't allow the opponent to get certain colors of mana: If the opponent gets blue and black, the can activate The Scarab God to affect the computation. If they get green mana, they can activate Temur Sabertooth. If they get red and blue, they can activate Master Transmuter...
Wow, this combo is REALLY restrictive. Can they also interfere with just their own basic lands, or do they need access to large quantities?
Also, I'm assuming March of the Machines goes infinite in some way that I'm missing, but on the off chance it doesn't, We could maybe give them the cards we need with Domineering Will, without letting them get Mox Sapphire. That would still leave unresolved the issue of generating black mana for The Scarab God, though.
March of the Machines should be okay, we cut it after it was no longer necessary; we just have to make sure that any further changes we made don't mess with it, e.g. Burnt Offering.
The reason for using Fractured Identity is explained in post #3858; basically, we can use Artificial Evolution to changed the text of Rotlung Reanimators and Dralnu's Crusades that are still on the stack, and then Worldfire can exile Artificial Evolution, allowing us to get it back. Then, if we can get an infinite computation going with using the text-changed permanents, we can end whenever we want with the Artificial Evolution still in our hand.
Iijil's solution was to use The Scarab God to create copies of Rotlung Reanimator, and Fractured Identity to bring it over to the opponent's side. I guess we have to check whether Domineering Will allows the opponent to go infinite.
Edit: Oh, about the basic lands: for the Vintage version, we assume that all the opponent's cards are Wastes, so if they can put those into play they have access to colorless mana.
Edit: Okay, I think Domineering Will and Dual Nature will go infinite, because we can change the text of a Rotlung Reanimator and a Dralnu's Crusade, then resolve those spells, and then Dual Nature can create non-hacked versions of Rotlung Reanimator, and we can make an infinite computation just like in Iijil's example.
But, if we remove Dual Nature, then I think we are fine. If we resolve a hacked Rotlung Reanimator, it will be the only one we have, and we can't go infinite with just that one. To make more copies, we have to use The Scarab God, which means putting our hacked Rotlung Reanimator in the graveyard, so all our copies will have normal text. So yeah, that should be fine.
We optimized our deck around having Dual Nature, so we may need to make some changes if we remove it. But, moxes now can't be given to the opponent, so perhaps the setup works now.
Hmmmm. Unfortunately, I think we can get one going that way, because we can use Domineering Will to give them the changed cards, with Artificial Evolution still in hand. Darn.
In that case, we either need a new way of replicating artifacts that the opponent can't mimic, a new way of producing blue and black mana which can't be donated, or a different key besides red mana and haste.
Perhaps we could do something similar with White mana and untapping, rather than just haste. If we swapped out Mobilize for something more restrictive, we could swap out Break through the Line for say High Alert, and make the stage key off white mana. Though now I realize that Axgard Armory taps for W as well... Foiled again. We need a land which enters the battlefield tapped, doesn't tap for white, and can tap and sacrifice itself to generate the effect we need. Could we maybe do something with Port of Karfell? That would be difficult to do in a way that can't be replicated by The Scarab God...
Another option would be something like Mirrorpool, but that has to create an infinite somewhere.
Edit: Actually, Mirrorpool has some potential if it somehow doesn't break the hyperstage wide open (I'm quite sure it's safe in the calculation itself, since we definitely have no way to keep it around or donate it). If the only way we could get back Guardian Zendikon were with something like Wargate or Genesis Wave set to X=3, we wouldn't be able to get it back with Spellweaver Helix, but we could with Mirrorpool. It would even be much more efficient by cards. Adding the decklist again because it helps me think:
edit: Is Thousand-Year Storm and the like also unsafe? In the Mirrorpool deck I just outlined I think an effect that can copy higher-cost sorceries is a necessity, but without Dual Nature I think we're limited to just one Temple Garden in a red mana deck, which wouldn't generate N green mana on its own from a Mobilize.
Keep in mind that we can only have two Arcbonded creatures in a computation. So the only way I see Mirrorpool being safe is if we have one Arcbond, and we get the second one using Mirrorpool. But, it looks like the plan is to only be able to recycle Mirrorpool in the additional stage, and of course if we do the computation that infrequently, we don't get a Busy Beaver hyperstage.
