Ah, right. It specifies "card". I think that might actually break the old Chalice deck (again).
In any case, as long as we can make sure we can't meaningfully interweave the spells to get mana after a Worldpurge it should be fine. We can switch from Red as the hyperstage color to green for Acorn Harvest and Deconstruct, probably. That might even save slots.
I also had an idea for how to allow creatures with tap abilities, though we'd still have to be careful: if the only unpreventable means of dealing damage is Excruciator, and we don't give ourselves any way to make our opponent gain life, then to deal more than 20 damage we would have to deal it by attacking with a lot of Excruciators, which would take over the combat phase and not let us use that time to deal damage and kick-start the TM without killing Vedalken Orrery.
Giving the opponent life is probably okay, since FortyTwo decided to change the scoring to the negative of the opponent's final life, as opposed to damage dealt.
I haven't come to grips with your combo yet, but I figured I would mention an important issue: We need to update our resources to the results of the BB computation continually. It looks like you are doing that with Mana Echoes to generate colorless mana, I just wanted to make sure that the colorless mana was filtering to everywhere else.
Edit: It looks like your explanation got messed up a bit, there is a part which goes "which lets us That gives us a stage", suggesting that something got left out.
Edit: Hmm, we can switch the hyperstage color over to green, but we would still need Inner Fire to get red mana for the stage. Unless we switch the stage over to green mana? I don't know how that would work with Desolation Giant.
I don't think Mana Echoes does enough: we rebuild our enchantments by resolving lots of copies of Satyr's Cunning, making lots and lots of tokens that can be Mirrorweaved into whatever enchantment we want. But that scales with red mana and instants/sorceries cast, which don't look like they keep up. I'm still quite shaky on the details of the Busy Beaver combo so that's probably not the last pitfall I've ran into with that list.
The issue with giving the opponent life isn't keeping it negative, it's making sure that the opponent dies to Arcbond triggers if we try to set off the loop via combat damage and not Goblin Boom Keg. If they can only get hit by Arcbond 7 times, that stops us from going infinite.
We've been updating using a buyback spell to update the Thousand-Year Storm count to the amount of colorless mana. Specifically, we've been using the sorcery Mystic Speculation so that it can recast Arcbond and Artificial Storm; if you want to avoid being able to recast those instants that cheaply, you can use an instant instead.
For the combat damage thing, it seems like it's the same whether then opponent has 20 life or BB(20); if the opponent keeps getting hit by Arcbond, he will eventually die after some fixed number if hits, so no infinite combo.
Edit: So, getting ahead of ourselves: What is the prospect for additional stages? The fact that Artificial Evolution and Arcbond can target any creature perhaps mean that further Bloodbond March stages are borked. But, there are other ways to make stages, as we have seen in our Standard decks. With the full collection of Vintage cards at our disposal, making similar stages should be easier?
That's about as far as a strategy like that can get, since there aren't any UW bounce spells (besides Time Wipe and Worldpurge, which bring their own problems), but 3 stages is still better than 0 and it fits fairly easily into the already-existing framework.
Stakfish is there anything in particular that is troubling you with the Turing Machine Combo?
Also, scoring by final negative life makes the most sense, as that is the final state of the game, is compatible with all previous versions of the deck, and also covers life loss. (We want Goblin Game to not work) "Most damage" is a bit ambiguous and disqualifies a deck that can infinitely cast lightning bolt and healing salve, but not one that casts lightning bolt and dromoka's command, even though both only get the opponent to -1.
However, the trick is to put those sorceries back into the library, without being able to have instants like Artificial Evolution and Mirrorweave in our hands during a computation.
Re negative life: It's not so much the principle of the change I'm worried about, it's the ability to use combat damage to start the computation instead of killing Goblin Boom Keg, which would let us keep Vedalken Orrery out mid-computation and potentially interrupt a non-halting execution for arbitrary damage. If we keep the conceit that the opponent will die if dropped to 0 life, then we can't deal arbitrary damage in a computation like that because they'd die before we could get arbitrary value out of it.
Re the combo: I mostly get the execution, but I'm not that well practiced with it so I'm unclear on what outputs it gives us on its own when we run it. I also have no idea about the math behind waterclocks, etc., so I'm slightly hazy on the details of how big we're getting (I assume from the fact we're iterating on it our number of states isn't capped by unique creature types, but I don't and may never understand exactly WHY).
