Re standard: I don't think that works. Slinn Voda, the Rising Deep triggers aren't enough to gain life with Teshar, you also need at least one Raging Swordtooth trigger above that, to actually kill the Paladin. So I don't think that works.
Re vintage: that makes sense to me, pending lijil's OK. Personally, I would lean toward Walking Atlas over Titania, Protector of Argoth, but it doesn't really matter: I don't think either go infinite, so it's a question of whether you want to add less than a rounding error worth more damage and have to explain shuffling the lands and milling them with Perpetual Timepiece, or forego the damage and explain making a Mimic Vat token or three of Walking Atlas.
Actually, one question: don't we still need two Skyshroud Archers to keep the hyperstage going? As is, we have one, which can either bounce or kill the Moggcatcher, but not both.
Ghitu Chronicler can turn red mana into energy, and can actually get two layers for it, since one red mana can generate lots of ETB triggers, and each ETB trigger can gain lots of energy. The problem is to find a way to turn energy into life or white mana.
Ah, I forgot to explain the Skyshroud Archer change - using Facevaulter allows us to get two uses of Goblin Dark-Dwellers per Moggcatcher ability, since we can put the Dark-Dwellers on a Mimic Vat after sacrificing it. Reducing the number of Skyshroud Archers to one should compensate for that.
Sadly, Ghitu Chronicler won't work here: we have no way to give it flash, and we can't bring it back with Teshar and still get the kicker ability. It's by far the hardest part of building the standard combo, the lack of flash.
In the Polyraptor version that I first built, of course, I just used Arcane Adaptation and Cherished Hatchling to at least extend flash to all creatures, but that's closed off here because Cherished Hatchling can make any dinosaur fight any creature. So we're pretty severely constrained in what spells we can actually use.
For the vintage one, I don't think that fixes the problem: we need to be able to store the Dark-Dwellers casts, which means making hasty Moggcatcher tokens, and one Archer isn't enough for that: we can spend a trigger to bounce it and get Bloodbond March triggers, but those will resolve before the remaining triggers we can use to kill it. We can kill it and get a token, but then we won't have a way to get it back to the battlefield.
My solution was to use two archers, so the first one could bounce the Moggcatcher, we could replay it, use a Bloodbond March trigger to get it back, kill it with the second Archer, and use the next Bloodbond March trigger down to rebuy it. Right now, we have a way to make the hasty token, but I don't see any way to bring the Moggcatcher back to the battlefield for the next hyperstage transition. So we still need that, and one that can't work for every Psychic Battle trigger on the Archer.
Does having two Archers cause an infinite higher up? I'm not sure if it does, because I don't know if I understand your megastage version well enough to get the math right, but it not, that would fix the problem.
One other issue: we can fairly easily make lots of token copies of Dearly Departed or Horobi over the course of a hyperstage, using spare Archer tokens, and if we have more than 3 other spirits out, we can just target Kodama of the Center Tree with Smite the Monstrous directly, so I think we have to go back to separate targeting spells, or find another soulshift creature. The only other soulshift legend has flying, so if we go that route we'll need to find another way to get our creatures back to the battlefield in the megastage.
Correct me if I am mistaken, but it looks like we can get two usages of Engineered Explosives per Moggcatcher token. We can use a hasted Moggcatcher token to bring Goblin Dark-Dwellers back to the battlefield, after first putting a Vedalken Orrery creation ability on the stack. This will activate the Goblin Dark-Dwellers ETB abilities, and we can cast Rebuild eventually. This gets us one Engineered Explosives. Then the Vedalken Orrery creation resolves, and we can put Mimic Vat back on the battlefield, and use the Explosives to create a Skyshroud Archer token, and bounce Moggcatcher back to our hand. Then, we can play Vedalken Orrery to put more creation abilities on the stack, sacrifice the Goblin Dark-Dwellers, imprint it on a Mimic Vat, create a Dark-Dwellers token, triggering more instant spells. We get Engineered Explosives back, which allows us to destroy Skyshroud Archer and create a token. Play Moggcatcher, getting Bloodbond March triggers, respond with Skyshroud Archer to destroy it, and we get our Moggcatcher token back.
