It looks to me like we need to not cast Worldfire to make the Recoup under Spellweaver Helix trick work. Not casting Worldfire means we don't get red mana back, so I think we don't go infinite with a red spell like Flash of Defiance.
Going with Radiant Fountain does lose us our access to green later on, so perhaps Call to the Netherworld would be better? Oh, but we need to gain life first without paying it, so rather than Urborg Syphon-Mage, we would need to use something else like the Draconian Cylix / Vampire Envoy combo. I think we can't gain life from discarding Vampire Envoy, since we need to have one on the battlefield to gain life. But, we can gain life from Xathrid Necromancer and K'rrik, Son of Yawgmoth, but only every other transition; we need to use one transition to cast them for Bloodbond March triggers, and then in the next transition we can discard them for life. So, maybe we need 4 black creatures besides Vampire Envoy to gain 4 life each transition? So we would need one more black creature, if my thinking is correct.
Edit: Or wait, maybe we don't need Bloodbond March triggers if we can put them into play with Soul Separator. So K'rrik and Necromancer is enough for 4 life each transition, and we need to replace Bala Ged Scorpion with something that isn't a black creature.
Worldfire involves more bouncing and using Wormfang behemoth to hide our hand. I think some versions needed to get the real wormfang under a token wormfang which was tricky but possible.
Sorry if that's a bit too handwave-y, I know that there were some other quirks like making the first token omniscience requiring some initial inefficiency.
I'm still kinda stumped on making the worldfire transition work the optimal way we want it to in the current decks, one hopeful thought is that channel (the mechanic not the card) is returning in the new kamigawa, So there's some potential for a black Ghost-Lit Redeemer style card.
Consider a cast-counter of Worldfire, getting TYS trigger. One TYS trigger turns into many Worldfire copies. Even if we prepare Wormfang behemoth before Worldfire resolving, first Worldfire clean the Wormfang behemoth, then our hand come back, then second Worldfire still clean our hand. We still fail to keep our hand.
If we simply cast a Worldfire without TYS trigger, then the card is on the stack. We cannot reuse Worldfire card to make multiple transitions on the stack.
Edit:
Oh I didn't noticed that: Spellweaver Helix casts a copy of a card. So with absence of Thousand-Year Storm we can put 1 copy of Worldfire on stack.
However, for the decks using 1 Worldfire plus 3 flashback sorcery cards, or 1 Worldfire plus 2 flashback sorcery cards plus Panharmonicon, Worldfire and another sorcery cast at the same time. Without Thousand-Year Storm (the requirement for only 1 Worldfire copy), there is only 1 copy of the "another sorcery".
group of Smite the Monstrous-TYS triggers (0 extra life)
Mirrorworks trigger for Vedalken Orrery
group of Metallurgeon-BM triggers (3 extra life)
Maybe other abilities
Child of Alara kills the life-gain creature, giving us 3 life. There is no Vedalken Orrery on board, so we simply wait for a Mirrorworks trigger for it, which allow us to act.
Usually the next Mirrorworks trigger for Vedalken Orrery should be at the top of "3 extra life" block. But here, since there are some space between targeting and killing Child of Alara, we meet the trigger earlier:
Top
Mirrorworks trigger for Vedalken Orrery
group of Smite the Monstrous-PB triggers
group of Smite the Monstrous-TYS triggers (0 extra life)
Mirrorworks trigger for Vedalken Orrery
group of Metallurgeon-BM triggers (3 extra life)
Maybe other abilities
Get the token Vedalken Orrery, and now we can again prepare things. And we have 3 life.
Top
group of Smite the Monstrous-PB triggers (partly consumed)
group of Smite the Monstrous-TYS triggers (3 extra life)
Yes, we use Spellweaver Helix to cast a copy of the Worldfire card. The intention is to do it without Thousand-Year Storm, and so we do have to make sure we can get enough from the beneficial sorcery. This no Thousand-Year Storm requirement is actually somewhat useful, since it prevents some shenanigans with having a copy of Worldfire higher on the stack, and then putting other sorceries beneath it.
Yeah, Flash is definitely a dangerous card, and we really have to watch for problems if we try to put it in a deck. If it's in one of the deck lists, it's probably there as a "let's check if this can work" kind of thing. Thanks for pointing out the infinite with Child of Alara.
However, there might be a problem. When first Spellweaver Helix trigger resolve, we cast a copy of Worldfire. Does it count as "whenever a player casts a card"?
I hadn't seen that, thanks.
Just skimming through, it appears to allow for arbitrarily large integers to be encoded, (and operation requires many turns) so growth rate doesn't apply in the same way as our challenge.
The competitive programming of the machine is interesting, though I don't think its possible (or profitable) to implement in our deck.
Yeah, that paper makes the mate-in-n problem one of high computational complexity, so it's not immediately related to fast-growing functions.
However, we could try to make something similar, in terms of creating a setup that can evaluate not just computable functions, but arithmetic sentences with alternating quantifiers. The more alternating quantifiers we can implement, the higher order busy beaver we could theoretically evaluate.
Of course, the MtG representation of arithmetic sentences in the paper required multiple turns, so that doesn't work with our turn 1 requirement. Also, we have to be careful of never being able to go infinite, something the paper doesn't have to worry about.
But, perhaps there's some setup that we could come up with for higher order Busy Beavers. I suspect this may be hard under current rules, with a deck invalidated if there are game paths that have arbitrarily high damage. Something like what I proposed earlier, where a deck is valid if the score for the deck is finite (this is different because the score presumes the opponent acts to reduce our damage as much as possible.) , might be more amenable to alternating quantifiers. I was thinking the alternating quantifiers could be represented by alternating decisions by the two players, one trying to maximize the damage, the other trying to minimize it.