Re the hyperstage: it needs some fixes, and in an edit in a few minutes I'll put the new version in this comment. Primarily, it needs a way to turn mana from Mana Echoes and/or generic computation output into Thousand-Year Storm count, and switch to green mana. Beyond that, I'm pretty sure it actually works
Re the layers: Heliod's Intervention can be used to kill a Goblin Boom Keg without killing Vedalken Orrery, allowing us to potentially interrupt a non-halting computation and deal arbitrary damage. Talon of Pain, however, can't, which is cool: we can only activate it when neither card is a creature, and with no Psychic Battle we can't change the target after playing March of the Machines and loop it back to the Boom Keg. Stream of Life will work, but I think we have to use green mana in the hyperstage or we'll be able to first resolve Worldpurge, then resolve Inner Fire, leaving the hyperstage with 7 R left to work with.
Edit: Is Hive Mind too dangerous for the computation, or would we be okay with giving our opponents copies? If so, we could use Sanguine Sacrament to give our opponents the life, which tucks itself on bottom and would let us draw cards safely.
Looking good. I don't think we can use Recruit the Worthy, since it creates a Soldier token, pumping up Soldiers on the opponent's side. It may not be obvious how this can halt an infinite computation, but by pumping up the opponent's Soldiers, it can delay when those Soldiers die, and the Arcbonded creatures may be relying on tokens produced from the Soldiers' deaths to pump their own toughness, so they may wind up dying when they would ordinarily stay alive.
I was wondering whether we could get enough Seismic Spikes that actually resolve, since all the copies of Thousand-Year Storm go on at the same time. But, we can now create a gajillion Thousand-Year Storms, so that should be okay I think.
Heliod's Intervention was a mistake, it should have been Energy Ray. Using green mana for Stream of Life should be fine; the limiter is the fact that it goes to the graveyard, and we can't cast it again. So we get 8 castings of life gain spells. Of course, it would be even better if we could find some limited way to recycle those spells.
This is quite nice, but perhaps we would have more opportunties for additional stages if we found a hyperstage without Cowardice.
Recruit the Worthy can create creatures during a computation which is bad; but if its purpose is just for storm count then Change of Heart works.
There's a decent explanation of how TWM and waterclocks work in this now-outdated writeup; the idea is we implement a specific UTM with a fixed number of waterclocks (as few as 6 is possible, though there are easier to understand constructions using more), in which its tape is encoded in the values of certain waterclocks; i.e. the number of creatures of certain types. And then the states + tape of any arbitrary TM are encoded within the tape of this UTM.
Unfortunately, Hive Mind allows the opponent to get copies of Arcbond, so he will have control of both Arcbond triggers and Xathrid Necromancer triggers, allowing him to control the order in which they resolve. Maybe if we somehow made it so that Hive Mind would not be around during a computation? If we could replace Goblin Boom Keg with an enchantment, then we could replace Vedalken Orrery with Leyline of Anticipation, and have enchantments gone during the computation instead of artifacts. I don't know if there is a suitable damaging enchantment though; probably not.
Still Sanguine Sacrament could be used. We would sadly give up the three layers at the start, but we could gain three layers from the end sequence I posted, and wind up with BB_{w^2+5} (BB_{w^2+5} (X)), where X is however many Finest Hours we can get prior to the first combat. It's too bad that we have to give up three layers one way or the other though.
@jfb1337: Out of curiosity, has there been any discussion of actually submitting this new deck to a journal? I'm struck by the Abstract section and formality of the language around the lemmas
The write-up was actually very helpful, and answers pretty much all of my questions on the math side. I'm a lot more confident now that the above hyperstage works. I would like to suggest that when we all next reach a short stopping point, we freeze the decklist for any improvements, besides maybe a megastage or a proof of the Flooding variant: there's basically always some improvement in efficiency or search space to make for a higher damage total on the output, and the delta between where we are now and where someone just reading the finished write-ups would be is gigantic (and far more so than it was just a few months ago, kudos again to you guys!).
I've also put some thought into cutting Cowardice from the hyperstage, but I'm honestly not sure it's possible in a way that still enables other stages: at the end of the day, we HAVE to be able to bounce Desolation Giant, and I can't find any way that would bounce it but not the other stage creatures in the above construction. We might be able to explore a trick similar to the Muldrotha, the Gravetide or Repeated Reverberation shenanigans we used for the standard decks, of course.
We could also switch to one of the other ways to destroy all our creatures, and just have something like Petrahydrox as our red producer. But then we still need to find a way to recur the new creature wipe.
There was actually some comments about submission to a journal the last time we posted the Standard stage combo, which was nice to hear. I didn't think that that combo would be interesting enough theoretically for a journal, but this deck could well be. However, there has already been an article published for the Alex Churchill Turing complete construction; the distinguishing factor here is that the deck can't go infinite. Is that enough to pique a journal's interest?