Edit: Okay, if taking out Dual Nature works, the next step is to fill out the ending layers. How can we generate a lot of red mana?
Edit: Hmm, without Fractured Identity, we can't give the opponent Treasure tokens or Gold tokens, so we can use those in the layer sequence. So, how about:
That would be 1 hyperstage + 1 stage + 5 layers, it looks like. I'm not sure that the changes don't require extra cards, so that's something we should look out for.
As for the actual number of hacked rotlungs and 2/2 tokens we need, it looks like X^^^X (of each) will be enough, as the nested conversions don't look as bad as I feared. I'm not sure on the absolute minimum size.
A very basic idea for flooding clocks is to have them chain into each other with the last one enabling halting when it finally trips. Even a very simple structure provides a lot of power.
For example say we have a simple heartbeat clock that triggers every other tick to prevent halting, and adds 1 to all of the flooding clocks
When flooding clock 1 trips it can increase that and allow the program to halt.
Flooding clock 2 doubles itself and adds to the first clock
FC3 doubles itself and adds to the first two FCs
FC4 doubles itself and adds to the first three FCs
....
FC250 or so doubles itself and adds itself to the other FCs (or wherever we run out of creature types)
Then we initialize the heartbeat at 1, halting clock at 2, and the flooding clocks at X - clock number. (maybe more optimal with everything but clock 1 at 1?)
This halts because clock 1 will eventually be the smallest clock, stopping the heatbeat and triggering the halting condition. The higher clocks eventually get too big to trigger enough.
I'm not sure exactly how long this takes, but it seems to get pretty big pretty fast.
If we add FC3, and FC3 adds 2X to FC1 and FC2, and 4X to FC3 when it hits 0, then it takes 952 steps starting at 100,1,1 for example. So it looks like this just multiplies by a small constant.
Edit: Oh, I guess the trick is to add 4X to FC1 instead of 2X when FC3 hits 0. That makes it much more powerful.
Does that implement an arrow system?
Edit: Hmm no, it looks like that just goes infinite actually.
Edit: Well, it might not actually go infinite, but the behavior seems rather chaotic. We can make things more orderly, by having each flooding clock imcrease every flooding clock by 4X, except for the one before it, which it can increase by 2X. That has a more orderly recursion, but it doesn't quite work right; the largest index FC's will trigger first, and then work it's way down, which is the opposite of what we want. What we would like is to have the lowest index trigger first, and have each FC trigger lead to many triggers of the one before it. I don't know how to do that.
Perhaps we could do something similar to what Iijil did, using one or two clocks per layer to signify state, except we do it with flooding clocks. We can sorta deal with the multiplicative nature of the flooding clocks by considering powers of 2 perhaps, where doubling becomes basically incrementing the exponent. If so, we could cut the number of creature types needed in half.
Edit: Hmmm, I wonder if Temple Garden and Ambush Commander can be partially replaced by Lair of the Hydra. Lair of the Hydra can come in untapped, but we use the green mana produced to turn it into a creature, so that it can be repeatedly untapped by Mobilize. We can not turn it into a creature, and just use the green mana to get ahead in the lower stage, but then Lair of the Hydra will be on the battlefield as a noncreature. The next time we resolve Worldfire, Lair of the Hydra will go into exile, where it will cost more to retrieve. So maybe that doesn't go infinite? I'm definitely not certain though, I could use some verification.
In the case that it does happen to work, we will still need a source of white mana. We can replace Mana Crypt with Mox Pearl again to get access to white mana. But, this seems to conflict with Avacyn, Guardian Angel. So, we would like to replace Avacyn with some other card that protects us from dying to a computation. Gravebreaker Lamia already protects the opponent from both the computation and Kaervek's Spite, so we only have to protect ourselves. But, Safe Passage doesn't seem to work, as that will prevent the damage we need to have dealt to ourselves in the layers.