(I know you guys rejected that rule change before, but in this case I'm just proposing it as an alternate ruleset with an eye towards achieving higher order Busy Beavers; the original challenge will still remain.)
So yeah, this is something we can think about, but I also want to continue to work towards finishing our current Busy Beaver deck.
Hmm, thats a good point about the rules change. Might be worth revisiting if there's actually a way to implement alternating quantifiers like in the paper, but it seems difficult to arrange. (Maybe with Hive mind? But that lets them cast all sorts of other spells we don't want them to...)
Edit: as for the current deck, I think we are still stuck on making the Worldfire transition work properly.
Yeah, the paper had the benefit that the "mate-in-n" problem was for a particular point in a game, and before that point you could set things up so that there were only n choices to make before the game end. I thought for a moment about the idea of measuring the score not just for a deck, but for a deck and a point in the game, like the paper - but that doesn't work, since if a deck can create arbitrarily many hasted creatures, then you can have different snapshots with N attacking monsters for infinitely many N.
For the Worldfire transition, I posted earlier about why I thought that Recoup under Spellweaver Helix could be okay. If we spend a red mana on Flash of Defiance, and we don't wind up resolving a Worldfire, then we don't get the red mana back, so we spent a red in the process. If we resolve a Recoup after a Worldfire, there's no graveyard left, so it doesn't do anything. If we resolve a Recoup before a Worldfire, that's not a problem, since any benefit from the Recoup is above the hyperstage transition. Is there something wrong with this reasoning?
Jin-Gitaxias, Progress Tyrant looks intriguing, posslbly substituting for 3 cards. But yeah, the triggering once per turn means we have to refresh its numbers fast enough, which may be difficult. We do have Soul Separator to keep making copies each time we resolve Pull From Eternity, but that doesn't seem to be fast enough for artifact copying. Still, seems like something to keep an eye on.
Hmm, yeah the recoup line works, its just not as deckspace efficient as before. (with something like Bala ged scorpion -> cruel cut? oops thats arena only krovikan rot? ) I was thinking there was some problem with flash of defiance's life payment being avoided with recoup? But it doesn't seem to be a problem.
Also the channel creatures appear to all be artifacts/enchantments which ruins my hopeful line with a black channeler.
Because of Harmonic prodigy, we only need to remake one Jin-Gitaxias at a time. Though we would need more than wrong turn for the cephalid shrine ability, as the opponent can legend rule the fresh one and keep the used one. Still seems like the timing is really awkward.
Using the wandering emperor probably requires more shenanigans than we can accommodate here.
I'm unsure whether Krovikan Rot is safe - it's an instant that we can return to our hand. But, it can only destroy a creature with power 2 or less, so it can only target a 2/2 creature that is not boosted via Coat of Arms by there being any other creature with the same creature type. Such a creature will die the next time Arcbond deals 3 damage to all but the triggering creature, but if we can preempt that with Krovikan Rot, we can have that 2/2 creature die prematurely, possibly causing more creatures to be created prematurely. So, it does look like we can affect the computation after an arbitrary amount of time, so long as we can create lone creatures of a creature type after an arbitrary amount of time, which certainly looks possible.
Edit: So, it looks like we can fix the problem of gaining life through the hyperstage transition by switching from double Recoup to double Flash of Defiance. That requires two Flash of Defiances and a Recoup, so we add an addtional card. But, paying 3 life is too much when we only gain 2 life from discarding Aegis Automaton. Adding another Aegis Automaton doesn't quite work out, since we can use all the TYS copies of Pull from Eternity to bring back Stern Proctor and create copies using Soul Separator, allowing us to bounce both copies of Aegis Automaton, discard one for life, cast the other and counter it to bring both back to the battlefield via Bloodbond March, and bounce them again.
So, we would want to add a different artifact creature. Adding one more gets us 3 life per Pull From Eternity cast, since we can get 2 life from whatever artifact creature we use as our stage creature, and 2 life every other cast for the other artifact creature. That matches the 3 life from Flash of Defiance, but doesn't work out with K'rrik, Son of Yawgmoth costing us 2 life per black mana. Adding another artifact creature could work; we gain 4 life in the hyperstage transition, going up to 5 life after Worldfire, but all we can do with the 5 life is one cast of Flash of Defiance, with the extra life being useless. So that seems to work. But, we add 1 sorcery and 2 artifact creatures, increasing our card count by 3.
A better option seems to be to use the recent card Hungry for More, which can be flashbacked for a red mana and a black mana, the latter of which we can pay for with 2 life. So that's exactly what we need to counter the 2 life gain. Again, we need 2 Hungry for More's and a Recoup, so we add another card, but no need to add any artifact creatures. So our card count goes up by 1.
which should get us to BB_{w^2 + w2 + 10} (BB_{w^2 + w2 + 10} (X)), barring mistakes.
Another idea is to take out K'rrik, and use Acorn Harvest for our Life stage sorcery. That saves a card with K'rrik, but we would add a card for the Flash of Defiance setup, and another card for another artifact creature to gain 3 life per sorcery cast. So it's again 1 more card slot, but we've mooted green mana, so it doesn't seem to be an improvement.