I dunno about the short stopping point; this feels like the last go-around for this challenge, so I feel like we might as well give it our all. We could stop at some intermediate point, write it up, then come back to it for more improvements, but I get the feeling that the motivation for the second go-around might not be there.
Of course, we don't want to just keep going on forever with the improvements, but right now we have just made a big breakthrough (thanks Stakfish), so I feel like we are still early in terms of developing this new paradigm. Before this, I was thinking maybe we would write it up in May at the latest, but if there are more major improvements, maybe even that could be delayed. It all depends on how you guys are feeling, though.
Ooh, Beacon of Immortality seems like just what we need! So we should be able to get to BB_{w^2 + 8} (BB_{w^2 + 8} (BB_{w^2 + 5} (9))).
Yeah, Desolation Giant looks troublesome for a Cowardice-less hyperstage. I was thinking of using either Mogg Infestation or Death Pit Offering instead. What are some nice ways to recycle enchantments and artifacts? Instant speed bouncing/destruction is dangerous due to possibly interfering with the computation.
It looks like adding Recoup would be quite convenient for the stage, although it messes up Flash of Defiance and Acorn Harvest in the hyperstage. We could still use K'rrik and black mana spells to transition to life.
Yeah,this might be worth submitting to a journal, Not only does this not go infinite but it also does all the computation on the same turn. Proving 1 turn is TC is stronger than proving the game TC. Whether or not there's interest remains to be seen.
I think being formal with the lemmas is important, as they are key to showing that the deck both works and doesn't go infinite.
As for what our TWM setup outputs: Given an input of N Necromancers and P plants for the opponent (P>N> 10^^^10 or so, lower bound to be more precisely determined),setting the computation off with a Goblin Boom keg gives us BB numbers of the following for creatures of whatever type(s): taking damage, tokens dying, and tokens ETB.
We have been using Mana Echoes+some creature we have (Desolation Giant) to turn that last one into colorless mana. Then a buyback spell to convert to storm for Thousand-Year Storm, and from there we can go whichever way we want.
That's fair, especially given how close we seem now (though you just know at some point someone's going to figure out how to stack a second UTM on top of the one in our list and we'll have to figure out some new notation: perhaps a new ordinal in the fast-growing hierarchy representing Busy Beaver itself, added by fiat?).
Looking at the options with K'rrik, Son of Yawgmoth, just replacing Desolation Giant as the destroy effect won't be enough: we'll also need a second trigger to replace its Bloodbond March triggers. Right now, it also breaks Seismic Smite since we can recoup it to get lots of Thousand-Year Storm triggers and get {r}{r} per trigger, but we might be able to use that to our advantage by making it harder for us to get back the land. Though the fact that Recoup, Seismic Smite, AND all the destroy-all-your-creatures sorceries cost {r} is a big problem. Damn.
What if we combined that with Petrahydrox as the red-producer, and a red sorcery to set things off? That way, you could produce red each time by targeting Petrahydrox when you cast an instant via Spellweaver Volute and bounce it to hand, where it can tap for an {r} via Chrome Mox and get brought back by a Bloodbond March below, and since the red spell you'd use at the topmost layer is still a sorcery, it would still let you bounce it. Or no, Petrahydrox is blue and would let us flashback Worldpurge.
Actually, we can't use Recoup and the Worldpurge hyperstage at all: we have to hardcast Omniscience to rebuild after a Worldpurge, which means we NEED the easy blue access that would let us flashback Worldpurge with Recoup in the first place. Damn.
There does seem to be the potential for more color stages, except that I couldn't figure out how to use red or black mana to return things at all. Also, there doesn't seem to be a nonblue version of Petrahydrox. So we get stages with blue, green, and white, with one of them taking up the main stage.
Hmm, I didn't see a different card from Temur Sabertooth that uses green to return a creature. Perhaps we could use blue or white in the stage, and just have one additional stage.
The idea behind Soulblast was to trigger it with Nightbird's Clutches, so that we wouldn't need Recoup. (I forgot to remove Recoup at first.) But yeah, the damage dealt from Soulblast is a problem.
Perhaps if we replaced Vedalken Orrery with a creature or creatures? There's Tidal Barracuda - if we Mirrorweave it over to the opponent, they won't have any Xathrid Necromancers. So we will just have our copy, and it will die from the Soulblast. But, how do we bring it back?
Edit: So, I can't find an alternate green stage for the first version. Mina and Denn, Wildborn looked kinda interesting, possibly using it in conjunction with Arixmethes, Slumbering Isle, or Dryad Arbor, but the fact that the return is part of the activation cost rather than the effect is a problem, I think. Living Twister returns a land as an effect, but we would have to avoid having red mana if we use it.
