I just remembered that Illusionist's Bracers has the equip ability to bounce our creatures with Cowardice. Only at sorcery speed, but that would still go infinite. So we kinda have to use Psychic Battle for the base stage and probably have to fall back on Kurkesh, Onakke Ancient + Feldon of the Third Path for the ability copies we need for the additional stages.
EDIT: generating blue or colorless mana to operate in the base stage without the opponent being able to do the same is one of those things that seem like they should be doable, but are pretty much impossible. So I'm not sure we can make the colors work when we need red to be a bit more expensive.
Since we don't have Tolarian Kraken, the Phabine version can just add an extra card, which I suppose would be God-Pharaoh's Gift. So that gets BB_{w^3 + w2 + 5} (BB_{w^3 + w2 + 5} (14,700)).
We do have Precursor Golem and Psychic Battle now, so we would be able to get many copies of our instants, and change the targets where we want.
Oh, but with Phabine we can also get the layer sequence above, with Sentry Oak. That gets us to layer 6, so that seems like the best. Calculating the exact value is definitely harder though, we have to figure out the start.
Oh, I was just thinking that having blue in the primary stage would be convenient, for casting Omniscience. There's no way to get blue mana just for ourselves? Phooey.
Unfortunately, there doesn't seem to be a way to just pay life for the hyperstage transition, other than with Flowstone Flood. But the buyback for Flowstone Flood only works if we don't counter it, which is not workable for the hyperstage.
Otherwise, we have Acorn Harvest and Flash of Defiance. We can't moot green, so we give up red... and also spend 3 life. Is there an enchantment aura for 1 or 3 life?
For a deck with FDZ and Psychic Possession, Riftsweeper is necessary, so that we can refill the opponent's deck. For a deck without Psychic Possession, we'd rather go with Mirror of Fate, but I guess Fated Infatuation isn't enough to get copies of it. (We can't bring the copies below the hyperstage transition.)
I thought about the Phabine version of this, considering what you suggested, but if we use Flash of Defiance for the hyperstage, how do we pay for the red mana cost?
Ah, right. We aren't really able to pay only life for flashback
We can gain 3 life from an Aura with Soaring Hope. I don't see anything for 1 life.
We can of course get any 2 colors with signets, but if we need 1 color each for the hyperstage, megastage and the additional stages we can't afford that. Other than that I think we can get red or black via instants and green or red via ETB triggers.
Of course, if the opponent can't get haste we have a lot more options, so maybe Anger is the way to go.
Another thing to keep in mind is that Feldon of the Third Path gives the opponent access to hasty creatures even if we usually get our haste from Anger. An alternative would be Saheeli's Artistry to get our artifact copies of creatures without haste. But for the purposes of going infinite that gives the opponent access to ETB triggers, which we might need to keep in mind.
Edit: Wait, Saheeli's Artistry can copy the Mox without animating it, so it goes infinite with the megastage transition. I don't know how we can get the additional stages then.
Feldon of the Third Path seems to nullify the advantage of Anger over Phabine. So I guess we're a little stuck at the moment.
What do we have at minimum? I think we can get by if we use a signet for the primary stage mana, so we would just lose the second additional stage I guess? Hopefully there is a way to salvage that as well.
Edit: Uh-oh. There's another problem: How do we keep the Mountain on the battlefield so that we can keep haste? Both transitions get rid of the Mountain, but we don't have a cheap way of bringing it back.
Edit: I suppose we could have Drownyard Temple and Blood Moon. That's another card slot lost, down to BB_{w^3 + w2 + 4}(X).
After resolving most of a megastage we get back to a batch of Kurkesh, Onakke Ancient triggers on Groundskeeper. We pay 3 red to bring our basics back to hand. Then we come to a layer of Kurkesh triggers on Firebrand Ranger. We pay 3 red to put the basics into play. We tap them for 1 green, 1 blue and X red. Use the blue to make a hasty artifact Groundskeeper. Use the green to activate it and refresh the batch we used up earlier. We have gained X-6 red mana to go through the megastage.
It looks like the white mana stage is okay. Due to targeting issues, we can only flicker one Forest per activation, but we can just wait around until we have a bunch of green mana before proceding upwards.
Do you understand how, starting from a bunch of hasted Armor Thrull tokens, we execute a stage to build up our resources?
Do you understand how, if we have a bunch of life (plus some other minimum resources), we can build up a hyperstage as a bunch of stages on top of each other?
Do you understand how, if we have a bunch of red mana, we can build up a megastage as a bunch of hyperstages?
After that, can you see how, at the top of the megastage, we can execute a computation, so that the basic function is the Busy Beaver function rather than doubling or multiplication?
Do you see how the additional stages and layers recurse over the megastage?
Let's try to narrow down where things get confusing.
Edit: Another possible issue:
We've set things up so that it is difficult to kill things, other than with Kaervek's Spite. It would be allowable to kill things higher up, but it doesn't look like we have anything that does that; Worldfire of course exiles everything.
For Smoke Spirits' Aid, there seems to be a problem with Xorn; if we need to resolve a Kaervek's Spite to destroy our creatures enchanted by SSA auras, then the Spite will destroy all of our Xorns. In addition, it will destroy our Vedalken Orrery, so that we won't be able to create any more Xorns, until all of the destruction triggers are resolved. So it doesn't look like we can take advantage of any Xorns, if Kaervek's Spite is required for destroying things.
For Floating-Dream Zubera, destroying an FDZ with Kaervek's Spite, and drawing a bunch of cards in a single go seems fine. But, if we have Harmonic Prodigy, we will get a bunch of draw triggers, and again we don't have Vedalken Orrery, so we don't have the ability to do much of anything between draws. So we may just deck ourselves.
For Psychic Possession, I don't see how we even destroy an FDZ on the opponent's side.
There may be card fixes for these problems, but with just a few card slots left, it may be better just to find new layer sequences. Our current 4 layer sequence just has a problem with Xorn; taking it out, we can get BB_{w^3 + w2 + 3}(~50) with a card slot left. Is there any card that can add a layer. If not, what's the best thing to do with it?