Edit: Hmm, it doesn't seem like there is a problem with adding Stakfish's additional stage using Surgespanner, so what about:
Right right. Okay, we can instead use just a sorcery that costs a red and a black mana, and use Recoup to give it flashback. I'd like to just use a sorcery that costs a black mana, like Essence Harvest, but the loophole that FortyTwo pointed out of having Recoup get cast from underneath Spellweaver Helix seems to make that not work. And unfortunately, it looks like the loophole doesn't work if we do our normal procedure of casting Worldfire, since Worldfire will get rid of our graveyard, preventing us from keeping a sorcery that we give flashback to. So, we can use a red and black spell like Blightning, but we have to get an extra red mana in our hyperstage transition, which I guess means two lands.
hopefully getting us to BB_{w^2 + w3 + 4} (X), where X is around 50.
Edit: I guess we don't really need Marchesa's Smuggler if we aren't going to use combat for the layers, since Essence Harvest can make them lose a lot of life. (I suppose Gravitic Punch is perhaps preferable to Essence Harvest, as damage was the measure in the original version of the challenge, so perhaps we should choose damage if there is no reason not to.) Taking out Marchesa's Smuggler leaves us short a blue creature though, so we have to add a blue creature. Floating-Dream Zubera is the best that I can think of:
So the initial usage of Floating-Dream Zubera should get us to BB_{w^2 + w3 + 3}(X) where X is almost 60, then 53 ETBs of Enlightened Sphinx should get us to more than BB_{w^2 + w3 + 4} (54).
I think we might want like thought hemorrhage instead of blightning to more easily keep the opponent alive? Though I'm not sure its needed.
So to refresh myself and to check the operations:
Use a few Spellweaver volute triggers setup opponent with BB waterclocks (they have Coat of arms, arcbonded k'rrik, some crusades, many necromancers and X zombies)
Play real copies of the artifacts and enchantments, each in response to their Dual Nature triggers. Cast death pit offering, in response to the trigger soul separator on stern proctor to bounce our real cards.
death pit offering kills our goblin boom keg, march of the machines and 2 opalescence, so only creatures are animated for the computation that just started.
Opponent ends computation with BB(X) zombies, we resolve dual nature triggers.
Repeat until out of spellweaver volute triggers.
Cheapest sorcery is Lingering Souls, which costs black or 2 life.
We can gain life by discarding Aegis Automaton to Draconian Cylix and regenerate Vampire Envoy from Bala Ged Scorpion
However to get AA back in hand from the grave we need to use a Bloodbond march trigger for it and then make a Stern Proctor to bounce it which needs pull from eternity to reuse soul separator. (the way the stack works we still also need Alhammarret's Archive to double the life gain.)
This makes the black/life stage, its a bit more complicated than before but still a stage.
Eventually we get low on life, and want to gravitic punch ourselves to gain life.
We do this by spending red to flashback recoup on Blightning and using that to trigger spellweaver helix with the punch on it. the helix also has worldfire, which is important to allow us to replay the lands via restore also from the helix.
We can nest these until we run out of red, and the worldfires required for rebuilding prevent us from smuggling life.
Do we still need blood moon? can those just be 2x mountain? or even Graven Cairns?
I believe the Blood Moon was there to stop being able to start the hyperstage transition with TYSes, and then resolve loops of Worldfire -> Pull from Eternity -> Restore to keep recycling a red mana land. Our hope was that having multiple Worldfires exile our hand kept us from safely doing this, but Iijil pointed out that we could cast our cards and have them sitting on the stack while this happened. I tried to think of a flaw in this reasoning, but I didn't see one.
Should get BB_{w^2 + w3 + 4} (BB_{w^2 + w3 + 3} (BB_{w^2 + w3 + 2} (X))) where X is almost 60.
Edit: Hmm, we can save a card by replacing one of the lands with Academy Ruins, allowing us to drop [c]Drafna's Restoration. This allows us to get 5 layers:
Making the follow up radiated rite of replication generate... well a number I'm not even able to comprehend. Effectively each radiated copy spell would create a nested exponential for as many permanents as there are in play. myfiosgateway.com mobdro
Going with Radiant Fountain does lose us our access to green later on, so perhaps Call to the Netherworld would be better? Oh, but we need to gain life first without paying it, so rather than Urborg Syphon-Mage, we would need to use something else like the Draconian Cylix / Vampire Envoy combo. I think we can't gain life from discarding Vampire Envoy, since we need to have one on the battlefield to gain life. But, we can gain life from Xathrid Necromancer and K'rrik, Son of Yawgmoth, but only every other transition; we need to use one transition to cast them for Bloodbond March triggers, and then in the next transition we can discard them for life. So, maybe we need 4 black creatures besides Vampire Envoy to gain 4 life each transition? So we would need one more black creature, if my thinking is correct.
Edit: Or wait, maybe we don't need Bloodbond March triggers if we can put them into play with Soul Separator. So K'rrik and Necromancer is enough for 4 life each transition, and we need to replace Bala Ged Scorpion with something that isn't a black creature.
How to get permanents on field after Child of Alara and Worldfire wipe everything? And what kind of preparation should we do before them?
Worldfire involves more bouncing and using Wormfang behemoth to hide our hand. I think some versions needed to get the real wormfang under a token wormfang which was tricky but possible.
Sorry if that's a bit too handwave-y, I know that there were some other quirks like making the first token omniscience requiring some initial inefficiency.
I'm still kinda stumped on making the worldfire transition work the optimal way we want it to in the current decks, one hopeful thought is that channel (the mechanic not the card) is returning in the new kamigawa, So there's some potential for a black Ghost-Lit Redeemer style card.
Generally, Thousand-Year Storm has 3 usages:
If we simply cast a Worldfire without TYS trigger, then the card is on the stack. We cannot reuse Worldfire card to make multiple transitions on the stack.