Edit: Okay, for the idea of donating flash-giving creatures to work, we need a repeatable way to keep exchanging control of the creatures. Using Akroan Horse might work, if we have a repeatable way to keep making copies of it, or maybe flickering it. I was thinking of using Flicker of Fate, but unfortunately we can hold that in our hand, and cast it to destroy a token during a computation. Anyone have any other ideas?
Hmm, but Mirozel getting targeted cheaply may be a problem. I think it's just Change of Heart that is the problem, since the other instants trigger off of Artful Dodge anyway. But, Change of Heart is pretty necessary. Hmm, that kind of kills my plan to have a repeatable sorcery both bounce our stage creature and trigger Soulblast.
Edit: Okay, so an issue: When we flicker Death Pit Offering, it has to be a creature. So when it comes back, all the enchantments will be creatures, and will die. So we need to get them back somehow. But, Argivian Find is not good, because we can bring the enchantments back to our hand, and discard them to Chrome Mox to get various colors of mana. How do we avoid this? Bringing back to the battlefield directly would be nice I guess. But, we can't use nonrepeatable sorceries for that. Possibly we could protect them using some exile effect - Leonin Relic-Warder again? But, we need to recycle LRW somehow.
In any case, as long as we can make sure we can't meaningfully interweave the spells to get mana after a Worldpurge it should be fine. We can switch from Red as the hyperstage color to green for Acorn Harvest and Deconstruct, probably. That might even save slots.
I also had an idea for how to allow creatures with tap abilities, though we'd still have to be careful: if the only unpreventable means of dealing damage is Excruciator, and we don't give ourselves any way to make our opponent gain life, then to deal more than 20 damage we would have to deal it by attacking with a lot of Excruciators, which would take over the combat phase and not let us use that time to deal damage and kick-start the TM without killing Vedalken Orrery.
I haven't come to grips with your combo yet, but I figured I would mention an important issue: We need to update our resources to the results of the BB computation continually. It looks like you are doing that with Mana Echoes to generate colorless mana, I just wanted to make sure that the colorless mana was filtering to everywhere else.
Edit: It looks like your explanation got messed up a bit, there is a part which goes "which lets us That gives us a stage", suggesting that something got left out.
Edit: Hmm, we can switch the hyperstage color over to green, but we would still need Inner Fire to get red mana for the stage. Unless we switch the stage over to green mana? I don't know how that would work with Desolation Giant.
If Desolation Giant proves troublesome, there are some other cards can destroy/sacrifice all creatures: Mogg Infestation, Death Pit Offering, Hellion Eruption, Soulblast.
The issue with giving the opponent life isn't keeping it negative, it's making sure that the opponent dies to Arcbond triggers if we try to set off the loop via combat damage and not Goblin Boom Keg. If they can only get hit by Arcbond 7 times, that stops us from going infinite.
For the combat damage thing, it seems like it's the same whether then opponent has 20 life or BB(20); if the opponent keeps getting hit by Arcbond, he will eventually die after some fixed number if hits, so no infinite combo.
Edit: So, getting ahead of ourselves: What is the prospect for additional stages? The fact that Artificial Evolution and Arcbond can target any creature perhaps mean that further Bloodbond March stages are borked. But, there are other ways to make stages, as we have seen in our Standard decks. With the full collection of Vintage cards at our disposal, making similar stages should be easier?
2 Bloodbond March
3 Thousand-Year Storm
4 Cephalid Shrine
5 Mirrorworks
6 Recoup
8 Nyxborn Seaguard
9 Nyxborn Seaguard
10 Reduce to Dreams
12 Xathrid Necromancer
13 Aether Mutation
14 Whirling Dervish
We weave together batches of Thousand-Year Storm triggers on the bounce spells and Bloodbond March triggers on the creatures. Reduce to Dreams lets us rebuy the enchantment creatures, netting a red mana to cast Recoup and get our Spellweaver Volute copies of the instants. Deny Reality can rebuy all our enchantment creatures, but also Xathrid Necromancer to generate black mana. Aether Mutation can do the same, but it can also target Whirling Dervish (Deny Reality can't since the Dervish has pro-black) to generate the green mana it needs.
That's about as far as a strategy like that can get, since there aren't any UW bounce spells (besides Time Wipe and Worldpurge, which bring their own problems), but 3 stages is still better than 0 and it fits fairly easily into the already-existing framework.