Edit: Hmm, I think we can cut out Mana Crypt and Tolarian Academy! We can exile Elvish Spirit Guide and put Temple Garden into play untapped, allowing us to cat Channel. Then, we can cast Mirrorworks, and get a copy of Mox Sapphire after we play it. Two blue mana is enough to cast Prosperity, and then cast Show and Tell to put Omniscience into play.
That's a lot of life spent though. If we play Radiant Fountain at the beginning, we can cast Prosperity for up to 11, and still have 2 life left over for Show and Tell. That's a lot fewer cards, but hopefully we can get started? If necessary, we can use one of the extra cards for Master Skald.
With the extra space, we can add World at War, taking us up to BB_{w^2+14} (BB_{w^2+13} (X)) I think.
Edit: Correction, we can't play Radiant Fountain at the beginning, because we need to play Temple Garden. So we have 1 card remaining in hand, and can draw up to 8.
https://pastebin.com/9cQf3d1B
I tried to keep the explanation brief (and failed) so I didn't include stuff like optimal ordering that doesn't end up making a noticeable difference in the final number. That's also why I kept the explanations for each step rather barebones; I can explain further if it isn't clear enough. I also included the method I used to estimate how effective it was, but I don't know where your most recent list is to compare with.
I'm just going to post your deck here, so that I can look at the cards more easily.
1 Tolarian Academy
1 Show and Tell
1 Omniscience
1 Opalescence
1 Doubling Season
1 Cackling Counterpart
1 Radiate
1 Thousand-Year Storm
1 Biblioplex Assistant
1 Chef's Kiss
1 Balance of Power
1 Fathom Mage
1 Nykthos Paragon
1 Death's Oasis
1 Gorilla Shaman
1 Mirrorworks
1 Scrap Trawler
1 Bolt Bend
1 Voracious Greatshark
1 Sleeper Dart
1 Rings of Brighthearth
1 Vedalken Orrery
1 Monkey Cage
1 Argentum Armor
1 Summoning Station
1 Possessed Portal
1 The Magic Mirror
1 Aladdin's Lamp
1 Rootwater Diver
1 Chainer, Nightmare Adept
1 Bloodbond March
1 Molten Echoes
1 Double Major
1 Magda, Brazen Outlaw
1 Rhys the Exiled
1 Mishra, Artificer Prodigy
1 Yarok, the Desecrated
1 Vorinclex, Monstrous Raider
1 Kamahl, Heart of Krosa
1 Myojin of Life's Web
1 Myojin of Infinite Rage
1 Iname as One
1 Draco
1 Goblin Welder
1 Battle Cry Goblin
1 Rousing Refrain
1 Douse
1 Circle of Dreams Druid
1 Worldsoul Colossus
1 King Crab
1 Spinal Villain
1 Southern Paladin
1 Western Paladin
1 Daretti, Ingenious Iconoclast
1 Elite Arcanist
1 Emergency Powers
1 Teferi, Temporal Archmage
Edit: A couple of problems I noticed: Note that the opponent is not a true goldfish, they will act in a way to minimize the amount of damage that you can deal. (But they will also cooperate to go infinite and invalidate your deck) So, they can mulligan down to 0 cards to prevent you from drawing cards from Balance of Power.
Also, it looks like Fathom Mage goes infinite, since it's Evolve ability will allow it to gain counters whenever a creature with higher power or toughness enters the battlefield, so there's no need for Nykthos Paragon.
Concerning the counting, why does Argentum Armor get (+8,38)? It seems like it should be +4 like the previous members of the sequence.
On the whole though, some very nice ideas! Getting a bunch of layers for each additional mana cost is quite powerful, and getting it for both artifacts and creatures is very nice.
You are right that Fathom Mage would go infinite, I can salvage that by switching to Benthic Biomancer, drawing EtI and discarding Show and Tell which we can easily get back before the next one. It only gives 1 draw trigger per counter-granting effect so we swap Nykthos to Archangel of Thune and don't benefit from Doubling Season's counter doubling, but it still ends up similarly good.
Argentum Armor is a typo from copying it. Change that to (+4, 34), Summoning Station to (+4, 38) and it adds up again after that.