If we use Swarm Intelligence instead of Thousand-Year Storm, we can exactly 1 Worldfire copy, but we cannot keep large number on it and Child of Alara.
Edit:
Oh I didn't noticed that: Spellweaver Helix casts a copy of a card. So with absence of Thousand-Year Storm we can put 1 copy of Worldfire on stack.
However, for the decks using 1 Worldfire plus 3 flashback sorcery cards, or 1 Worldfire plus 2 flashback sorcery cards plus Panharmonicon, Worldfire and another sorcery cast at the same time. Without Thousand-Year Storm (the requirement for only 1 Worldfire copy), there is only 1 copy of the "another sorcery".
More detailed, suppose we have Child of Alara, Goblin Dark-Dwellers, Vedalken Orrery and its token copy on board. We cannot spend life anymore. The stack looks like:
Play Vedalken Orrery, getting its Mirrorworks trigger.
Flash Child of Alara, getting its dying trigger.
Usually the next Mirrorworks trigger for Vedalken Orrery should be at the top of "3 extra life" block. But here, since there are some space between targeting and killing Child of Alara, we meet the trigger earlier:
Yeah, Flash is definitely a dangerous card, and we really have to watch for problems if we try to put it in a deck. If it's in one of the deck lists, it's probably there as a "let's check if this can work" kind of thing. Thanks for pointing out the infinite with Child of Alara.
Flashback Spider Spawning, which triggers the Spellweaver Helix imprinting another Spider Spawning and a Worldfire. We manage to take Thousand-Year Storm off, then the trigger resolve to cast a Worldfire copy.
This triggers another Spellweaver Helix imprinting another Worldfire and the life gain card (Multani's Decree). We put Thousand-Year Storm on, then the trigger resolve to cast the Multani's Decree, getting its TYS trigger.
However, there might be a problem. When first Spellweaver Helix trigger resolve, we cast a copy of Worldfire. Does it count as "whenever a player casts a card"?
How high can it (non-arithmetic) be, in terms of growth rate?
Just skimming through, it appears to allow for arbitrarily large integers to be encoded, (and operation requires many turns) so growth rate doesn't apply in the same way as our challenge.
The competitive programming of the machine is interesting, though I don't think its possible (or profitable) to implement in our deck.
However, we could try to make something similar, in terms of creating a setup that can evaluate not just computable functions, but arithmetic sentences with alternating quantifiers. The more alternating quantifiers we can implement, the higher order busy beaver we could theoretically evaluate.
Of course, the MtG representation of arithmetic sentences in the paper required multiple turns, so that doesn't work with our turn 1 requirement. Also, we have to be careful of never being able to go infinite, something the paper doesn't have to worry about.
But, perhaps there's some setup that we could come up with for higher order Busy Beavers. I suspect this may be hard under current rules, with a deck invalidated if there are game paths that have arbitrarily high damage. Something like what I proposed earlier, where a deck is valid if the score for the deck is finite (this is different because the score presumes the opponent acts to reduce our damage as much as possible.) , might be more amenable to alternating quantifiers. I was thinking the alternating quantifiers could be represented by alternating decisions by the two players, one trying to maximize the damage, the other trying to minimize it.
(I know you guys rejected that rule change before, but in this case I'm just proposing it as an alternate ruleset with an eye towards achieving higher order Busy Beavers; the original challenge will still remain.)
So yeah, this is something we can think about, but I also want to continue to work towards finishing our current Busy Beaver deck.
Edit: as for the current deck, I think we are still stuck on making the Worldfire transition work properly.
Edit2: neo Kamigawa has a couple interesting cards previewed so far,
the saga/creatures we can flip with proliferate,
The wandering emperor gives us our first fully instant speed walker.
Jin-gitaxias, progress tyrant is a weird swarm intelligence/mirrorworks/cephalid shrine mixture. Only triggering once per turn is sad though.
The reality chip + a card like leonin shikari gives a blue gated way to cast off the top.
And I'm still waiting for a black channel creature that drains life, Which I think fixes our whole trouble with the worldfire transition line.
For the Worldfire transition, I posted earlier about why I thought that Recoup under Spellweaver Helix could be okay. If we spend a red mana on Flash of Defiance, and we don't wind up resolving a Worldfire, then we don't get the red mana back, so we spent a red in the process. If we resolve a Recoup after a Worldfire, there's no graveyard left, so it doesn't do anything. If we resolve a Recoup before a Worldfire, that's not a problem, since any benefit from the Recoup is above the hyperstage transition. Is there something wrong with this reasoning?
Jin-Gitaxias, Progress Tyrant looks intriguing, posslbly substituting for 3 cards. But yeah, the triggering once per turn means we have to refresh its numbers fast enough, which may be difficult. We do have Soul Separator to keep making copies each time we resolve Pull From Eternity, but that doesn't seem to be fast enough for artifact copying. Still, seems like something to keep an eye on.
The Wandering Emperor doesn't seem to have abilities that we can use, but The Reality Chip and the flipping sagas are also interesting.
Also the channel creatures appear to all be artifacts/enchantments which ruins my hopeful line with a black channeler.
Because of Harmonic prodigy, we only need to remake one Jin-Gitaxias at a time. Though we would need more than wrong turn for the cephalid shrine ability, as the opponent can legend rule the fresh one and keep the used one. Still seems like the timing is really awkward.
Using the wandering emperor probably requires more shenanigans than we can accommodate here.