Also, scoring by final negative life makes the most sense, as that is the final state of the game, is compatible with all previous versions of the deck, and also covers life loss. (We want Goblin Game to not work) "Most damage" is a bit ambiguous and disqualifies a deck that can infinitely cast lightning bolt and healing salve, but not one that casts lightning bolt and dromoka's command, even though both only get the opponent to -1.
Layer 0: Each time we tap an Angelheart Vial, we gain a lot of life, with the aid of Boon Reflection.
Layer 1: Each time we tap a Talon of Pain, we can damage ourselves and charge up a lot of Angelheart Vials.
Layer 2: Each time we damage the opponent with Flame Rift / Talon of Pain, we can charge up a lot of Talons of Pain.
Layer 3: Each time we resolve a Stream of Life or a Heliod's Intervention, the opponent gains a lot of life.
Layer 4: Each time we cast a Stream of Life or a Heliod's Intervention, we get a lot of copies.
So with 4 Stream of Life and 4 Heliod's Intervention, we can reach BB_{w^2 + 5}(9).
Of course, we would like to make use of all the other colors besides red at least, but I didn't see an easy way to do that.
Edit: If there is a way to put the life gain spells back in the library, we can continue with more layers like
Dragon Appeasement
Pitchstone Wall
Curse of Chaos
Finest Hour
World at War
However, the trick is to put those sorceries back into the library, without being able to have instants like Artificial Evolution and Mirrorweave in our hands during a computation.
Re the combo: I mostly get the execution, but I'm not that well practiced with it so I'm unclear on what outputs it gives us on its own when we run it. I also have no idea about the math behind waterclocks, etc., so I'm slightly hazy on the details of how big we're getting (I assume from the fact we're iterating on it our number of states isn't capped by unique creature types, but I don't and may never understand exactly WHY).
Re the hyperstage: it needs some fixes, and in an edit in a few minutes I'll put the new version in this comment. Primarily, it needs a way to turn mana from Mana Echoes and/or generic computation output into Thousand-Year Storm count, and switch to green mana. Beyond that, I'm pretty sure it actually works
Re the layers: Heliod's Intervention can be used to kill a Goblin Boom Keg without killing Vedalken Orrery, allowing us to potentially interrupt a non-halting computation and deal arbitrary damage. Talon of Pain, however, can't, which is cool: we can only activate it when neither card is a creature, and with no Psychic Battle we can't change the target after playing March of the Machines and loop it back to the Boom Keg. Stream of Life will work, but I think we have to use green mana in the hyperstage or we'll be able to first resolve Worldpurge, then resolve Inner Fire, leaving the hyperstage with 7 R left to work with.
Edit:
2 Xathrid Necromancer
3 Arcbond
4 Artificial Evolution
5 Mirrorweave
6 Dowsing Dagger
7 Dralnu's Crusade
8 Starfield of Nyx
9 Desolation Giant
10 March of the Machines
11 Goblin Boom Keg
12 Vedalken Orrery
14 Pull from Eternity
15 Cowardice
16 Mirrorworks
17 Bloodbond March
18 Cephalid Shrine
19 Thousand-Year Storm
20 Invulnerability
21 Satyr's Cunning
22 Mana Echoes
24 Chrome Mox
25 Spellweaver Helix
26 Seismic Spike
27 Worldpurge
28 Gerrard's Verdict
29 Flash of Defiance
30 Flash of Defiance
31 Panharmonicon
32 Drownyard Temple
We actually don't have to switch to green, and can't: Deconstruct can nuke Goblin Boom Keg, so we use Seismic Spike destroying Drownyard Temple to generate the R.
Edit: Is Hive Mind too dangerous for the computation, or would we be okay with giving our opponents copies? If so, we could use Sanguine Sacrament to give our opponents the life, which tucks itself on bottom and would let us draw cards safely.
But, there are plenty of alternatives, like Change of Heart or Invulnerability.
I was wondering whether we could get enough Seismic Spikes that actually resolve, since all the copies of Thousand-Year Storm go on at the same time. But, we can now create a gajillion Thousand-Year Storms, so that should be okay I think.
Heliod's Intervention was a mistake, it should have been Energy Ray. Using green mana for Stream of Life should be fine; the limiter is the fact that it goes to the graveyard, and we can't cast it again. So we get 8 castings of life gain spells. Of course, it would be even better if we could find some limited way to recycle those spells.
This is quite nice, but perhaps we would have more opportunties for additional stages if we found a hyperstage without Cowardice.