But yeah, I don't see yet how to get back our soulshift creature.
In the case of the Soulshift creature, if we just switch back to Gloomwidow's Feast and Brace for Impact, that should fix it at the cost of a card slot.
We also need Breeding Pool over Island as the extra land, or we won't produce green in the gigastage. Past that... This looks good to me!
Edit: for the standard deck, we need to kill Thorn Lieutenant before any Slinn Voda triggers, and resolving that many Swordtooth triggers will also kill Raff Capashen, Ship's Mage. I'm not quite sure how to deal with that, since toughness pumping is so dangerous here.
Actually never mind, if we can discard arbitrary cards, it should be fine to bounce the Lieutenant. I don't think there's any value specifically to dumping cards from our hand directly to the graveyard, when the only real way we care about it is with Muldrotha. So Key to the City should work as a one-card fix. God knows there's room for it, with all creatures, enchantments, artifacts, Instants/Sorceries resolving, and four colors of Mana closed off, with nearly half the deck open or packed with flex slots.
We also can't use Rashmi to get the two sorceries on the stack, because the triggers resolve separately. I think Djinn of Wishes is safe, though: we can use it three times per cast, but can only recast it at sorcery speed, at which point all our critical spells will be in our library. It can get three of them out, but by my count there are five different Instants and sorceries we need on the stack to go off:
And that's even cutting Onward//Victory, which is possible but makes the combo harder to follow. Either way, that's more than can be drawn with the Djinn, so each time we do that we need to consume some other resource (draw 2).
re Standard: Why can't you use Rashmi to get two sorceries on the stack? Cast Shrewd Negotiation, Rashmi triggers, cast Switcheroo.
re Vintage: I think the problem with switching from Smite the Monstrous to Gloomwidow's Feast is that we then get one more Explosives per megastage transition, but are draining two. So we will lose a Moggcatcher token each megastage transition, which I'm pretty sure is too much. On the other hand, if we remove one Explosives drain, then the megastage transition will be fine, but then the gigastage transition gets one more Explosives usage, because it bounces Goblin Dark-Dwellers as well. So we have to figure out how to drain another Explosives in the gigastage transition. Since that has proven to be difficult, the idea I had was to somehow bounce Goblin Dark-Dwellers in the megastage as well, so we would get the same number of extra Explosive uses in both n-stages, and the same draining strategy would work for both. (It is fine if the gigastage drains more, but that's not the way it is turning out.) My initial idea was to use Smite the Monstrous to bounce Goblin Dark-Dwellers; if that is out, I'm not sure how to replace it.
Rashmi only lets us cast the revealed spell if its converted Mana cost is less than the first spell, so since they both have CMC 5, we can't cast one with the other.
For the Soulshift creature, we can make it green and just include a second one: then, when the first dies, we can tutor the second one from library to battlefield, and shuffle the first one from graveyard to library. Harbinger of Spring is probably the safest. I think that works!
So, what's left for us to build on from there? Goblins are out, as are 1-drops, 2-drops, humans, flyers, creatures with power 4+, legends, and green or white creatures. We have 8 slots to work with, but that's really 7 because at least one will be taken up with whatever spell we use to draw everything. That actually closes off most of the stages I've seen: we can't use any paragons, because they pump each other into smite range. We can't use Ghosthelm Courier because it's a human, Tidewater Minion and all the possessed creatures for their power, etc.
More stages are probably possible, but as you can see we are out of deck space.
The next problem is... how do we get the combo off the ground? This deck will be much tougher than the megastage one, because of our need to do the whole Mimic Vat / Mirror of Fate / Perpetual Timepiece shebang just to generate one enchantment, as opposed to Dual Nature / Copy Enchantment where we could generate them very quickly. I see mana perhaps being a serious issue.