How many layers does your current combo have?
Edit the maybe last: I can cut Sleeper's Dart, Rousing Refrain, and Douse for Fire Diamond and Filigree Sages, without even losing a layer since the Sages are Rings-viable. So -Dart, Refrain, Douse, Fathom Mage, Nykthos Paragon, +Fire Diamond, Filigree, Benthic, Archangel actually leaves me at 99 layers and 59 cards. I'm not sure of any 1-card additional layers I can add at the moment, so I'll leave it there for now.
MetroidComposite's first deck, reaching 23 layers
MetroidComposite's second deck, reaching 36 layers
Sadistic Mystic's first deck, reaching 113 layers
The first deck I had much of a hand in, reaching 263 layers
The most recent writeup, reaching 408 layers - I contributed to this as well, but plopfill was the biggest contributor here.
Shortly after that last deck was written up, an expansion came out that had Gideon, Ally of Zendikar, which can be used to replace both Soul of New Phyrexia and Thran Dynamo, leaving us with another card slot. That card slot can be used to add Devout Lightcaster, which can come in at the end of the sequence, adding 9 layers. That yields a final total of more than 2 -> 19 -> 417.
Not too long after that, we had a major breakthrough.
We've defined a "stage combo" to be a combo that takes X of some resource (like X creature tokens or X green mana, etc) and turns it into X layers. So the first stage basically performs the Ackermann function. This goes far beyond what you can do by individually adding layers, if you can execute the combo more than once; for example, if you start by using the stage combo with 10 green mana, you'll wind up with something like 10 -> 10 -> 10 green mana. (This is Conway chained arrow notation; in A -> B -> C, C is the number of layers, and B is the amount of the resource, i.e. how many times you perform the previous layer.) If we use the stage combo again, we'll get something like 10 -> 10 -> (10 -> 10 -> 10) green mana, so the number of layers is itself 10 -> 10 -> 10. Obviously we can never match that with a deck that adds layers individually. So this was a major paradigm shift.
The stage combo is somewhat complicated, and can be implemented in many different ways.
Here is an unfinished writeup of one of our decks. section 7 has a description of the stage combo.
More recently, stage combos have found their way into Standard decks for this challenge.
This was the first Standard deck to have a stage combo. It has a very nice section talking about the mechanics behind creating a stage combo. This was also the first stage combo that veered away markedly from the Bloodbond March / Psychic Battle paradigm at the time.
After that, you can go to 1 stage + 1 layer, which takes X of a resource and performs the stage combo X times; 1 stage + 2 layers, which performs the 1 stage + 1 layer combo X times; and so on. Then, the second stage takes X of a resource and turns it into 1 stage + X layers.
You can keep adding more stages like this; I think the furthest we got with individual stages was 26 stages. The next step then is to create a combo that takes X of a resource and creates X stages, which we called a hyperstage combo. We were able to create a deck with 2 hyperstages plus a few stages afterwards. Then, we defined a megastage as a combo that creates X hyperstages, and a gigastage as a deck that creates X megastages. We have a megastage deck that I'm quite confident works, and a few gigastage deck entries that still need verification to see whether everything is kosher or not.
That work was more or less abandoned when jfb came in with the idea of implementing the Busy Beaver function, using a deck that implemented The Waterfall Model, which is a Turing-complete computational model. The Busy Beaver function blows the pants off anything we can do with recursion. (This next part gets pretty mathy, so you can skip if you're not that into math.) To give an idea of how fast it grows, we can represent our numbers using the fast-growing hierarchy, which is indexed by ordinals. The smallest ordinals are polynomials over w (actually the greek letter omega, but I'm too lazy to cut and paste that in). A layer increments the ordinal index by 1, a stage increments the ordinal index by w, and a hyperstage/megastage/gigastage increments the ordinal by w^2, w^3, and w^4, respectively. So the largest numbers we have been able to reach with recursion is about F_{w^4}. BB(85) has been shown to be bigger than F_{epsilon_0}(1908), which is equal to F_{w^w^w^w^w...^w}(1908) with 1908 w's in the exponential tower! So even at relatively low values, the Busy Beaver function beats anything we could conceivably do with just recursion.