There's a decent explanation of how TWM and waterclocks work in this now-outdated writeup; the idea is we implement a specific UTM with a fixed number of waterclocks (as few as 6 is possible, though there are easier to understand constructions using more), in which its tape is encoded in the values of certain waterclocks; i.e. the number of creatures of certain types. And then the states + tape of any arbitrary TM are encoded within the tape of this UTM.
Still Sanguine Sacrament could be used. We would sadly give up the three layers at the start, but we could gain three layers from the end sequence I posted, and wind up with BB_{w^2+5} (BB_{w^2+5} (X)), where X is however many Finest Hours we can get prior to the first combat. It's too bad that we have to give up three layers one way or the other though.
The write-up was actually very helpful, and answers pretty much all of my questions on the math side. I'm a lot more confident now that the above hyperstage works. I would like to suggest that when we all next reach a short stopping point, we freeze the decklist for any improvements, besides maybe a megastage or a proof of the Flooding variant: there's basically always some improvement in efficiency or search space to make for a higher damage total on the output, and the delta between where we are now and where someone just reading the finished write-ups would be is gigantic (and far more so than it was just a few months ago, kudos again to you guys!).
Re Hive Mind: yeah that makes sense. In that case, we could use Beacon of Immortality to get the same effect.
I've also put some thought into cutting Cowardice from the hyperstage, but I'm honestly not sure it's possible in a way that still enables other stages: at the end of the day, we HAVE to be able to bounce Desolation Giant, and I can't find any way that would bounce it but not the other stage creatures in the above construction. We might be able to explore a trick similar to the Muldrotha, the Gravetide or Repeated Reverberation shenanigans we used for the standard decks, of course.
We could also switch to one of the other ways to destroy all our creatures, and just have something like Petrahydrox as our red producer. But then we still need to find a way to recur the new creature wipe.
I dunno about the short stopping point; this feels like the last go-around for this challenge, so I feel like we might as well give it our all. We could stop at some intermediate point, write it up, then come back to it for more improvements, but I get the feeling that the motivation for the second go-around might not be there.
Of course, we don't want to just keep going on forever with the improvements, but right now we have just made a big breakthrough (thanks Stakfish), so I feel like we are still early in terms of developing this new paradigm. Before this, I was thinking maybe we would write it up in May at the latest, but if there are more major improvements, maybe even that could be delayed. It all depends on how you guys are feeling, though.
Ooh, Beacon of Immortality seems like just what we need! So we should be able to get to BB_{w^2 + 8} (BB_{w^2 + 8} (BB_{w^2 + 5} (9))).
Yeah, Desolation Giant looks troublesome for a Cowardice-less hyperstage. I was thinking of using either Mogg Infestation or Death Pit Offering instead. What are some nice ways to recycle enchantments and artifacts? Instant speed bouncing/destruction is dangerous due to possibly interfering with the computation.
It looks like adding Recoup would be quite convenient for the stage, although it messes up Flash of Defiance and Acorn Harvest in the hyperstage. We could still use K'rrik and black mana spells to transition to life.
I think being formal with the lemmas is important, as they are key to showing that the deck both works and doesn't go infinite.
As for what our TWM setup outputs: Given an input of N Necromancers and P plants for the opponent (P>N> 10^^^10 or so, lower bound to be more precisely determined),setting the computation off with a Goblin Boom keg gives us BB numbers of the following for creatures of whatever type(s): taking damage, tokens dying, and tokens ETB.
We have been using Mana Echoes+some creature we have (Desolation Giant) to turn that last one into colorless mana. Then a buyback spell to convert to storm for Thousand-Year Storm, and from there we can go whichever way we want.
2 Xathrid Necromancer
3 Arcbond
4 Artificial Evolution
5 Mirrorweave
6 Dowsing Dagger
7 Dralnu's Crusade
8 Starfield of Nyx
9 Mogg Infestation
10 March of the Machines
11 Goblin Boom Keg
12 Vedalken Orrery
14 Pull from Eternity
15 Recoup
16 Argivian Find
17 Mana Echoes
18 Change of Heart
19 Thousand-Year Storm
20 Temur Sabertooth
21 Barbarian Horde
22 Bloodbond March
23 Cephalid Shrine
24 Rings of Brighthearth
25 Chrome Mox
27 Worldpurge
28 Sealed Fate
29 Sealed Fate
30 Panharmonicon
31 Cinder Glade
32 Radiant Fountain
33 K'rrik, Son of Yawgmoth
34 Walker of Secret Ways
35 Aegis Automaton
36 Guardian of the Guildpact
Looking at the options with K'rrik, Son of Yawgmoth, just replacing Desolation Giant as the destroy effect won't be enough: we'll also need a second trigger to replace its Bloodbond March triggers. Right now, it also breaks Seismic Smite since we can recoup it to get lots of Thousand-Year Storm triggers and get {r}{r} per trigger, but we might be able to use that to our advantage by making it harder for us to get back the land. Though the fact that Recoup, Seismic Smite, AND all the destroy-all-your-creatures sorceries cost {r} is a big problem. Damn.