I've been following this thread for some time, and it's been pretty exciting to see it evolve. I don't play MTG, but I'm a mathematician and so it's nice to see the fast growing hierarchy in the wild, as it were. On top of that, the atmosphere of collaboration reminds me of a Polymath project; I enjoy following the breakthroughs, even if I don't understand them.
I was wondering whether you think you are "close" to the optimal solution. I realize that this is a very difficult question and also that any time you can add a layer, the damage gets astronomically larger. However, right now it seems like you are on the verge of getting f_{w^4+C}(N) damage where C is an ordinal smaller than w^4 and N is a rather large number. How far do you think the rabbit hole goes? f_{w^5}, f_{w^7}, f_{w^w}, f_{e_0}...? I'm would be surprised if it's the latter, but I'm very surprised by the progress you have already made.
Sorry to distract from your work. I was just curious.
Very interesting question! One has to be very careful answering these types of questions; in fact, those with more expertise on the subject can very often be more wrong in their predictions about what is possible, because there knowledge of the subject puts them in a certain box that they aren't aware that they can escape from. Pre-2015, SadisticMystic was of the opinion that an Ackermann combo was impossible, and I was quite unsure myself - we had thought about it for a while, and had come up with nothing. I was even very surprised that we made it to 417 arrows - metroidcomposite's original 36-arrow construction already looked very big to me. So initially I wouldn't have expected even 100 arrows, much less higher levels of the fast-growing hierarchy. So take everything I say next with a huge grain of salt.
It does seem to me we are somewhere close to the end. Even at w^4 there are many little issues to take care of, and they seem to grow worse as the number of stage types increases. For example, there is the problem of each successive transition collecting more benefits as you go up, and you have to pay all those out. So for example in our current deck, at the hyperstage (the w^2) level, we get Engineered Explosives back in our hand after the transition, which we promptly use to restart the hyperstage. At the megastage (w^3) level, we get a green mana, but we can't avoid being able to bounce Engineered Explosives back to our hand, so we have to be forced to pay that out as well as the green mana. Then, at the gigastage (w^4) level, we get 3 life, but we also get the green mana and the Explosives back that we have to pay out. At w^5 it gets even worse.
The other problem is we are running out of both different stage resources to use, and different ways to implement the stage. In the previous deck, we had the hyperstage use triggered instants, and have mana as the resource; in the megastage, we used triggered sorceries, with life as the resource. Stakfish was able to brilliantly come up with the idea for a new stage level that used instants with limited CMC (at most 3 or 4) and had creature tokens as the stage resource. But, I'm not sure how many more implementations we could come up with. And of course, we are running up against the 60-card limit. (Although it is interesting to wonder how far we could get with no card limit.) Right now, w^5 seems very remote, and w^6 looks pretty much impossible.
But, maybe I am thinking too much in my particular box. Stakfish had an interesting idea for an extensible structure that could chain different stage levels in succession; had the idea worked, w^20 might have been a possibility. But, there is a fundamental flaw in the idea that seems unfixable. However, that doesn't mean there isn't an idea that actually does work.
One thing that I have no idea how we would implement is a w^w structure. Our basic idea for the w^n structures is that you have groups of alternating abilities for the w-stage, then to go from w^n to w^(n+1) you add a beneficial ability to recreate the w^n stage, and a destructive ability that differentiates the w^n stages. These abilities have to be different of course, and it seems there are only finitely many distinct abilities in Magic, so how could you get w^w? One possibility is that we could create infinitely many different abilities that differ by some X component; that seems like a pretty big stretch though, that at each opportunity to make an X ability you would have exactly the right amount of resources to activate an ability with X but not X+1. But I can't say it is impossible, nor can I see there can't be some other completely different method that I haven't thought of.