Our latest decks have managed to perform some pretty heavy recursion over the Busy Beaver function itself. We can define a fast-growing hierarchy BB_a, except instead of starting with a weak function, we start from the Busy Beaver function. So BB_0 = the Busy Beaver function, BB_1(N) applies the Busy Beaver function N times starting from N, BB_2(N) applies X -> BB_1(X) N times starting from N, and so on. Then we can have Busy Beaver stages and hyperstages as well. Our best is a Busy Beaver hyperstage plus a few layers, with the final damage number being BB_{w^2+14} (BB_{w^2+13} (X)), with X not quite having been worked out yet.
Edit: Going back to the busy beaver deck, I suppose switching out Mox Sapphire for Eye of Ramos will allow us to pay 3 mana instead of 7 for Mirrorworks, so we wind up with 14 cards rather than 9. 14 cards are hopefully enough to get started.
Edit: I think we can get 14 layers with fewer cards, by using Thrummingbird and proliferating Invasion of the Giants rather than Tymarret. Invasion of the Giants only does 2 damage, but we can use Gisela, Blade of Goldnight to double the damage. That's 2 extra cards rather than 3, so we have an extra card we can use to help out with the opening, perhaps Master Skald.
We know we can generalize all or most stages into five instructions across two states:
1. Use 1 resource I to push X trigger A.
2. Iff the state is unlocked, pop one Trigger A to push X trigger B and lock the state.
3. Pop 1 trigger B to generate X resource O.
4. Pop 1 trigger B to unlock the state.
5. Iff the state is unlocked, pop 1 trigger A to generate 1 resource I and lock the state.
Traditionally, the state has been "this creature is not in the graveyard" in vintage, but it doesn't have to be. If we can find a new State tracker and free up a resource (white mana through Safe Passage or red mana, maybe by using Sacrifice and Coat of Arms to store our progress), and some linking triggers that don't get wrecked by Temur Sabertooth.
I've done some digging, and I think we can get some mileage out of taplands with a sacrifice ability and an aura which animates them. Because they come in tapped, we have to animate them with our one aura to untap them with Mobilize. If their activation cost requires we spend mana of our input resource, we can make it too expensive to activate them without leaving them a creature and using something like Primal Beyond. And when they activate, the aura falls off and goes to the graveyard, locking the state since we currently have no way to revive an aura from there.
Putting it all together, I get the following:
Break through the Line
Primal Beyond
Guardian Zendikon
Axgard Armory
Tinder Farm
Viridian Harvest
Additionally, we replace Marchesa's Smuggler with Thopter Engineer, and swap our red rituals for Sacrifice.
We spend a red mana to grant Axgard Armory (enchanted) haste, untap it, and activate it, each time grabbing Viridian Harvest to gain life. Then, when it comes time to regenerate red mana, we put the Zendikon on Tinder Farm, give it haste, untap it, sac it for an R, and use an Axgard trigger below to get back the Zendikon. We could move the Zendikon after untapping Tinder Farm but before sacrificing it, but without Zendikon bouncing it back to hand we'd have no way to rebuy it later, and the stage would fizzle.
I can only imagine this has an infinite somewhere earlier in the chain, but I can't find it with my still limited understanding and either way this should hopefully be a good starting point for a stage discussion!
Edit: putting a full decklist here sans final layers as a point of reference, since it doesn't look like there's a full list on this page of the thread.