What if we combined that with Petrahydrox as the red-producer, and a red sorcery to set things off? That way, you could produce red each time by targeting Petrahydrox when you cast an instant via Spellweaver Volute and bounce it to hand, where it can tap for an {r} via Chrome Mox and get brought back by a Bloodbond March below, and since the red spell you'd use at the topmost layer is still a sorcery, it would still let you bounce it. Or no, Petrahydrox is blue and would let us flashback Worldpurge.
Actually, we can't use Recoup and the Worldpurge hyperstage at all: we have to hardcast Omniscience to rebuild after a Worldpurge, which means we NEED the easy blue access that would let us flashback Worldpurge with Recoup in the first place. Damn.
2 Xathrid Necromancer
3 Arcbond
4 Artificial Evolution
5 Mirrorweave
6 Dowsing Dagger
7 Dralnu's Crusade
8 Starfield of Nyx
9 Soulblast
10 March of the Machines
11 Goblin Boom Keg
12 Vedalken Orrery
14 Pull from Eternity
15 Argivian Find
16 Mana Echoes
17 Change of Heart
18 Thousand-Year Storm
19 Nightbird's Clutches
20 Petrahydrox
21 Bloodbond March
22 Cephalid Shrine
23 Rings of Brighthearth
24 Chrome Mox
26 Worldpurge
27 Gerrard's Verdict
28 Flash of Defiance
29 Flash of Defiance
30 Panharmonicon
31 Drownyard Temple
32 Temur Sabertooth
33 Rotfeaster Maggot
34 Aegis Automaton
35 Voice of Duty
There does seem to be the potential for more color stages, except that I couldn't figure out how to use red or black mana to return things at all. Also, there doesn't seem to be a nonblue version of Petrahydrox. So we get stages with blue, green, and white, with one of them taking up the main stage.
(still has problems)
Soulblast isn't safe anymore, because we can give our opponent a Goblin Boom Keg and March of the Machines, ignore our own March of the Machines, and use Soulblast to kill their Boom Keg without killing our Vedalken Orrery. We can just swap it out for either of the other red sorceries.
Petrahydrox+Recoup lets us generate blue mana, which lets us Recoup Worldpurge outside the hyperstage. I'm pretty much blank on how to fix that.
Temur Sabertooth doesn't target, so it can still bounce Voice of Duty.
The first attempt looks like it would work, except that Temur Sabertooth can also return Walker of Secret Ways, and that we can still loop by using Recoup to directly flashback Rude Awakening. I think that if we swap Rude Awakening for Hidden Strings and find a different stage from Walker of Secret Ways (or a Sabertooth equivalent that targets), the first draft is workable!
The idea behind Soulblast was to trigger it with Nightbird's Clutches, so that we wouldn't need Recoup. (I forgot to remove Recoup at first.) But yeah, the damage dealt from Soulblast is a problem.
Perhaps if we replaced Vedalken Orrery with a creature or creatures? There's Tidal Barracuda - if we Mirrorweave it over to the opponent, they won't have any Xathrid Necromancers. So we will just have our copy, and it will die from the Soulblast. But, how do we bring it back?
Hmmm... instead of Tidal Barracuda, what if we have a combination of Hypersonic Dragon, Prophet of Kruphix, and Raff Capashen, Ship's Mage? Then, we can Cytoshape or donate copies over to the opponent. After a computation, we can resolve a Brand on the stack to gain control back. We won't own Dowsing Dagger Plant tokens though - Akroan Horse and a couple of copies maybe? I think it could work.
Edit: So, I can't find an alternate green stage for the first version. Mina and Denn, Wildborn looked kinda interesting, possibly using it in conjunction with Arixmethes, Slumbering Isle, or Dryad Arbor, but the fact that the return is part of the activation cost rather than the effect is a problem, I think. Living Twister returns a land as an effect, but we would have to avoid having red mana if we use it.
I'll take another look at the Soulblast version. There's also Death Pit Offering as a possibility.
Edit: Okay, for the idea of donating flash-giving creatures to work, we need a repeatable way to keep exchanging control of the creatures. Using Akroan Horse might work, if we have a repeatable way to keep making copies of it, or maybe flickering it. I was thinking of using Flicker of Fate, but unfortunately we can hold that in our hand, and cast it to destroy a token during a computation. Anyone have any other ideas?