Edit: I guess we can use a computation to destroy an FDZ on the opponent's side, but we won't have any Psychic Possessions when that happens.
Do you understand how, starting from a bunch of hasted Armor Thrull tokens, we execute a stage to build up our resources?
Do you understand how, if we have a bunch of life (plus some other minimum resources), we can build up a hyperstage as a bunch of stages on top of each other?
Do you understand how, if we have a bunch of red mana, we can build up a megastage as a bunch of hyperstages?
After that, can you see how, at the top of the megastage, we can execute a computation, so that the basic function is the Busy Beaver function rather than doubling or multiplication?
Do you see how the additional stages and layers recurse over the megastage?
Let's try to narrow down where things get confusing.
Roughly. Cowardice and Psychic Battle let us bounce our creatures, then we can replay them, counter them with Cephalid Shrine, and let each Bloodbond March bring them back. Although I'm not sure how to keep the tokens from bouncing the original Armor Thrull and doing this infinitely.
Sort of. Hungry for More lets us use life and red mana to trigger Spellweaver Helix and recast Worldfire to exile our cards, so Mirror of Fate can retrieve them, then use Chromescale Drake, Academy Rector, and Weathered Wayfarer to get the artifacts, enchantments, and lands back out of our library. I'm not sure what order we build back up to make effective use of this, though, or where the lifegain cards are incorporated to avoid going infinite.
I found this explanation, which to my understanding precedes a discovery of needing to use a third set of creature types. I'm not sure where it fits into the rest of the process, though.
Vaguely. I can see how the cards get recycled, but I'm not sure when to reuse them or what some parts are proportional to.
1. Actually, bouncing the original Armor Thrull is like the primary purpose of tapping a hasted token; of course, we bounce other stuff as well to make progress. We use up a hasted token, but by bouncing the nontoken Armor Thrull, we can imprint it onto a Soul Foundry, and get a token back.
So, why doesn't this go infinite? The key is, after imprinting the Thrull onto a Soul Foundry, we need to resolve a Bloodbond March trigger on the stack to bring the Thrull back to the battlefield. So that's one BM trigger from the top BM layer on the stack. We can get more BM triggers, by bouncing and recasting the Thrull, but that costs us a PB trigger from further down the stack. And so on; refreshing each layer costs us at least one ability from the next layer down, so it doesn't go infinite.
3. Heh - I haven't worked out all the details yet myself. My lazy ass is kinda waiting for Iijil to work it all out, since I don't trust myself to get it all right.
For lifegain, we put Divine Congregation on the stack above Worldfire. We then resolve the DC, giving us enough life to set up a higher hyperstage, while leaving Worldfire on the stack. The Worldfire prevents life from getting through that barrier to the hyperstage below. The dangerous thing is if we put Divine Congregation below Worldfire, allowing us to gain life in the lower hyperstage. But, right after Worldfire, there are no creatures on the battlefield, so DC won't gain us any life in that case.
4. The computations mainly go at the top; it may be optimal to incorporate computations in lower hyperstage transitions, but we don't need these other computations to reach our estimate. So the highest stage performs a computation, which turns X creatures into BB(X) creatures. Since we have Precursor Golem currently, we can get on the order of BB(X) copies of our targeting instants, like Fated Infatuation, to turn the result of the computation into usable resources. Everything below the top hyperstage transition is support for that top stage.
5. The basic idea behind the additional stages is that we have pairs of abilities, each one costing one mana of a certain color. With N mana of that particular color, we can build up N alternating layers of those two abilities. We can regain one mana of that color by resolving each of the abilities; for example, for the green mana stage, we can get a green mana by resolving a Groundskeeper ability to move the Forest from the graveyard to our hand, followed by resolving a Firebrand Ranger to put it onto the battlefield, so that we can tap it for a green mana.
So the mechanism winds up working in much the same way as for our Armor Thrull stage; we can refresh layers, but each refresh costs us one ability from the next layer down. You may wonder why we need linked abilities; the answer is, if we only needed one ability, then we could get a bunch of green mana by just resolving all the abilities in a single layer for that ability. The need to resolve both abilities to get the green mana back means that it only pays to resolve one ability from a layer at a time, so the stage structure works out as intended.
Edit: Honestly, there should be some set of 3 cards that can get us to layer 2, so that Sphinx of Enlightenment + Consecrated Sphinx can get us to layer 4. I haven't been able to find one though.
We've set things up so that it is difficult to kill things, other than with Kaervek's Spite. It would be allowable to kill things higher up, but it doesn't look like we have anything that does that; Worldfire of course exiles everything.
For Smoke Spirits' Aid, there seems to be a problem with Xorn; if we need to resolve a Kaervek's Spite to destroy our creatures enchanted by SSA auras, then the Spite will destroy all of our Xorns. In addition, it will destroy our Vedalken Orrery, so that we won't be able to create any more Xorns, until all of the destruction triggers are resolved. So it doesn't look like we can take advantage of any Xorns, if Kaervek's Spite is required for destroying things.
For Floating-Dream Zubera, destroying an FDZ with Kaervek's Spite, and drawing a bunch of cards in a single go seems fine. But, if we have Harmonic Prodigy, we will get a bunch of draw triggers, and again we don't have Vedalken Orrery, so we don't have the ability to do much of anything between draws. So we may just deck ourselves.
For Psychic Possession, I don't see how we even destroy an FDZ on the opponent's side.
There may be card fixes for these problems, but with just a few card slots left, it may be better just to find new layer sequences. Our current 4 layer sequence just has a problem with Xorn; taking it out, we can get BB_{w^3 + w2 + 3}(~50) with a card slot left. Is there any card that can add a layer. If not, what's the best thing to do with it?
Edit: I guess we can use a computation to destroy an FDZ on the opponent's side, but we won't have any Psychic Possessions when that happens.
This is something that worries me about getting deep into abstractions, if it can obscure what specific things some of the cards are doing or when.