2 Opalescence
3 Dual Nature
4 Vedalken Orrery
5 Riftsweeper
6 Gravebreaker Lamia
7 Thopter Engineer
8 Coat of Arms
9 Rotlung Reanimator
10 Dralnu's Crusade
11 Arcbond
12 Arcbond
13 Fractured Identity
14 Artificial Evolution
15 Sea Gate Stormcaller
16 Kaervek's Spite
17 Avacyn, Guardian Angel
18 Goblin Boom Keg
20 Mirrorworks
21 Master Transmuter
22 Mox Sapphire
23 Bloodbond March
24 The Scarab God
25 Rings of Brighthearth
26 Temur Sabertooth
27 Elvish Spirit Guide
28 Mobilize
29 Ambush Commander
30 Worldfire
31 Worldpurge
32 Acorn Harvest
33 Acorn Harvest
34 Radiant Fountain
35 Akoum Refuge
36 Eureka
38 Wood Elves
39 Panharmonicon
40 Spellweaver Helix
41 Wormfang Behemoth
42 Deathgrip
43 Mystical Teachings
44 Temple Garden
45 Mana Crypt
46 Tolarian Academy
47 Channel
48 Break through the Line
49 Primal Beyond
50 Guardian Zendikon
51 Axgard Armory
52 Tinder Farm
53 Viridian Harvest
After that we'd add layers starting with red mana and/or mana of any color.
Oh, very interesting! Still thinking about the the new stage, but one thing I noticed: I don't see how Sacrifice can allow us to store the progress of a computation. Previously we have used Brightstone Ritual, because the opponent can output Goblins, or Battle Hymn, which we can resolve after killing off a large number of opponent's creatures by bouncing Coat of Arms, while having a Rotlung Reanimator on our side, so that we get lots of creatures. Sacrifice gets us the mana value of the creature, which can only get really large if we cast a creature with an X in its casting cost, timees the number of Sea Gate Stormcallers which can't be really large until we have taken advantage of the result of an opponent's computation, so that's kind of a Catch-22.
Previously, we used Mana Echoes to get mana commensurable with the output of the computation, but we won't have that with Kaevek's Spite.
Oh, we are also currently using Master Transmuter to convert mana into artifacts like Rings of Brighthearth, so if we require red mana for haste, then we need to turn the result of the computation into red mana. Maybe the best way to salvage your idea then is to find a way to turn the result of a computation into usable resources without using red mana or Master Transmuter.
As for giving Master Transmuter haste, it should innately get it from Thopter Engineer, which would give the Transmuter haste but not any of our other creatures.
Edit: How about Vile Redeemer? I know tokens are dying in the computation, but if we could find a creature that repeatedly dies and reanimates whenever some part of the computation happens, we could tie the computation to nontoken creatures dying and build a huge number of spawns.
Edit: How about Solemnity and an inoffensive creature with undying, like Young Wolf. It would constantly die and reanimate, but I guess it would probably break the Turing Machine model...
Edit: Hmmm... For Vile Redeemer, I don't think we want a creature that triggers in our graveyard, since then it would trigger the same time as our Arcbonds, so we could decide whether or not the creature goes back to the battlefield immediately, or gets buried underneath Arcbond triggers. But, a creature that triggers when it dies could work.
Edit: I think Young Wolf probably doesn't work, since it will come back on our side, and from then on it will be our trigger, which we can choose how we resolve compared to the Arcbonds.
Illusionist's Bracers would have the same problem, I think. Kurkesh, Onakke Ancient costs red mana, which we don't want.
Edit: I thought about using Mana Echoes on the opponent's side, and then using Drain Power to steal the colorless mana, but then I remembered that Mana Echoes goes infinite with Mirrorworks and Master Transmuter.
Cryptic Trilobite
Immaculate Magistrate
Dimir Signet
and Rage Weaver in addition to Thopter Engineer
We can use Coat of Arms/Rotlung Reanimator shenanigans to generate plentiful elf tokens on our side, then give an Immaculate Magistrate haste, put that many counters on Cryptic Trilobite, and from there we can filter the colorless mana through Dimir Signet to generate blue and black. To keep doing so, we need more Immaculate Magistrates, which means reshuffling them back into the library and milling them, which requires some green mana input to bounce Riftsweeper and Gravebreaker Lamia.