Needless to say, I don't have much hope for e_0. There are some mathematical e_0 constructions; some involve trees, which I have no idea how we would implement in Magic. The most approachable construction is probably Beklemishev worms, which involve sequences of finite numbers. You could implement that with say the stack. But if you look at how the procedure is implemented, there seems like no way you could force a player to do that with the stack. (For example, you would have to resolve part of the stack, then have to remake the stack using the same abilities you just resolved...) So none of these seem likely.
That's a tricky question. I haven't been here for nearly as long as either Deedlit or lijil, but I believe we're close to the optimal solution for our current approach. I highly doubt we can fit anything beyond w^4 with this current strategy.
With that said, I still have hopes for an extensible n-stage framework, which might get as high as f_{w^25}, but would probably require a complete rethinking of the deck. Even that approach, though is fundamentally flawed: we're building our deck recursively, through layers, stages, and eventually n-stages. That makes sense, because the number of permutations of a 60 card vintage-legal magic deck is impossible to evaluate and this lets us break the deck down into components like a hyperstage, a megastage, etc. That's the best strategy for maximizing the damage we can find, but I believe that even if we can get to the optimal recursive deck, we'll ultimately have found a local maximum.
I believe that the true optimal solution is probably not made of a coherently organized deck like ours is, possibly avoiding shortcuts like Omniscience altogether. I think the optimal deck probably implements a non-universal Turing machine whose ruleset implements some fast-growing function in the f_{e_0} range, and that we will never find it, because the individual cards have to be so precise, assuming that we are even aware of that fast-growing function's existence. It may be a function we have not yet discovered. It may require algorithmic knowledge we don't yet have (for instance, a machine that is Turing-equivalent to one which solves the problem, but can be implemented safely in magic while the "primary" machine cannot). I have no doubt that it exists.
My best guess is that this hypothetical deck implements a derivative function to the Kirby-Paris Hydra or the Beklemishev Worm. Both are functions that seem simple enough to implement in a Turing machine that something equivalent to that machine could be constructed in magic, and both get to f_{e_o}, but even if we focused all our efforts on the problem, I doubt we'd ever find it.
For practical purposes, I think we're getting there. We may never find the exact optimum, but I don't think we'll get higher than f_{w^4}, even if we find n-stages that can be added to this structure: there won't be enough space in 60 cards for them. I think we'll top out at f_{w^4+w*m+k}, for some m between 5 and 10, and that's where the challenge will stabilize, until someone figures out an extensible n-stage framework, someone finds a way to cut a large number of cards from the current n-stages, or Wizards prints multiple double-purpose cards that let us combine slots we currently need multiples for.
Turning back to the vintage deck, to what extent is Reality Spasm necessary? If we could replace that with some mechanism to generate a lot of colorless, that would go a long way towards helping the startup. Mana Echoes, Thran Dynamo, even Mana Reflection. Especially with Mirrorworks also costing mana, I don't see how we get started at all with that exact configuration.
Edit: Deedlit, what you were saying about your old work on the Ackermann combo is fascinating to me. It's a shame Wizards closed down their forums, I would have liked to read the old discussions. I've gotten a kick out of seeing the progress made in this thread, but it's a story that begins in the middle.
Reality Spasm is very nice, because it allows us to create a hasted Disciple of Tevesh Szat token, which we need, and also contributes a layer. In our last deck Goblin Bushwhacker was ridiculous - it created hasted tokens, and managed to create three layers per card draw, with the aid of Black Lotus and Dual Nature; unfortunately, we don't have either of those. With a creature stage at the end, it looks like we need at least a token creator and an initial card drawer to get started, and probably more. Hopefully we can optimize the main combo list to allow for whatever cards we need to get started.
One problem with generating mana is that we have to watch for infinities in each of the stage transitions. For the hyperstage transition, for instance, we have to watch out for being able to restart the hyperstage without any Metallurgeon tokens, just with mana and the artifacts that we can use, and generate more mana each time we repeat. Mana Echoes and Mana Reflection look very dangerous to me, since if the deck works we would be able to have gajillions of those, and could get a lot of mana without any Metallurgeons. Thran Dynamo / Mana Vault may be safe, since then there is a limit to how much mana we can get starting from no artifacts.