Another thing I was wondering - is there a way for the most recent list to get multiple Kurkesh, Onakke Ancient triggers on one activation, with no Harmonic Prodigy and no way around the legend rule?
Yes, Archon of Falling Stars can only be triggered by Kaervek's Spite, which is the plan; we want to make sure we can't put Pious Interdiction into play too easily to gain cheap life. So this seems to be working as intended.
As for three types per clock during a computation, That original post of mine ignores some of the unfortunate side effects of what a proper setup would actually look like: Namely that our Xathrid necromancers pump the tokens with coat of arms by different amounts (Dralnu's crusade pumps too). Fixing it by adding more coat of arms doesn't work, and adding more necromancers changes the program.
Post #4018 shows how to add necromancers (and a few tokens) to convert any TWM program into one we can actually run, though at the cost of using an additional type per clock to fix the problems along the diagonal.
This process does make the resulting program even bigger, but because we are already needing approximately x^^^x or x^^^^x to set up the chain of UTMs (Its something like: Waterfall implementing Spiral Rise running a Tag System emulating a Minsky Machine evaluating the UTM on our input x) the conversion being O(x^2) or so is no trouble.
Edit: also that's just a lower bound, there are almost definitely TWM programs that exhibit direct BB(X) growth instead of BB(XvvvvX) growth with XvvX being the inverse of X^^X. The issue is finding them and proving they are runnable in the 2 creature type clocks version. Since at least one of those is impossible, we just use the worse bound.
Ruh roh... Flash can get us cheap deaths, particularly for Archon of Falling Stars. I don't see an impediment to bouncing the creature that Pious Interdiction enchants to put PI in the graveyard, then Flash the Archon to bring PI back and gain 2 life.
So... new start? We can try the Replenish start again. We have very few slots left though.
I think Mox Sapphire can replace Island; when we run the megastage, we will accumulate a lot of blue mana, which should be plenty for what we need to execute the first additional stage.
We only have one card left after Omniscience is played, so we may need a big draw card like Wheel of Fortune. That only gives us 7 uses though, so we probably wind up with something like BB_{w^3 + w2 + 3}(X) where is probably between 5 and 7. Assuming this works, of course.
Edit: It looks like we can make do with Sphinx of Enlightenment rather than Wheel of Fortune. So Replenish can get back Omniscience and Thousand-Year Storm, and then with Sphinx of Enlightenment we can draw Consecrated Sphinx, Comeuppance, and Fated Infatuation, which we play in order. We will get three copies of Fated Infatuation, of which we can use the first two to copy Consecrated Sphinx, and the last one to copy Sphinx of Enlightenment, drawing 9 cards. With 9 cards, I think there is a good chance that we can create enough copies of Consecrated Sphinx so that with the next Sphinx of Enlightenment trigger, we can access the full deck and still have at least 51 Consecrated Sphinx triggers left; this would allow us to get to BB_{w^3 + w2 + 3}(51). If not, we should definitely be able to get BB_{w^3 + w2 + 3}(50).
Edit: It looks like we can make do with Sphinx of Enlightenment rather than Wheel of Fortune. So Replenish can get back Omniscience and Thousand-Year Storm, and then with Sphinx of Enlightenment we can draw Consecrated Sphinx, Comeuppance, and Fated Infatuation, which we play in order. We will get three copies of Fated Infatuation, of which we can use the first two to copy Consecrated Sphinx, and the last one to copy Sphinx of Enlightenment, drawing 9 cards. With 9 cards, I think there is a good chance that we can create enough copies of Consecrated Sphinx so that with the next Sphinx of Enlightenment trigger, we can access the full deck and still have at least 51 Consecrated Sphinx triggers left; this would allow us to get to BB_{w^3 + w2 + 3}(51). If not, we should definitely be able to get BB_{w^3 + w2 + 3}(50).
Edit: Aww, fiddlesticks. Mirrorworks can copy Mox Ruby, giving us too much red.
We can go back to Twinning Glass, but we won't be able to make copies of Mirror of Fate cheaply enough to sacrifice them when needed; more generally, I don't know how we would be able to make cheap enough Mirror of Fate copies without also being able to make copies of Mox Ruby or Armor Thrull too cheaply.
We can go back to using Riftsweeper, but that prevents us from using Sphinx of Enlightenment or any of the other card drawers that depend on the opponent's library being limited. Memory Jar won't work, since the opponent's hand will be empty after a Worldfire, and we can use Memory Jar then. Does Garruk, Primal Hunter go infinite? It does seem like we could be able to put Garruk back into the library after activating it to be drawn by its own ability. That requires a Worldfire, so I'm not totally sure the infinite works out.
Oh, I have another idea. We aaondoned the idea of getting additional creature token stages, because Fated Infatuation can make copies of any creature, and with Soul Foundry we have to use an outside source for haste. But, we still have the potential to make additional *mana* stages that use creatures. I'm thinking of stages that get mana by discarding or exiling cards, using cards like Elvish Spirit Guide, Simian Spirit Guide, Skirge Familiar, Chrome Mox, and the various land cycling cards.
These all involve creatures (except maybe Chrome Mox, but we've only really used Chrome Mox with creatures), so we need to remove Cowardice to get these additional stages. Of course, these could cause problems, but let's hope that these can all be addressed. We have the potential for another three stages, for the three colors we may have left; even two stages will probably be an improvement, since our current deck can barely fit in two stages.
The main problem, at first glance, is finding the right abilities to bounce the creatures, without Cowardice. There's Escape Routes, is basically a general targeter, due to Dralnu's Crusade and Artifical Evolution. Kogla, the Titan Ape can't be used, since it can fight opponent's creatures. Tetsuo Umezawa uses red, black, and blue mana to destroy tapped creatures, which can turn into a bounce with Together Forever. But that can't be combined with Escape Routes. Still looking, but not there yet.
We can combine these mana/creature stages with our old mana/land stages, so there is perhaps a good chance to get two or three stages out of this.