Edit: So, it looks like we can fix the problem of gaining life through the hyperstage transition by switching from double Recoup to double Flash of Defiance. That requires two Flash of Defiances and a Recoup, so we add an addtional card. But, paying 3 life is too much when we only gain 2 life from discarding Aegis Automaton. Adding another Aegis Automaton doesn't quite work out, since we can use all the TYS copies of Pull from Eternity to bring back Stern Proctor and create copies using Soul Separator, allowing us to bounce both copies of Aegis Automaton, discard one for life, cast the other and counter it to bring both back to the battlefield via Bloodbond March, and bounce them again.
So, we would want to add a different artifact creature. Adding one more gets us 3 life per Pull From Eternity cast, since we can get 2 life from whatever artifact creature we use as our stage creature, and 2 life every other cast for the other artifact creature. That matches the 3 life from Flash of Defiance, but doesn't work out with K'rrik, Son of Yawgmoth costing us 2 life per black mana. Adding another artifact creature could work; we gain 4 life in the hyperstage transition, going up to 5 life after Worldfire, but all we can do with the 5 life is one cast of Flash of Defiance, with the extra life being useless. So that seems to work. But, we add 1 sorcery and 2 artifact creatures, increasing our card count by 3.
A better option seems to be to use the recent card Hungry for More, which can be flashbacked for a red mana and a black mana, the latter of which we can pay for with 2 life. So that's exactly what we need to counter the 2 life gain. Again, we need 2 Hungry for More's and a Recoup, so we add another card, but no need to add any artifact creatures. So our card count goes up by 1.
So, we could try the following:
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 Death Pit Offering
10 Goblin Boom Keg
11 March of the Machines
Utility: Enchantments/Artifacts
12 Vedalken Orrery
13 Dual Nature
14 Copy Enchantment
15 Stern Proctor
Utility: Spells
16 Thousand-Year Storm
17 Spellweaver Volute
18 Pull from Eternity
Utility: Creatures
19 Bloodbond March
20 Cephalid Shrine
21 Soul Separator
22 Mana Echoes
23 Draconian Cylix
24 Vampire Envoy
25 Alhammarret's Archive
26 Bala Ged Scorpion
27 K'rrik, Son of Yawgmoth
28 Lingering Souls
R Transition
29 Spellweaver Helix
30 Worldfire
31 Essence Harvest
32 Hungry for More
33 Hungry for More
34 Restore
35 Blood Moon
36 Panharmonicon
37 Wormfang Behemoth
U stage
38 Stolen By the Fae
39 Recoup
40 Chrome Mox
W stage
41 Guardian of the Guildpact
42 Aegis Automaton
43 Illusionist's bracers
44 Magnetic Theft
45 Black Lotus
46 Show and Tell
47 Omniscience
48 Floating-Dream Zubera
49 Marchesa's Smuggler
Layers
50 Crucible of the Spirit Dragon
51 Amulet of Vigor
52 Animist's Awakening
53 Angelheart Vial
54 Justice
55 Brallin, Skyshark Rider
56 Pain Magnification
57 Moraug, Fury of Akoum
Misc utility
58 Doubling Season
59 Drafna's Restoration
60 Uproot
which should get us to BB_{w^2 + w2 + 10} (BB_{w^2 + w2 + 10} (X)), barring mistakes.
Another idea is to take out K'rrik, and use Acorn Harvest for our Life stage sorcery. That saves a card with K'rrik, but we would add a card for the Flash of Defiance setup, and another card for another artifact creature to gain 3 life per sorcery cast. So it's again 1 more card slot, but we've mooted green mana, so it doesn't seem to be an improvement.
Edit: Hmm, it doesn't seem like there is a problem with adding Stakfish's additional stage using Surgespanner, so what about:
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 Death Pit Offering
10 Goblin Boom Keg
11 March of the Machines
Utility: Enchantments/Artifacts
12 Vedalken Orrery
13 Dual Nature
14 Copy Enchantment
15 Stern Proctor
Utility: Spells
16 Thousand-Year Storm
17 Spellweaver Volute
18 Pull from Eternity
Utility: Creatures
19 Bloodbond March
20 Cephalid Shrine
21 Soul Separator
22 Mana Echoes
23 Draconian Cylix
24 Vampire Envoy
25 Alhammarret's Archive
26 Bala Ged Scorpion
27 K'rrik, Son of Yawgmoth
28 Lingering Souls
R Transition
29 Spellweaver Helix
30 Worldfire
31 Essence Harvest
32 Hungry for More
33 Hungry for More
34 Restore
35 Blood Moon
36 Panharmonicon
37 Wormfang Behemoth
U stage
38 Stolen By the Fae
39 Recoup
40 Chrome Mox
G stage
41 Surgespanner
42 Storm Front
43 Shiv's Embrace
44 Karoo Meerkat
45 Karoo Meerkat
46 Guardian of the Guildpact
47 Aegis Automaton
48 Illusionist's bracers
49 Magnetic Theft
Initial setup
50 Black Lotus
51 Show and Tell
52 Omniscience
53 Floating-Dream Zubera
54 Marchesa's Smuggler
Layers
55 Blue Sun's Zenith
56 Angelheart Vial
57 Justice
58 Moraug, Fury of Akoum
Misc utility
59 Doubling Season
60 Drafna's Restoration
hopefully getting us up to BB_{w^2 + w3 + 5} (BB_{w^2 + w3 + 5} (X)).
Hungry for more unfortunately is a source of haste and therefore combat and arcbond triggers.