I think it is: otherwise, every Slinn Voda trigger will wipe out all the tokens on our opponent's side. We use Shrewd Negotiation to swap out an elf token for a noncreature artifact, which won't get bounced.
Though actually, if we could just keep a Merfolk alive through the Swordtooth triggers, we could donate that instead of an artifact, and Slinn Voda wouldn't bounce it. We only need one card on our opponent's board to persist at a time: we can exchange that one with Lieutenant, exchange Lieutenant with Swordtooth, making a token, then exchange the token with this Merfolk.
If we replaced Merfolk Branchwalker from the setup with Jadelight Ranger, a ranger brought back right before or after a Swordtooth entering would always have more than twice as much toughness as Swordtooth triggers. Then, whenever we get back Teshar, we can also bring back the Jadelight Ranger? Though I'm not sure how we'd get back the Ranger then. Damn.
Well, there's a huge amount of free slots we can't productively fill in the deck, since the Ackermann stage shuts out all creatures, four colors of Mana, our life total, Instants and Sorceries, artifacts, and enchantments, so we can just add a Jungle Delver or a Deeproot Champion and call it a day. That makes the stack setup a lot easier. I wonder if it allows us to use The Antiquities War to double up on Gravestones again....
Edit: I just realized: for the vintage combo, we no longer have a way to recur Moriok Scavenger during the hyperstage, so that construction is out. That should be fine though, because I don't see us ever using Explosives for 2.
We also need Orzhov Keyrune, not Azorius, because Facevaulter requires black Mana.
Edit: much bigger problem: I think I've found an infinite. If we use a Smite the Monstrous to bounce Child instead of killing it, we can use the next one down to bounce and replay Dark-Dwellers, put all the prep we need to rebuild the megastage on the stack, then use a Dwellers trigger to flash in... Wait, never mind. We can't stack a Flash cast above the artifact triggers we need to rebuild, and we can't make those triggers after Child of Alara. So I think that's actually safe.
That has turned me into a cut, though: we no longer need Goryo's Vengeance. We can't pull off any shenanigans between Volute triggers and therefore underneath the death trigger, but we can still do stuff after Child dies, but before the trigger resolves. If we kill the Child, and put a Verdant Succession trigger under its own death trigger, with the destroy-all-nonlands still on the stack we can respond by shuffling Child back in with Timepiece. Then the death trigger resolves like usual, and the Verdant Succession trigger below it still has something to search for.
One more edit: I'm still unclear on how generating too much generic/white/lower resources of Mana can go infinite. We can use it to generate more copies of enchantments, or pay for Mirrorworks triggers, but the only resources we can't let ourselves generate are untapped hasty tokens of our various creatures, green Mana, and life, and I don't see any way lots of colorless Mana would change the amounts of those that we have. What am I missing?
Standard: Hmm, so I take it we can only put Baffling End onto the battlefield and destroy it when the stack is empty, since Baffling End doesn't have flash, and that is the only way to give the enemy a creature when it has none. After that, we can never allow Slinn Voda to get rid of all the enemy creatures while there are still objects on the stack. Do I have that right?
Vintage: Nice removal of Goryo's Vengeance. If we can use colorless mana to generate more and more copies of enchantments, then if we have an infinite mana loop, we can leave the loop with as many enchantments as we want. Then, for example, we can resolve a layer of Bloodbond March triggers for Metallurgeon to create a gajillion tapped hasted Metallurgeons, cast Battle Cry to untap them, and attack.
That's exactly the idea in standard. I've explored other options that could give our opponents tokens/creatures, but every single one in standard goes infinite.
For Vintage, a few other observations:
1. Right now, we have no way to generate specifically colorless in the deck, and need it for Vile Redeemer, so we'll need to adjust the low-level mana rocks a bit.
4. We no longer need to kill Llanowar Scout, so we can save a card (and fix the issue I mentioned with Moriok Scavenger) by going back to Metallurgeon/Battle Cry. We have no white creature stages, so that's fine.