Edit: Hmm, not being able to copy our landcycling abilities is an issue; I tried to deal with it by increasing the number of our landcycling creatures, but that winds up costing several card slots. So I still wound up with only a few card slots left.
which appears to get BB_{w^3 + w2 + 3}(~50), whereas the previous deck with land stages gets BB_{w^3 + w2 + 3}(14,760). Of course, things can and probably will change with more improvements/fixes, so it's still worth keeping both of these decks in mind, assuming there's no fatal flaw.
With these new stages actually rather cumbersome, I was thinking it would be worth it to switch to one mana/creature stage and one mana/land stage; but if we try to keep the smaller blue mana stage, then the white mana stage would presumably include Ruin Ghost, which would naturally be paired with an Archaeological Dig that comes into play tapped. But, we need to put the Island we fetch with Jhessian Zombies to come into play untapped. I suppose we could have a way to put basic lands into play untapped and other lands tapped, but we also need to mess with World Shaper, so I haven't figured out how to make this idea work.
which appears to get BB_{w^3 + w2 + 3}(~50), whereas the previous deck with land stages gets BB_{w^3 + w2 + 3}(14,760). Of course, things can and probably will change with more improvements/fixes, so it's still worth keeping both of these decks in mind, assuming there's no fatal flaw.
Even killing 4 FDZ's would get us 16 cards, which would probably be enough to get going - provided we can recover from the Kaervek's Spite. So that seems to be the big issue.
Even killing 4 FDZ's would get us 16 cards, which would probably be enough to get going - provided we can recover from the Kaervek's Spite. So that seems to be the big issue.
How would it mess with the enchantments? The only thing that gets copied is the life loss, right?
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I don't think we would be able to stack up the delayed triggers from Repeated Reverberation. Without Thousand-Year Storm we wouldn't get more than one cast out of Spellweaver Volute per transition.
I just remembered that Illusionist's Bracers has the equip ability to bounce our creatures with Cowardice. Only at sorcery speed, but that would still go infinite. So we kinda have to use Psychic Battle for the base stage and probably have to fall back on Kurkesh, Onakke Ancient + Feldon of the Third Path for the ability copies we need for the additional stages.
EDIT: generating blue or colorless mana to operate in the base stage without the opponent being able to do the same is one of those things that seem like they should be doable, but are pretty much impossible. So I'm not sure we can make the colors work when we need red to be a bit more expensive.
Maybe if we free up black by not using K'rrik, Son of Yawgmoth for the hyperstage?
For Anger we can have
1 Smoke Spirits' Aid
2 Psychic Possession
3 Floating-Dream Zubera
4 Xorn
5 Precursor Golem
6 Harmonic Prodigy
for BB_{w^3 + w2 + 5}(14,700).
Since we don't have Tolarian Kraken, the Phabine version can just add an extra card, which I suppose would be God-Pharaoh's Gift. So that gets BB_{w^3 + w2 + 5} (BB_{w^3 + w2 + 5} (14,700)).
We do have Precursor Golem and Psychic Battle now, so we would be able to get many copies of our instants, and change the targets where we want.
Oh, but with Phabine we can also get the layer sequence above, with Sentry Oak. That gets us to layer 6, so that seems like the best. Calculating the exact value is definitely harder though, we have to figure out the start.
Is maybe Repeated Reverberation possible with Precursor Golem and Psychic Battle to help with the instants? What about Garruk, Primal Hunter, does that go infinite?
Unfortunately, there doesn't seem to be a way to just pay life for the hyperstage transition, other than with Flowstone Flood. But the buyback for Flowstone Flood only works if we don't counter it, which is not workable for the hyperstage.
Otherwise, we have Acorn Harvest and Flash of Defiance. We can't moot green, so we give up red... and also spend 3 life. Is there an enchantment aura for 1 or 3 life?
For a deck with FDZ and Psychic Possession, Riftsweeper is necessary, so that we can refill the opponent's deck. For a deck without Psychic Possession, we'd rather go with Mirror of Fate, but I guess Fated Infatuation isn't enough to get copies of it. (We can't bring the copies below the hyperstage transition.)
Edit: Anger deck:
1 Coat of Arms
2 Xathrid Necromancer
3 Artificial Evolution
4 Arcbond
5 Comeuppance
6 Dralnu's Crusade
Starting a Computation
7 Opalescence
8 Wrong Turn
9 Kaervek's Spite
10 Goblin Boom Keg
11 March of the Machines
Utility: Enchantments/Artifacts
12 Vedalken Orrery
13 Silver Myr
14 Mirrorworks
Utility: Spells
15 Thousand-Year Storm
Utility: Creatures
16 Bloodbond March
17 Cephalid Shrine
Foundry Stage
18 Armor Thrull
19 Soul Foundry
20 Anger
21 Volcanic Island
22 Cowardice
23 Psychic Battle
24 Mirror of Fate
25 Shriekhorn
26 Spellweaver Volute
27 Spellweaver Volute
28 Fated Infatuation
29 Lingering Souls
30 K'rrik, Son of Yawgmoth
31 Archon of Falling Stars
32 Pious Interdiction
Helix Megastage
33 Spellweaver Helix
34 Worldfire
35 Divine Congregation
36 Hungry for More
37 Hungry for More
38 Mox Ruby
39 Panharmonicon
40 Chromescale Drake
Green Mana Stage
41 Groundskeeper
42 Firebrand Ranger
43 Forest
44 Feldon of the Third Path
45 Kurkesh, Onakke Ancient
46 Ruin Ghost
47 Weathered Wayfarer
48 Archaeological Dig
49 Patron of the Moon
50 Dryad Arbor
Start
51 Flash
52 Academy Rector
53 Omniscience
54 Sphinx of Enlightenment
Layers
55 Smoke Spirits' Aid
56 Warden of the Woods
57 Entangling Trap
58 Sentry Oak
59 Precursor Golem
60 Xorn
for BB_{w^3 + w2 + 5}(X), X to be determined.
I thought about the Phabine version of this, considering what you suggested, but if we use Flash of Defiance for the hyperstage, how do we pay for the red mana cost?