On the bright side, this accounts for both blue and black mana, though we may still need Mox Sapphire for the start. That would get us a decklist like this:
2 Opalescence
3 Dual Nature
4 Vedalken Orrery
5 Riftsweeper
6 Gravebreaker Lamia
7 Thopter Engineer
8 Coat of Arms
9 Rotlung Reanimator
10 Dralnu's Crusade
11 Arcbond
12 Arcbond
13 Fractured Identity
14 Artificial Evolution
15 Sea Gate Stormcaller
16 Kaervek's Spite
17 Avacyn, Guardian Angel
18 Goblin Boom Keg
20 Immaculate Magistrate
21 Dimir Signet
22 Rage Weaver
23 Mirrorworks
24 Master Transmuter
25 Mox Sapphire
26 Bloodbond March
27 The Scarab God
28 Rings of Brighthearth
29 Temur Sabertooth
30 Elvish Spirit Guide
31 Mobilize
32 Ambush Commander
33 Worldfire
34 Worldpurge
35 Acorn Harvest
36 Acorn Harvest
37 Radiant Fountain
38 Akoum Refuge
39 Eureka
41 Wood Elves
42 Panharmonicon
43 Spellweaver Helix
44 Wormfang Behemoth
45 Deathgrip
46 Mystical Teachings
47 Temple Garden
48 Mana Crypt
49 Tolarian Academy
50 Channel
51 Break through the Line
52 Primal Beyond
53 Guardian Zendikon
54 Axgard Armory
55 Tinder Farm
56 Viridian Harvest
How do we break this?
Oh wait, with Thopter Engineer they don't need the red mana, so Master Transmuter + Thopter Engineer + Mox Sapphire + Fractured Identity is no good.
Also, I'm assuming March of the Machines goes infinite in some way that I'm missing, but on the off chance it doesn't, We could maybe give them the cards we need with Domineering Will, without letting them get Mox Sapphire. That would still leave unresolved the issue of generating black mana for The Scarab God, though.
The reason for using Fractured Identity is explained in post #3858; basically, we can use Artificial Evolution to changed the text of Rotlung Reanimators and Dralnu's Crusades that are still on the stack, and then Worldfire can exile Artificial Evolution, allowing us to get it back. Then, if we can get an infinite computation going with using the text-changed permanents, we can end whenever we want with the Artificial Evolution still in our hand.
Iijil's solution was to use The Scarab God to create copies of Rotlung Reanimator, and Fractured Identity to bring it over to the opponent's side. I guess we have to check whether Domineering Will allows the opponent to go infinite.
Edit: Oh, about the basic lands: for the Vintage version, we assume that all the opponent's cards are Wastes, so if they can put those into play they have access to colorless mana.
Edit: Okay, I think Domineering Will and Dual Nature will go infinite, because we can change the text of a Rotlung Reanimator and a Dralnu's Crusade, then resolve those spells, and then Dual Nature can create non-hacked versions of Rotlung Reanimator, and we can make an infinite computation just like in Iijil's example.
But, if we remove Dual Nature, then I think we are fine. If we resolve a hacked Rotlung Reanimator, it will be the only one we have, and we can't go infinite with just that one. To make more copies, we have to use The Scarab God, which means putting our hacked Rotlung Reanimator in the graveyard, so all our copies will have normal text. So yeah, that should be fine.
We optimized our deck around having Dual Nature, so we may need to make some changes if we remove it. But, moxes now can't be given to the opponent, so perhaps the setup works now.
In that case, we either need a new way of replicating artifacts that the opponent can't mimic, a new way of producing blue and black mana which can't be donated, or a different key besides red mana and haste.
Perhaps we could do something similar with White mana and untapping, rather than just haste. If we swapped out Mobilize for something more restrictive, we could swap out Break through the Line for say High Alert, and make the stage key off white mana. Though now I realize that Axgard Armory taps for W as well... Foiled again. We need a land which enters the battlefield tapped, doesn't tap for white, and can tap and sacrifice itself to generate the effect we need. Could we maybe do something with Port of Karfell? That would be difficult to do in a way that can't be replicated by The Scarab God...
Another option would be something like Mirrorpool, but that has to create an infinite somewhere.