5. I think that if we can squeeze a Consecrated Sphinx in there, we can start okay.
The next copy draws 4 total cards. We use the token Timepiece to mill Acorn Harvest and Dearly Departed, then play Spellweaver Helix, exiling Rude Awakening and Acorn Harvest. We play the second Acorn Harvest from our hand, and get 3 Rude Awakenings to get us to 12 mana. We can spend 2 of that and the 3 life we gained earlier to flashback Acorn Harvest, which gets us to 22 mana floating. I think that's probably enough to get the combo going.
Looking good. We probably want to carry out the start somewhat further, as creating enchantments is actually quite expensive; we have to pay three mana to destroy an enchantment, two mana to create a copy of Mimic Vat after we bounce it, three mana to use Mimic Vat, two mana each for copies of Mirror of Fate, Perpetual Timepiece, and Skull of Orm, and five mana to use Skull of Orm. So that's 19 mana per enchantment in addition to Metallurgeon bounces, pretty steep.
With the extra card slot, we could add Precursor Golem for another layer; however, this does make it harder to generate mana via Rude Awakening, so I don't know if we can still get started.
Doesn't Panharmonicon cause it to gain way more than 3 life? Though I do think we can shave a card there another way: Starlit Sanctum. The only cleric we have in the deck is Rector, which is a 2/3 when accounting for Dearly Departed in the graveyard. If we replace Rude Awakening with... Cut the Tethers? Hmmm, thst doesn't quite work, we need a way to get Spirit Guide to the battlefield from there exile. Point being, if we can find a way to generate green without untapping lands, one Worldpurge corresponds to one cleric sacrifice, which in turn corresponds to 3 life.
One interesting observation: we're actually very close to having Dual Nature be safe again. In the past, it was dangerous with Spell Queller and champion creatures, but it's safe with the new Dark-Dwellers version of the hyperstage. The only problem is that, since Child of Alara is legendary, we can bounce it instead of killing it, do our preprocessing, replay it, and then use the Dual Nature triggers to kill the nontoken via legend rule. I don't think that problem is solvable, but it's certainly worth keeping in mind.
Actually, one major problem: I don't see how we can actually generate a net positive Mana at all. We generate Mana by bouncing Tooth of Ramos with Metallurgeon, but to bounce it, we need March out, and to tap it, we need to remove March. That means we need to spend 3 for each Psychic Battle trigger, on Allay, but we only generate 1 Mana from that line. We might have to bite the bullet and go back to Su-chi and a priest or something...
Yeah, it does look like Tooth of Ramos is not enough. So we may need to eliminate the Giant Oyster stage, which will give us two more cards, with which we could really get going quickly probably. Although we'll probably want to use one of them on Precursor Golem, so I guess we'll just get that plus a card that generates a lot of easy mana.
It occurs to me that we don't have to worry about generating too much colorless/white mana, provided we can require something else to use a Metallurgeon PB targeting. I thought about Millikin, but that seems to require using it just to mill itself, so I don't think that works. But Gold Myr should be fine, as we can't cast Engineered Explosives without the white mana. So that opens up arbitrary cards that can generate lots of colorless mana.
I think I've just found a flaw in our approach: Smite the Monstrous can just bounce Dark-Dwellers. So we can use the Moggcatcher token we generate in the megastage transition to build up a new hyperstage instead of grabbing Goblin Turncoat, leaving Dwellers on the battlefield. Then we can start up the megastage. Use the topmost Smite to bounce Dark-Dwellers and replay it, which gets us explosives 1, making a hasty Archer token, which we then use to bounce Moggcatcher. Then we use the second Smite down to bounce Dark-Dwellers again, generate another explosives and another Archer token, use the second token to generate a hasty Moggcatcher like before, and use that one to grab Turncoat. It doesn't actually fix the problem.
Now, if we had a spell that tutored Dwellers from the library or graveyard directly...