We can gain 3 life from an Aura with Soaring Hope. I don't see anything for 1 life.
We can of course get any 2 colors with signets, but if we need 1 color each for the hyperstage, megastage and the additional stages we can't afford that. Other than that I think we can get red or black via instants and green or red via ETB triggers.
Of course, if the opponent can't get haste we have a lot more options, so maybe Anger is the way to go.
Another thing to keep in mind is that Feldon of the Third Path gives the opponent access to hasty creatures even if we usually get our haste from Anger. An alternative would be Saheeli's Artistry to get our artifact copies of creatures without haste. But for the purposes of going infinite that gives the opponent access to ETB triggers, which we might need to keep in mind.
Edit: Wait, Saheeli's Artistry can copy the Mox without animating it, so it goes infinite with the megastage transition. I don't know how we can get the additional stages then.
I'm not seeing much in the way of ETB triggers in the deck - there's Spellweaver Helix, but the opponent can't do anything with that. If we replace Feldon of the Third Path with Saheeli's Artistry in the above deck, does it work?
Maybe there is another option to turn creatures into artifacts that I overlooked?
EDIT: Maybe Neurok Transmuter, if blue mana is expensive enough that the targeting doesn't hurt, but available for the additional stages.
So from the above deck remove Silver Myr, Volcanic Island and Feldon of the Third Path, replace them with Palladium Myr, Island and Neurok Transmuter.
We can get Omniscience onto the battlefield after transitions via Archon of Falling Stars, so we don't need blue for that.
Feldon of the Third Path seems to nullify the advantage of Anger over Phabine. So I guess we're a little stuck at the moment.
What do we have at minimum? I think we can get by if we use a signet for the primary stage mana, so we would just lose the second additional stage I guess? Hopefully there is a way to salvage that as well.
Edit: So, we're looking at:
1 Coat of Arms
2 Xathrid Necromancer
3 Artificial Evolution
4 Arcbond
5 Comeuppance
6 Dralnu's Crusade
Starting a Computation
7 Opalescence
8 Wrong Turn
9 Kaervek's Spite
10 Goblin Boom Keg
11 March of the Machines
Utility: Enchantments/Artifacts
12 Vedalken Orrery
13 Palladium Myr
14 Mirrorworks
Utility: Spells
15 Thousand-Year Storm
Utility: Creatures
16 Bloodbond March
17 Cephalid Shrine
Foundry Stage
18 Armor Thrull
19 Soul Foundry
20 Anger
21 Mountain
22 Cowardice
23 Psychic Battle
24 Mirror of Fate
25 Shriekhorn
26 Spellweaver Volute
27 Spellweaver Volute
28 Fated Infatuation
29 Lingering Souls
30 K'rrik, Son of Yawgmoth
31 Archon of Falling Stars
32 Pious Interdiction
Helix Megastage
33 Spellweaver Helix
34 Worldfire
35 Divine Congregation
36 Hungry for More
37 Hungry for More
38 Mox Ruby
39 Panharmonicon
40 Chromescale Drake
Green Mana Stage
41 Groundskeeper
42 Firebrand Ranger
43 Forest
44 Neurok Transmitter
45 Kurkesh, Onakke Ancient
46 Ruin Ghost
47 Weathered Wayfarer
48 Archaeological Dig
49 Patron of the Moon
50 Dryad Arbor
Start
51 Flash
52 Academy Rector
53 Omniscience
54 Sphinx of Enlightenment
Layers
55 Smoke Spirits' Aid
56 Warden of the Woods
57 Entangling Trap
58 Sentry Oak
59 Precursor Golem
60 Xorn
Edit: Uh-oh. There's another problem: How do we keep the Mountain on the battlefield so that we can keep haste? Both transitions get rid of the Mountain, but we don't have a cheap way of bringing it back.
Edit: I suppose we could have Drownyard Temple and Blood Moon. That's another card slot lost, down to BB_{w^3 + w2 + 4}(X).
I think we need Forest, Mountain, Island and Gauntlet of Might for the green mana stage to operate.
After resolving most of a megastage we get back to a batch of Kurkesh, Onakke Ancient triggers on Groundskeeper. We pay 3 red to bring our basics back to hand. Then we come to a layer of Kurkesh triggers on Firebrand Ranger. We pay 3 red to put the basics into play. We tap them for 1 green, 1 blue and X red. Use the blue to make a hasty artifact Groundskeeper. Use the green to activate it and refresh the batch we used up earlier. We have gained X-6 red mana to go through the megastage.
Our precious card slots!
So that takes us down to... 4 card slots for layers? I suppose we go Smoke Spirits' Aid, Xorn, Precursor Golem, Psychic Possession into Sphinx of Enlightenment, for BB_{w^3 + w2 + 4}(~50).
It looks like the white mana stage is okay. Due to targeting issues, we can only flicker one Forest per activation, but we can just wait around until we have a bunch of green mana before proceding upwards.
Edit:
1 Coat of Arms
2 Xathrid Necromancer
3 Artificial Evolution
4 Arcbond
5 Comeuppance
6 Dralnu's Crusade
Starting a Computation
7 Opalescence
8 Wrong Turn
9 Kaervek's Spite
10 Goblin Boom Keg
11 March of the Machines
Utility: Enchantments/Artifacts
12 Vedalken Orrery
13 Palladium Myr
14 Mirrorworks
Utility: Spells
15 Thousand-Year Storm
Utility: Creatures
16 Bloodbond March
17 Cephalid Shrine
Foundry Stage
18 Armor Thrull
19 Soul Foundry
20 Emblem of the Warmind
21 Cowardice
22 Psychic Battle
23 Mirror of Fate
24 Shriekhorn
25 Spellweaver Volute
26 Spellweaver Volute
27 Fated Infatuation
28 Lingering Souls
29 K'rrik, Son of Yawgmoth
30 Archon of Falling Stars
31 Pious Interdiction
Helix Megastage
32 Spellweaver Helix
33 Worldfire
34 Divine Congregation
35 Hungry for More
36 Hungry for More
37 Mox Ruby
38 Panharmonicon
39 Chromescale Drake
Green Mana Stage
40 Groundskeeper
41 Firebrand Ranger
42 Forest
43 Island
44 Mountain
45 Gauntlet of Might
46 Neurok Transmitter
47 Kurkesh, Onakke Ancient
48 Ruin Ghost
49 Weathered Wayfarer
50 Archaeological Dig
51 Patron of the Moon
52 Dryad Arbor
Start
53 Flash
54 Academy Rector
55 Omniscience
56 Sphinx of Enlightenment
Layers
57 Smoke Spirits' Aid
58 Psychic Possession
Misc Utility
59 Precursor Golem
60 Xorn
Edit: Any chance Floating-Dream Zubera + Psychic Possession, rather than Sphinx of Enlightenment, can get off the ground? (Or Floating-Dream Zubera + Harmonic Prodigy)
Before we worry about the computations:
Do you understand how, starting from a bunch of hasted Armor Thrull tokens, we execute a stage to build up our resources?