So:
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 Death Pit Offering
10 Goblin Boom Keg
11 March of the Machines
Utility: Enchantments/Artifacts
12 Vedalken Orrery
13 Dual Nature
14 Copy Enchantment
15 Stern Proctor
Utility: Spells
16 Thousand-Year Storm
17 Spellweaver Volute
18 Pull from Eternity
Utility: Creatures
19 Bloodbond March
20 Cephalid Shrine
21 Soul Separator
22 Mana Echoes
23 Draconian Cylix
24 Vampire Envoy
25 Alhammarret's Archive
26 Bala Ged Scorpion
27 K'rrik, Son of Yawgmoth
28 Lingering Souls
R Transition
29 Spellweaver Helix
30 Worldfire
31 Essence Harvest
32 Blightning
33 Blightning
34 Recoup
35 Restore
36 Blood Moon
37 Crucible of the Spirit Dragon
38 Crucible of the Spirit Dragon
39 Panharmonicon
40 Wormfang Behemoth
U stage
41 Distorting Wake
42 Chrome Mox
43 Surgespanner
44 Storm Front
45 Shiv's Embrace
46 Karoo Meerkat
47 Karoo Meerkat
W stage
48 Guardian of the Guildpact
49 Aegis Automaton
50 Illusionist's bracers
51 Magnetic Theft
Initial setup
52 Black Lotus
53 Show and Tell
54 Omniscience
55 Enlightened Sphinx
56 Marchesa's Smuggler
Layers
57 Blue Sun's Zenith
58 Consecrated Sphinx
Misc utility
59 Doubling Season
60 Drafna's Restoration
hopefully getting us to BB_{w^2 + w3 + 4} (X), where X is around 50.
Edit: I guess we don't really need Marchesa's Smuggler if we aren't going to use combat for the layers, since Essence Harvest can make them lose a lot of life. (I suppose Gravitic Punch is perhaps preferable to Essence Harvest, as damage was the measure in the original version of the challenge, so perhaps we should choose damage if there is no reason not to.) Taking out Marchesa's Smuggler leaves us short a blue creature though, so we have to add a blue creature. Floating-Dream Zubera is the best that I can think of:
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 Death Pit Offering
10 Goblin Boom Keg
11 March of the Machines
Utility: Enchantments/Artifacts
12 Vedalken Orrery
13 Dual Nature
14 Copy Enchantment
15 Stern Proctor
Utility: Spells
16 Thousand-Year Storm
17 Spellweaver Volute
18 Pull from Eternity
Utility: Creatures
19 Bloodbond March
20 Cephalid Shrine
21 Soul Separator
22 Mana Echoes
23 Draconian Cylix
24 Vampire Envoy
25 Alhammarret's Archive
26 Bala Ged Scorpion
27 K'rrik, Son of Yawgmoth
28 Lingering Souls
R Transition
29 Spellweaver Helix
30 Worldfire
31 Gravitic Punch
32 Blightning
33 Blightning
34 Recoup
35 Restore
36 Blood Moon
37 Crucible of the Spirit Dragon
38 Crucible of the Spirit Dragon
39 Panharmonicon
40 Wormfang Behemoth
U stage
41 Distorting Wake
42 Chrome Mox
43 Surgespanner
44 Storm Front
45 Shiv's Embrace
46 Karoo Meerkat
47 Karoo Meerkat
W stage
48 Guardian of the Guildpact
49 Aegis Automaton
50 Illusionist's bracers
51 Magnetic Theft
Initial setup
52 Black Lotus
53 Show and Tell
54 Omniscience
55 Enlightened Sphinx
56 Floating-Dream Zubera
Layers
57 Blue Sun's Zenith
58 Consecrated Sphinx
Misc utility
59 Doubling Season
60 Drafna's Restoration
So the initial usage of Floating-Dream Zubera should get us to BB_{w^2 + w3 + 3}(X) where X is almost 60, then 53 ETBs of Enlightened Sphinx should get us to more than BB_{w^2 + w3 + 4} (54).
Edit: We can a little better by switching in Sphinx of the Second Sun and World at War for card draws:
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 Death Pit Offering
10 Goblin Boom Keg
11 March of the Machines
Utility: Enchantments/Artifacts
12 Vedalken Orrery
13 Dual Nature
14 Copy Enchantment
15 Stern Proctor
Utility: Spells
16 Thousand-Year Storm
17 Spellweaver Volute
18 Pull from Eternity
Utility: Creatures
19 Bloodbond March
20 Cephalid Shrine
21 Soul Separator
22 Mana Echoes
23 Draconian Cylix
24 Vampire Envoy
25 Alhammarret's Archive
26 Bala Ged Scorpion
27 K'rrik, Son of Yawgmoth
28 Lingering Souls
R Transition
29 Spellweaver Helix
30 Worldfire
31 Gravitic Punch
32 Blightning
33 Blightning
34 Recoup
35 Restore
36 Blood Moon
37 Crucible of the Spirit Dragon
38 Crucible of the Spirit Dragon
39 Panharmonicon
40 Wormfang Behemoth
U stage
41 Distorting Wake
42 Chrome Mox
43 Surgespanner
44 Storm Front
45 Shiv's Embrace
46 Karoo Meerkat
47 Karoo Meerkat
W stage
48 Guardian of the Guildpact
49 Aegis Automaton
50 Illusionist's bracers
51 Magnetic Theft
Initial setup
52 Black Lotus
53 Show and Tell
54 Omniscience
55 Floating-Dream Zubera
Layers
56 Blue Sun's Zenith
57 Sphinx of the Second Sun
58 World at War
Misc utility
59 Doubling Season
60 Drafna's Restoration
Still 4 layers, but up to BB_{w^2 + w3 + 4} (BB_{w^2 + w3 + 3} (X)) where X is almost 60.