On a slightly lighter note, I think I've found a way to invert the mana-life ordering of the higher stages that shaves multiple cards:
Hmm, yes I see. So, we need another way to soak up a usage of Engineered Explosives. If we have to, we could go back to Doomed Necromancer or Pulsemage Advocate, just to make soaking up easier, but of course those two cards eliminate creature stages later. Just something to keep in mind.
Boggart Birth Rite can tutor Goblins, but of course that's a sorcery so it's out of bounds. Goblin Matron and Goblin Ringleader can on ETB as well, but I suppose those are not what you are looking for.
One idea that we used in the past was to make the gigastage transition require going down to the next megastage transition with Knight of the Reliquary. If we can no longer use that, perhaps we could use Dralnu, Lich Lord to give a spell flashback in the gigastage. Of course, we would have to be careful with our spell colors, making sure we can't cast a spell too cheaply.
What we need is a way to tutor Dark-Dwellers but NOT our sacrificer. One idea I found was Ethereal Ambush, so we can manifest the Dwellers, flip it, sacrifice it with Turncoat to get a mimic Vat token, shuffle it back in, repeat. The problem there is that we'll need a new way to target Child of Alara. Brace for Impact only works with the Mana version since Centaur Safeguard is also multi, but that itself causes a new problem: we can just bounce the Turncoat with Worldpurge and replay it, no tutoring necessary.
So, it would be nice at this point to have an alternative life megastage - we need to switch out either Brace for Impact or Centaur Safeguard, or both. Brace for Impact would be nice to keep since it can target Dralnu, Lich Lord at the cost of a megastage transition. Getting life seems hard though - Tarpans are too cheap, we can destroy them at CMC 1. Abyssal Maw would have to be bounced somehow, as would Watcher of the Roost. All the ETB creatures are of course out of bounds.
It really uses a lot of card slots, although we can probably cut some cards down. Perhaps switching Worldpurge out with Worldfire can save a few slots, but we still have to get that working. The first question is whether this can work.
Edit: Something I just noticed: For the Standard version, not only do we need to cast Raff Capashen, we also need to cast Samut, Voice of Dissent after Slinn Voda brings everything back to our hand. Does this give Firesong and Sunspeaker any possibility?
Right now, we need to have a way to get Child of Alara from the graveyard to the library, while its death trigger is on the stack, or we fizzle. This way, we have to crack explosives for 3 while the trigger is on the stack, so that we can kill the Shaman, exile it with Vat, bring it back by sacrificing a mirror, and search it out again with Verdant Succession, shuffling Child back into the library. That means Explosives can't be in our hand when we resolve it, so that's one explosives down. We can play or tutor for Dark-Dwellers to save it afterwards, but then we won't have any way to bounce it before the trigger resolves, so even if that doesn't fizzle us outright, it will cost a later explosives to get back explosives.
Then, once the Alara trigger resolves, we need a way to get back Loaming Shaman. If we don't include a second copy, the only way to do that will be to, before starting the megastage transition, spend an explosives on 3 to blow up the Shaman and put a lot of Succession triggers on the stack, under our sorcery cast and some other busywork. That accounts for the second (first?) Explosives, so of the two we could originally store, we have to spend one to keep the Child of Alara around later, and one to keep the Shaman around after the megastage transition resolves.
Edit: Shaman has the same cmc as Safeguards, but Nantuko Tracer should work fine. Then we switch to Rust Tick. One other nice thing: since Smite the Monstrous is safe, we don't need to copy Volute anymore and can cut Copy Enchantment, because we only ever need to enchant one instant at a time.
Hmm, very nice idea! I wonder if Nantuko Tracer and Metallurgeon is actually a problem or not. We have to blow up one Tracer early, so I think that creating a Metallurgeon token at that pont is not a problem, since it will be destroyed by Child of Alara. The second one, I'm not so sure about - does there have to be a Rebuild between when we create the Metallurgeon token and when we restart the megastage, or does the token survive?