Do you understand how, if we have a bunch of life (plus some other minimum resources), we can build up a hyperstage as a bunch of stages on top of each other?
Do you understand how, if we have a bunch of red mana, we can build up a megastage as a bunch of hyperstages?
After that, can you see how, at the top of the megastage, we can execute a computation, so that the basic function is the Busy Beaver function rather than doubling or multiplication?
Do you see how the additional stages and layers recurse over the megastage?
Let's try to narrow down where things get confusing.
Edit: Another possible issue:
We've set things up so that it is difficult to kill things, other than with Kaervek's Spite. It would be allowable to kill things higher up, but it doesn't look like we have anything that does that; Worldfire of course exiles everything.
This may cause a problem for things that depend on killing, in particular, Smoke Spirits' Aid and Floating-Dream Zubera.
For Smoke Spirits' Aid, there seems to be a problem with Xorn; if we need to resolve a Kaervek's Spite to destroy our creatures enchanted by SSA auras, then the Spite will destroy all of our Xorns. In addition, it will destroy our Vedalken Orrery, so that we won't be able to create any more Xorns, until all of the destruction triggers are resolved. So it doesn't look like we can take advantage of any Xorns, if Kaervek's Spite is required for destroying things.
For Floating-Dream Zubera, destroying an FDZ with Kaervek's Spite, and drawing a bunch of cards in a single go seems fine. But, if we have Harmonic Prodigy, we will get a bunch of draw triggers, and again we don't have Vedalken Orrery, so we don't have the ability to do much of anything between draws. So we may just deck ourselves.
For Psychic Possession, I don't see how we even destroy an FDZ on the opponent's side.
There may be card fixes for these problems, but with just a few card slots left, it may be better just to find new layer sequences. Our current 4 layer sequence just has a problem with Xorn; taking it out, we can get BB_{w^3 + w2 + 3}(~50) with a card slot left. Is there any card that can add a layer. If not, what's the best thing to do with it?
Edit: I guess we can use a computation to destroy an FDZ on the opponent's side, but we won't have any Psychic Possessions when that happens.
So, why doesn't this go infinite? The key is, after imprinting the Thrull onto a Soul Foundry, we need to resolve a Bloodbond March trigger on the stack to bring the Thrull back to the battlefield. So that's one BM trigger from the top BM layer on the stack. We can get more BM triggers, by bouncing and recasting the Thrull, but that costs us a PB trigger from further down the stack. And so on; refreshing each layer costs us at least one ability from the next layer down, so it doesn't go infinite.
3. Heh - I haven't worked out all the details yet myself. My lazy ass is kinda waiting for Iijil to work it all out, since I don't trust myself to get it all right.
For lifegain, we put Divine Congregation on the stack above Worldfire. We then resolve the DC, giving us enough life to set up a higher hyperstage, while leaving Worldfire on the stack. The Worldfire prevents life from getting through that barrier to the hyperstage below. The dangerous thing is if we put Divine Congregation below Worldfire, allowing us to gain life in the lower hyperstage. But, right after Worldfire, there are no creatures on the battlefield, so DC won't gain us any life in that case.
4. The computations mainly go at the top; it may be optimal to incorporate computations in lower hyperstage transitions, but we don't need these other computations to reach our estimate. So the highest stage performs a computation, which turns X creatures into BB(X) creatures. Since we have Precursor Golem currently, we can get on the order of BB(X) copies of our targeting instants, like Fated Infatuation, to turn the result of the computation into usable resources. Everything below the top hyperstage transition is support for that top stage.
5. The basic idea behind the additional stages is that we have pairs of abilities, each one costing one mana of a certain color. With N mana of that particular color, we can build up N alternating layers of those two abilities. We can regain one mana of that color by resolving each of the abilities; for example, for the green mana stage, we can get a green mana by resolving a Groundskeeper ability to move the Forest from the graveyard to our hand, followed by resolving a Firebrand Ranger to put it onto the battlefield, so that we can tap it for a green mana.
So the mechanism winds up working in much the same way as for our Armor Thrull stage; we can refresh layers, but each refresh costs us one ability from the next layer down. You may wonder why we need linked abilities; the answer is, if we only needed one ability, then we could get a bunch of green mana by just resolving all the abilities in a single layer for that ability. The need to resolve both abilities to get the green mana back means that it only pays to resolve one ability from a layer at a time, so the stage structure works out as intended.
Edit: Honestly, there should be some set of 3 cards that can get us to layer 2, so that Sphinx of Enlightenment + Consecrated Sphinx can get us to layer 4. I haven't been able to find one though.
This is something that worries me about getting deep into abstractions, if it can obscure what specific things some of the cards are doing or when.
Another thing I was wondering - is there a way for the most recent list to get multiple Kurkesh, Onakke Ancient triggers on one activation, with no Harmonic Prodigy and no way around the legend rule?
Oh oops, yeah it does seem like we need to have either Harmonic Prodigy or Mirror Gallery/Mirror Box to get multiple triggers from Kurkesh, Onakke Ancient.