So to refresh myself and to check the operations:
Use a few Spellweaver volute triggers setup opponent with BB waterclocks (they have Coat of arms, arcbonded k'rrik, some crusades, many necromancers and X zombies)
Play real copies of the artifacts and enchantments, each in response to their Dual Nature triggers. Cast death pit offering, in response to the trigger soul separator on stern proctor to bounce our real cards.
death pit offering kills our goblin boom keg, march of the machines and 2 opalescence, so only creatures are animated for the computation that just started.
Opponent ends computation with BB(X) zombies, we resolve dual nature triggers.
Repeat until out of spellweaver volute triggers.
Cheapest sorcery is Lingering Souls, which costs black or 2 life.
We can gain life by discarding Aegis Automaton to Draconian Cylix and regenerate Vampire Envoy from Bala Ged Scorpion
However to get AA back in hand from the grave we need to use a Bloodbond march trigger for it and then make a Stern Proctor to bounce it which needs pull from eternity to reuse soul separator. (the way the stack works we still also need Alhammarret's Archive to double the life gain.)
This makes the black/life stage, its a bit more complicated than before but still a stage.
Eventually we get low on life, and want to gravitic punch ourselves to gain life.
We do this by spending red to flashback recoup on Blightning and using that to trigger spellweaver helix with the punch on it. the helix also has worldfire, which is important to allow us to replay the lands via restore also from the helix.
We can nest these until we run out of red, and the worldfires required for rebuilding prevent us from smuggling life.
Do we still need blood moon? can those just be 2x mountain? or even Graven Cairns?
Edit: Another minor improvement, subbing in Moraug, Fury of Akoum to lead into Arbiter of the Ideal:
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 Death Pit Offering
10 Goblin Boom Keg
11 March of the Machines
Utility: Enchantments/Artifacts
12 Vedalken Orrery
13 Dual Nature
14 Copy Enchantment
15 Stern Proctor
Utility: Spells
16 Thousand-Year Storm
17 Spellweaver Volute
18 Pull from Eternity
Utility: Creatures
19 Bloodbond March
20 Cephalid Shrine
21 Soul Separator
22 Mana Echoes
23 Draconian Cylix
24 Vampire Envoy
25 Alhammarret's Archive
26 Bala Ged Scorpion
27 K'rrik, Son of Yawgmoth
28 Lingering Souls
R Transition
29 Spellweaver Helix
30 Worldfire
31 Gravitic Punch
32 Blightning
33 Blightning
34 Recoup
35 Restore
36 Blood Moon
37 Crucible of the Spirit Dragon
38 Crucible of the Spirit Dragon
39 Panharmonicon
40 Wormfang Behemoth
U stage
41 Distorting Wake
42 Chrome Mox
43 Surgespanner
44 Storm Front
45 Shiv's Embrace
46 Karoo Meerkat
47 Karoo Meerkat
W stage
48 Guardian of the Guildpact
49 Aegis Automaton
50 Illusionist's bracers
51 Magnetic Theft
Initial setup
52 Black Lotus
53 Show and Tell
54 Omniscience
55 Floating-Dream Zubera
Layers
56 Arbiter of the Ideal
57 Moraug, Fury of Akoum
58 World at War
Misc utility
59 Doubling Season
60 Drafna's Restoration
Should get BB_{w^2 + w3 + 4} (BB_{w^2 + w3 + 3} (BB_{w^2 + w3 + 2} (X))) where X is almost 60.
Edit: Hmm, we can save a card by replacing one of the lands with Academy Ruins, allowing us to drop [c]Drafna's Restoration. This allows us to get 5 layers:
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 Death Pit Offering
10 Goblin Boom Keg
11 March of the Machines
Utility: Enchantments/Artifacts
12 Vedalken Orrery
13 Dual Nature
14 Copy Enchantment
15 Stern Proctor
Utility: Spells
16 Thousand-Year Storm
17 Spellweaver Volute
18 Pull from Eternity
Utility: Creatures
19 Bloodbond March
20 Cephalid Shrine
21 Soul Separator
22 Mana Echoes
23 Draconian Cylix
24 Vampire Envoy
25 Alhammarret's Archive
26 Bala Ged Scorpion
27 K'rrik, Son of Yawgmoth
28 Lingering Souls
R Transition
29 Spellweaver Helix
30 Worldfire
31 Gravitic Punch
32 Blightning
33 Blightning
34 Recoup
35 Restore
36 Blood Moon
37 Crucible of the Spirit Dragon
38 Academy Ruins
39 Panharmonicon
40 Wormfang Behemoth
U stage
41 Distorting Wake
42 Chrome Mox
43 Surgespanner
44 Storm Front
45 Shiv's Embrace
46 Karoo Meerkat
47 Karoo Meerkat
W stage
48 Guardian of the Guildpact
49 Aegis Automaton
50 Illusionist's bracers
51 Magnetic Theft
Initial setup
52 Black Lotus
53 Show and Tell
54 Omniscience
55 Enlightened Sphinx
Layers
56 Arbiter of the Ideal
57 Awakening
58 Sphinx of the Second Sun
59 World at War
Misc utility
60 Doubling Season
This should get our damage up to BB_{w^2 + w3 + 5} (BB_{w^2 + w3 + 2} (X)).
Something like Thousand winds is maybe along the right lines?