Edit: Actually, Mirrorpool has some potential if it somehow doesn't break the hyperstage wide open (I'm quite sure it's safe in the calculation itself, since we definitely have no way to keep it around or donate it). If the only way we could get back Guardian Zendikon were with something like Wargate or Genesis Wave set to X=3, we wouldn't be able to get it back with Spellweaver Helix, but we could with Mirrorpool. It would even be much more efficient by cards. Adding the decklist again because it helps me think:
2 Opalescence
3 Dual Nature
4 Vedalken Orrery
5 Riftsweeper
6 Gravebreaker Lamia
7 Marchesa's Smuggler
8 Coat of Arms
9 Rotlung Reanimator
10 Dralnu's Crusade
11 Arcbond
12 Arcbond
13 Fractured Identity
14 Artificial Evolution
15 Sea Gate Stormcaller
16 Kaervek's Spite
17 Safe Passage
18 Goblin Boom Keg
20 Mirrorworks
21 Master Transmuter
22 Mox Sapphire
23 Bloodbond March
24 The Scarab God
25 Rings of Brighthearth
26 Temur Sabertooth
27 Elvish Spirit Guide
28 Hunting Wilds
29 Ambush Commander
30 Worldfire
31 Worldpurge
32 Acorn Harvest
33 Acorn Harvest
34 Radiant Fountain
35 Akoum Refuge
36 Eureka
38 Everbark Shaman
39 Burnt Offering
40 Panharmonicon
41 Spellweaver Helix
42 Wormfang Behemoth
43 Deathgrip
44 Mystical Teachings
45 Overgrown Tomb
46 Mana Crypt
47 Tolarian Academy
48 Channel
49 High Alert
50 Mirrorpool
51 Wargate
52 Guardian Zendikon
53 Tinder Farm
edit: Is Thousand-Year Storm and the like also unsafe? In the Mirrorpool deck I just outlined I think an effect that can copy higher-cost sorceries is a necessity, but without Dual Nature I think we're limited to just one Temple Garden in a red mana deck, which wouldn't generate N green mana on its own from a Mobilize.
Edit: question answered. Arcbond is rough.
Edit: Yeah, Thousand-Year Storm is also unsafe, because it can copy Arcbond. We're relying on Sea Gate Stormcaller to get copies of Artificial Evolution and Mobilize. (and Prosperity from the layer sequence)
Edit: Okay, if taking out Dual Nature works, the next step is to fill out the ending layers. How can we generate a lot of red mana?
Edit: Hmm, without Fractured Identity, we can't give the opponent Treasure tokens or Gold tokens, so we can use those in the layer sequence. So, how about:
2 Opalescence
3 Vedalken Orrery
4 Riftsweeper
5 Gravebreaker Lamia
6 Thopter Engineer
7 Coat of Arms
8 Rotlung Reanimator
9 Dralnu's Crusade
10 Arcbond
11 Arcbond
12 Domineering Will
13 March of the Machines
14 Artificial Evolution
15 Sea Gate Stormcaller
16 Kaervek's Spite
17 Avacyn, Guardian Angel
18 Goblin Boom Keg
19 Cryptic Trilobite
20 Immaculate Magistrate
21 Mox Jet
22 Rage Weaver
23 Mirrorworks
24 Master Transmuter
25 Mox Sapphire
27 The Scarab God
28 Rings of Brighthearth
29 Temur Sabertooth
30 Elvish Spirit Guide
31 Mobilize
32 Ambush Commander
33 Worldfire
34 Worldpurge
35 Acorn Harvest
36 Acorn Harvest
37 Radiant Fountain
38 Akoum Refuge
39 Eureka
40 Show and Tell
41 Panharmonicon
42 Spellweaver Helix
43 Wormfang Behemoth
44 Deathgrip
45 Wild Research
47 Channel
48 Break through the Line
49 Primal Beyond
50 Guardian Zendikon
51 Axgard Armory
52 Tinder Farm
53 Viridian Harvest
54 King Macar, the Gold-Cursed
55 Tolarian Kraken
56 Prosperity
57 Sylvan Echoes
58 Sentry Oak
59 Harmonic Prodigy
60 Kinnan, Bonder Prodigy
That would be 1 hyperstage + 1 stage + 5 layers, it looks like. I'm not sure that the changes don't require extra cards, so that's something we should look out for.