So, it may be that BB_{w^3 + w2 + 3}(~50) (using say Smoke Spirits' Aid / Precursor Golem / Psychic Possession / Sphinx of Enlightenment) is the best we can do. Harmonic Prodigy isn't necessarily just a wasted card for our final layers though, it can add layers for other cards. So it can help other starting cards, like Arbiter of the Ideal and Prosper, Tome-Bound... maybe there is something better.
Post #4018 shows how to add necromancers (and a few tokens) to convert any TWM program into one we can actually run, though at the cost of using an additional type per clock to fix the problems along the diagonal.
This process does make the resulting program even bigger, but because we are already needing approximately x^^^x or x^^^^x to set up the chain of UTMs (Its something like: Waterfall implementing Spiral Rise running a Tag System emulating a Minsky Machine evaluating the UTM on our input x) the conversion being O(x^2) or so is no trouble.
Edit: also that's just a lower bound, there are almost definitely TWM programs that exhibit direct BB(X) growth instead of BB(XvvvvX) growth with XvvX being the inverse of X^^X. The issue is finding them and proving they are runnable in the 2 creature type clocks version. Since at least one of those is impossible, we just use the worse bound.
So... new start? We can try the Replenish start again. We have very few slots left though.
Edit: Maybe
1 Coat of Arms
2 Xathrid Necromancer
3 Artificial Evolution
4 Arcbond
5 Comeuppance
6 Dralnu's Crusade
Starting a Computation
7 Opalescence
8 Wrong Turn
9 Kaervek's Spite
10 Goblin Boom Keg
11 March of the Machines
Utility: Enchantments/Artifacts
12 Vedalken Orrery
13 Palladium Myr
14 Mirrorworks
Utility: Spells
15 Thousand-Year Storm
Utility: Creatures
16 Bloodbond March
17 Cephalid Shrine
Foundry Stage
18 Armor Thrull
19 Soul Foundry
20 Emblem of the Warmind
21 Cowardice
22 Psychic Battle
23 Mirror of Fate
24 Shriekhorn
25 Spellweaver Volute
26 Spellweaver Volute
27 Fated Infatuation
28 Lingering Souls
29 K'rrik, Son of Yawgmoth
30 Archon of Falling Stars
31 Pious Interdiction
Helix Megastage
32 Spellweaver Helix
33 Worldfire
34 Divine Congregation
35 Hungry for More
36 Hungry for More
37 Mox Ruby
38 Panharmonicon
39 Chromescale Drake
Green Mana Stage
40 Groundskeeper
41 Firebrand Ranger
42 Forest
43 Mountain
44 Gauntlet of Might
45 Neurok Transmitter
46 Kurkesh, Onakke Ancient
47 Harmonic Prodigy
48 Ruin Ghost
49 Weathered Wayfarer
50 Archaeological Dig
51 Patron of the Moon
52 Dryad Arbor
Start
53 Mana Crypt
54 Mox Sapphire
55 Replenish
56 Omniscience
57 Wheel of Fortune
Layers
58 Smoke Spirits' Aid
59 Precursor Golem
60 Consecrated Sphinx
I think Mox Sapphire can replace Island; when we run the megastage, we will accumulate a lot of blue mana, which should be plenty for what we need to execute the first additional stage.
We only have one card left after Omniscience is played, so we may need a big draw card like Wheel of Fortune. That only gives us 7 uses though, so we probably wind up with something like BB_{w^3 + w2 + 3}(X) where is probably between 5 and 7. Assuming this works, of course.
Edit: It looks like we can make do with Sphinx of Enlightenment rather than Wheel of Fortune. So Replenish can get back Omniscience and Thousand-Year Storm, and then with Sphinx of Enlightenment we can draw Consecrated Sphinx, Comeuppance, and Fated Infatuation, which we play in order. We will get three copies of Fated Infatuation, of which we can use the first two to copy Consecrated Sphinx, and the last one to copy Sphinx of Enlightenment, drawing 9 cards. With 9 cards, I think there is a good chance that we can create enough copies of Consecrated Sphinx so that with the next Sphinx of Enlightenment trigger, we can access the full deck and still have at least 51 Consecrated Sphinx triggers left; this would allow us to get to BB_{w^3 + w2 + 3}(51). If not, we should definitely be able to get BB_{w^3 + w2 + 3}(50).
Archaeological Dig
Mox Ruby
Mox Sapphire
Mana Crypt
Shriekhorn
Replenish
Sphinx of Enlightenment
Edit: Aww, fiddlesticks. Mirrorworks can copy Mox Ruby, giving us too much red.
We can go back to Twinning Glass, but we won't be able to make copies of Mirror of Fate cheaply enough to sacrifice them when needed; more generally, I don't know how we would be able to make cheap enough Mirror of Fate copies without also being able to make copies of Mox Ruby or Armor Thrull too cheaply.
We can go back to using Riftsweeper, but that prevents us from using Sphinx of Enlightenment or any of the other card drawers that depend on the opponent's library being limited. Memory Jar won't work, since the opponent's hand will be empty after a Worldfire, and we can use Memory Jar then. Does Garruk, Primal Hunter go infinite? It does seem like we could be able to put Garruk back into the library after activating it to be drawn by its own ability. That requires a Worldfire, so I'm not totally sure the infinite works out.
If it does, we can go back to Caress of Phyrexia. Without Psychic Possession or other help though, it will only get us 9 cards total. Is that enough to get Floating-Dream Zubera going?
These all involve creatures (except maybe Chrome Mox, but we've only really used Chrome Mox with creatures), so we need to remove Cowardice to get these additional stages. Of course, these could cause problems, but let's hope that these can all be addressed. We have the potential for another three stages, for the three colors we may have left; even two stages will probably be an improvement, since our current deck can barely fit in two stages.
The main problem, at first glance, is finding the right abilities to bounce the creatures, without Cowardice. There's Escape Routes, is basically a general targeter, due to Dralnu's Crusade and Artifical Evolution. Kogla, the Titan Ape can't be used, since it can fight opponent's creatures. Tetsuo Umezawa uses red, black, and blue mana to destroy tapped creatures, which can turn into a bounce with Together Forever. But that can't be combined with Escape Routes. Still looking, but not there yet.