Another small improvement: replacing Arbiter of the Ideal and Awakening with Blue Sun's Zenith and Midnight Oil:
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 Death Pit Offering
10 Goblin Boom Keg
11 March of the Machines
Utility: Enchantments/Artifacts
12 Vedalken Orrery
13 Dual Nature
14 Copy Enchantment
15 Stern Proctor
Utility: Spells
16 Thousand-Year Storm
17 Spellweaver Volute
18 Pull from Eternity
Utility: Creatures
19 Bloodbond March
20 Cephalid Shrine
21 Soul Separator
22 Mana Echoes
23 Draconian Cylix
24 Vampire Envoy
25 Alhammarret's Archive
26 Bala Ged Scorpion
27 K'rrik, Son of Yawgmoth
28 Lingering Souls
R Transition
29 Spellweaver Helix
30 Worldfire
31 Gravitic Punch
32 Thought Hemorrhage
33 Thought Hemorrhage
34 Recoup
35 Restore
36 Blood Moon
37 Crucible of the Spirit Dragon
38 Academy Ruins
39 Panharmonicon
40 Wormfang Behemoth
U stage
41 Distorting Wake
42 Chrome Mox
43 Thousand Winds
44 Storm Front
45 Shiv's Embrace
46 Karoo Meerkat
47 Karoo Meerkat
W stage
48 Guardian of the Guildpact
49 Aegis Automaton
50 Illusionist's bracers
51 Magnetic Theft
Initial setup
52 Black Lotus
53 Show and Tell
54 Omniscience
55 Floating-Dream Zubera
Layers
56 Blue Sun's Zenith
57 Midnight Oil
58 Sphinx of the Second Sun
59 World at War
Misc utility
60 Doubling Season
This should get our damage up to BB_{w^2 + w3 + 5} (BB_{w^2 + w3 + 3} (X)).
Edit: Oh, I think can replace Alhammarret's Archive with a second Aegis Automaton, allowing us to use Mirror Gallery again. So, replace Illusionist's Bracers and Magnetic Theft with Kurkesh, Onakke Ancient and Mirror Gallery, allowing us to have multiple Legends at no cost in cards.
So, we go with:
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 Death Pit Offering
10 Goblin Boom Keg
11 March of the Machines
Utility: Enchantments/Artifacts
12 Vedalken Orrery
13 Dual Nature
14 Copy Enchantment
15 Stern Proctor
Utility: Spells
16 Thousand-Year Storm
17 Spellweaver Volute
18 Pull from Eternity
Utility: Creatures
19 Bloodbond March
20 Cephalid Shrine
21 Soul Separator
22 Mana Echoes
23 Draconian Cylix
24 Vampire Envoy
25 Aegis Automaton
26 Bala Ged Scorpion
27 K'rrik, Son of Yawgmoth
28 Lingering Souls
R Transition
29 Spellweaver Helix
30 Worldfire
31 Gravitic Punch
32 Thought Hemorrhage
33 Thought Hemorrhage
34 Recoup
35 Restore
36 Blood Moon
37 Crucible of the Spirit Dragon
38 Academy Ruins
39 Panharmonicon
40 Wormfang Behemoth
U stage
41 Distorting Wake
42 Chrome Mox
43 Thousand Winds
44 Storm Front
45 Shiv's Embrace
46 Karoo Meerkat
47 Karoo Meerkat
W stage
48 Guardian of the Guildpact
49 Aegis Automaton
50 Kurkesh, Onakke Ancient
51 Mirror Gallery
Initial setup
52 Black Lotus
53 Show and Tell
54 Omniscience
55 Floating-Dream Zubera
Layers
56 Blue Sun's Zenith
57 Tamiyo's Journal
58 Sphinx of the Second Sun
59 World at War
Misc utility
60 Doubling Season
getting us up to BB_{w^2 + w3 + 6} (BB_{w^2 + w3 + 3} (X)), where X is almost 60.
mobdro
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 Death Pit Offering
10 Goblin Boom Keg
11 March of the Machines
Utility: Enchantments/Artifacts
12 Vedalken Orrery
13 Dual Nature
14 Copy Enchantment
15 Stern Proctor
Utility: Spells
16 Thousand-Year Storm
17 Spellweaver Volute
18 Pull from Eternity
Utility: Creatures
19 Bloodbond March
20 Cephalid Shrine
21 Soul Separator
22 Mana Echoes
23 Urborg Syphon-Mage
24 Thousand-Year Elixir
25 Alhammarret's Archive
26 K'rrik, Son of Yawgmoth
27 Lingering Souls
R Transition
28 Spellweaver Helix
29 Worldfire
30 Gravitic Punch
31 Flash of Defiance
32 Flash of Defiance
33 Recoup
34 Restore
35 Environmental Sciences
36 Blood Moon
37 Academy Ruins
38 Panharmonicon
39 Wormfang Behemoth
U stage
40 Distorting Wake
41 Chrome Mox
G stage
42 Thousand Winds
43 Storm Front
44 Shiv's Embrace
45 Karoo Meerkat
46 Karoo Meerkat
47 Guardian of the Guildpact
48 Aegis Automaton
49 Kurkesh, Onakke Ancient
50 Bolt Bend
Initial setup
51 Black Lotus
52 Show and Tell
53 Omniscience
54 Floating-Dream Zubera
55 Marchesa Smugger
Layers
56 Blue Sun's Zenith
57 Angelheart Vial
58 Justice
59 Moraug, Fury of Akoum
Misc utility
60 Doubling Season
getting us up to BB_{w^2 + w3 + 5} (BB_{w^2 + w3 + 5} (BB_{w^2 + w3 + 3} (X))) where X is almost 60.
Welcome RaulR and nastasiya09.