Edit: So, here's what I got:
2 Xathrid Necromancer
3 Arcbond
4 Artificial Evolution
5 Mirrorweave
6 Dowsing Dagger
7 Dralnu's Crusade
8 Starfield of Nyx
9 Soulblast
10 Hypersonic Dragon
11 Prophet of Kruphix
12 Raff Capashen, Ship's Mage
13 Akroan Horse
14 Cackling Counterpart
15 Brand
17 Pull from Eternity
18 Argivian Find
19 Mana Echoes
20 Change of Heart
21 Thousand-Year Storm
22 Nightbird's Clutches
23 Petrahydrox
24 Bloodbond March
25 Cephalid Shrine
26 Rings of Brighthearth
27 Chrome Mox
29 Worldpurge
30 Gerrard's Verdict
31 Flash of Defiance
32 Flash of Defiance
33 Panharmonicon
34 Drownyard Temple
35 Aegis Automaton
36 Centaur Peacemaker
It's hard to believe that we can't get more than two color stages - hopefully I'm just missing something.
Edit: Okay, I may have figured out how to add a green stage in:
2 Xathrid Necromancer
3 Arcbond
4 Artificial Evolution
5 Mirrorweave
6 Dowsing Dagger
7 Dralnu's Crusade
8 Starfield of Nyx
9 Soulblast
10 Hypersonic Dragon
11 Prophet of Kruphix
12 Raff Capashen, Ship's Mage
13 Akroan Horse
14 Cackling Counterpart
15 Brand
17 Pull from Eternity
18 Argivian Find
19 Mana Echoes
20 Change of Heart
21 Thousand-Year Storm
22 Artful Dodge
23 Mirozel
24 Bloodbond March
25 Cephalid Shrine
26 Rings of Brighthearth
27 Chrome Mox
28 Rude Awakening
29 Worldpurge
30 Sealed Fate
31 Sealed Fate
32 Panharmonicon
33 Svylenuite Temple
34 Radiant Fountain
35 K'rrik, Son of Yawgmoth
37 Arixmethes, the Slumbering Isle
38 Aegis Automaton
39 Centaur Peacemaker
40 Nyxbloom Ancient
Hey, it's post #3000!
Mirozel can be targeted by Change of Heart as well as Arcbond and Artificial Evolution; is that an issue?
Living Twister can damage creatures which starts a computation, which is an issue.
Raff Capashen, Ship's Mage may not be necessary - with Vedalken Orrery gone, I don't think we need to recycle artifacts. Enchantments and artifacts can be protected by Leonin Relic-Warder.
Hmm, but Mirozel getting targeted cheaply may be a problem. I think it's just Change of Heart that is the problem, since the other instants trigger off of Artful Dodge anyway. But, Change of Heart is pretty necessary. Hmm, that kind of kills my plan to have a repeatable sorcery both bounce our stage creature and trigger Soulblast.
Edit: Oh wait, I think we can use Invulnerability instead.
Edit: Hmmm, I guess it's just simpler to use Death Pit Offering instead:
2 Xathrid Necromancer
3 Arcbond
4 Artificial Evolution
5 Mirrorweave
6 Dowsing Dagger
7 Dralnu's Crusade
8 Starfield of Nyx
9 Death Pit Offering
10 March of the Machines
11 Goblin Boom Keg
12 Vedalken Orrery
13 Cloudshift
15 Pull from Eternity
16 Argivian Find
17 Mana Echoes
18 Invulnerability
19 Thousand-Year Storm
20 Artful Dodge
21 Mirozel
22 Bloodbond March
23 Cephalid Shrine
24 Rings of Brighthearth
25 Chrome Mox
27 Worldpurge
28 Sealed Fate
29 Sealed Fate
30 Panharmonicon
31 Svylenuite Temple
32 Radiant Fountain
33 K'rrik, Son of Yawgmoth
34 Living Twister
35 Arixmethes, the Slumbering Isle
36 Aegis Automaton
37 Centaur Peacemaker
38 Nyxbloom Ancient
Edit: Okay, so an issue: When we flicker Death Pit Offering, it has to be a creature. So when it comes back, all the enchantments will be creatures, and will die. So we need to get them back somehow. But, Argivian Find is not good, because we can bring the enchantments back to our hand, and discard them to Chrome Mox to get various colors of mana. How do we avoid this? Bringing back to the battlefield directly would be nice I guess. But, we can't use nonrepeatable sorceries for that. Possibly we could protect them using some exile effect - Leonin Relic-Warder again? But, we need to recycle LRW somehow.