We can combine these mana/creature stages with our old mana/land stages, so there is perhaps a good chance to get two or three stages out of this.
Edit: Hmm, not being able to copy our landcycling abilities is an issue; I tried to deal with it by increasing the number of our landcycling creatures, but that winds up costing several card slots. So I still wound up with only a few card slots left.
First attempt at a deck using landcyclers:
1 Coat of Arms
2 Xathrid Necromancer
3 Artificial Evolution
4 Arcbond
5 Comeuppance
6 Dralnu's Crusade
Starting a Computation
7 Opalescence
8 Wrong Turn
9 Kaervek's Spite
10 Goblin Boom Keg
11 March of the Machines
Utility: Enchantments/Artifacts
12 Vedalken Orrery
13 Golgari Signet
14 Mirrorworks
Utility: Spells
15 Thousand-Year Storm
Utility: Creatures
16 Bloodbond March
17 Cephalid Shrine
Foundry Stage
18 Possessed Aven
19 Soul Foundry
20 Emblem of the Warmind
21 Illusionist's Bracers
22 Sigarda's Aid
23 Mirror of Fate
24 Shriekhorn
25 Stern Proctor
26 Spellweaver Volute
27 Spellweaver Volute
28 Fated Infatuation
29 Acorn Harvest
30 Radiant Fountain
31 Glimmerpost
32 World Shaper
Helix Megastage
33 Spellweaver Helix
34 Worldfire
35 Divine Congregation
36 Flash of Defiance
37 Flash of Defiance
38 Taiga
39 Panharmonicon
40 Wood Elves
White Mana Stage
41 Igneous Pouncer
42 Igneous Pouncer
43 Igneous Pouncer
44 Igneous Pouncer
45 Southern Paladin
46 Simian Spirit Guide
47 Plateau
48 Walking Atlas
49 Jhessian Zombies
50 Jhessian Zombies
51 Escape Routes
52 Island
Start
53 Mana Crypt
54 Flash
55 Academy Rector
56 Omniscience
57 Sphinx of Enlightenment
Layers
58 Smoke Spirits' Aid
59 Consecrated Sphinx
60 Precursor Golem
which appears to get BB_{w^3 + w2 + 3}(~50), whereas the previous deck with land stages gets BB_{w^3 + w2 + 3}(14,760). Of course, things can and probably will change with more improvements/fixes, so it's still worth keeping both of these decks in mind, assuming there's no fatal flaw.
With these new stages actually rather cumbersome, I was thinking it would be worth it to switch to one mana/creature stage and one mana/land stage; but if we try to keep the smaller blue mana stage, then the white mana stage would presumably include Ruin Ghost, which would naturally be paired with an Archaeological Dig that comes into play tapped. But, we need to put the Island we fetch with Jhessian Zombies to come into play untapped. I suppose we could have a way to put basic lands into play untapped and other lands tapped, but we also need to mess with World Shaper, so I haven't figured out how to make this idea work.
Which deck was that?
1 Coat of Arms
2 Xathrid Necromancer
3 Artificial Evolution
4 Arcbond
5 Comeuppance
6 Dralnu's Crusade
Starting a Computation
7 Opalescence
8 Wrong Turn
9 Kaervek's Spite
10 Goblin Boom Keg
11 March of the Machines
Utility: Enchantments/Artifacts
12 Vedalken Orrery
13 Palladium Myr
14 Twinning Glass
Utility: Spells
15 Thousand-Year Storm
Utility: Creatures
16 Bloodbond March
17 Cephalid Shrine
Foundry Stage
18 Armor Thrull
19 Soul Foundry
20 Emblem of the Warmind
21 Cowardice
22 Psychic Battle
23 Riftsweeper
24 Shriekhorn
25 Spellweaver Volute
26 Spellweaver Volute
27 Fated Infatuation
28 Lingering Souls
29 K'rrik, Son of Yawgmoth
30 Archon of Falling Stars
31 Pious Interdiction
Helix Megastage
32 Spellweaver Helix
33 Worldfire
34 Divine Congregation
35 Hungry for More
36 Hungry for More
37 Mox Ruby
38 Panharmonicon
39 Chromescale Drake
Green Mana Stage
40 Groundskeeper
41 Firebrand Ranger
42 Forest
43 Mountain
44 Gauntlet of Might
45 Neurok Transmitter
46 Kurkesh, Onakke Ancient
47 Harmonic Prodigy
48 Ruin Ghost
49 Weathered Wayfarer
50 Archaeological Dig
51 Patron of the Moon
52 Dryad Arbor
Start
53 Mana Crypt
54 Mox Sapphire
55 Replenish
56 Omniscience
57 Caress of Phyrexia
Layers
58 Smoke Spirits' Aid
59 Precursor Golem
60 Floating-Dream Zubera
Although, I still haven't figured out how to get started off of Caress of Phyrexia, so we may need to amend this.
So we start with Mox Ruby, Mox Sapphire, Mana Crypt, Archaeological Dig, Shriekhorn, Replenish, and Caress of Phyrexia. We mill Omniscience and Thousand-Year Storm, reanimate them, and draw Floating-Dream Zubera, Fated Infatuation, Spellweaver Helix, Kaervek's Spite, and both Hungry for More. Play one Hungry and the Zubera, use the Helix to exile Hungry and Replenish, play the second Hungry to get Replenish on the stack, then play Infatuation to make another six Zuberas, play Kaervek's Spite to kill them all... and then we have seven copies of it on the stack, which have to resolve before we can play sorceries again. Not sure what to do about that.
Edit: Draw Floating-Dream Zubera, Fated Infatuation, Kaervek's Spite, Archon of Falling Stars, and Divine Congregation? That leaves one spot for something else.
Even killing 4 FDZ's would get us 16 cards, which would probably be enough to get going - provided we can recover from the Kaervek's Spite. So that seems to be the big issue.