The problem with Rust Tick is that the combos I know of use Goryo's Vengeance, which can bring back Child of Alara without any problem. Perhaps there is another Rust Tick setup?
Re vintage: that makes sense to me, pending lijil's OK. Personally, I would lean toward Walking Atlas over Titania, Protector of Argoth, but it doesn't really matter: I don't think either go infinite, so it's a question of whether you want to add less than a rounding error worth more damage and have to explain shuffling the lands and milling them with Perpetual Timepiece, or forego the damage and explain making a Mimic Vat token or three of Walking Atlas.
Actually, one question: don't we still need two Skyshroud Archers to keep the hyperstage going? As is, we have one, which can either bounce or kill the Moggcatcher, but not both.
Ghitu Chronicler can turn red mana into energy, and can actually get two layers for it, since one red mana can generate lots of ETB triggers, and each ETB trigger can gain lots of energy. The problem is to find a way to turn energy into life or white mana.
Ah, I forgot to explain the Skyshroud Archer change - using Facevaulter allows us to get two uses of Goblin Dark-Dwellers per Moggcatcher ability, since we can put the Dark-Dwellers on a Mimic Vat after sacrificing it. Reducing the number of Skyshroud Archers to one should compensate for that.
In the Polyraptor version that I first built, of course, I just used Arcane Adaptation and Cherished Hatchling to at least extend flash to all creatures, but that's closed off here because Cherished Hatchling can make any dinosaur fight any creature. So we're pretty severely constrained in what spells we can actually use.
For the vintage one, I don't think that fixes the problem: we need to be able to store the Dark-Dwellers casts, which means making hasty Moggcatcher tokens, and one Archer isn't enough for that: we can spend a trigger to bounce it and get Bloodbond March triggers, but those will resolve before the remaining triggers we can use to kill it. We can kill it and get a token, but then we won't have a way to get it back to the battlefield.
My solution was to use two archers, so the first one could bounce the Moggcatcher, we could replay it, use a Bloodbond March trigger to get it back, kill it with the second Archer, and use the next Bloodbond March trigger down to rebuy it. Right now, we have a way to make the hasty token, but I don't see any way to bring the Moggcatcher back to the battlefield for the next hyperstage transition. So we still need that, and one that can't work for every Psychic Battle trigger on the Archer.
Does having two Archers cause an infinite higher up? I'm not sure if it does, because I don't know if I understand your megastage version well enough to get the math right, but it not, that would fix the problem.
One other issue: we can fairly easily make lots of token copies of Dearly Departed or Horobi over the course of a hyperstage, using spare Archer tokens, and if we have more than 3 other spirits out, we can just target Kodama of the Center Tree with Smite the Monstrous directly, so I think we have to go back to separate targeting spells, or find another soulshift creature. The only other soulshift legend has flying, so if we go that route we'll need to find another way to get our creatures back to the battlefield in the megastage.
But yeah, I don't see yet how to get back our soulshift creature.
In the case of the Soulshift creature, if we just switch back to Gloomwidow's Feast and Brace for Impact, that should fix it at the cost of a card slot.
We also need Breeding Pool over Island as the extra land, or we won't produce green in the gigastage. Past that... This looks good to me!
Edit: for the standard deck, we need to kill Thorn Lieutenant before any Slinn Voda triggers, and resolving that many Swordtooth triggers will also kill Raff Capashen, Ship's Mage. I'm not quite sure how to deal with that, since toughness pumping is so dangerous here.
Actually never mind, if we can discard arbitrary cards, it should be fine to bounce the Lieutenant. I don't think there's any value specifically to dumping cards from our hand directly to the graveyard, when the only real way we care about it is with Muldrotha. So Key to the City should work as a one-card fix. God knows there's room for it, with all creatures, enchantments, artifacts, Instants/Sorceries resolving, and four colors of Mana closed off, with nearly half the deck open or packed with flex slots.
We also can't use Rashmi to get the two sorceries on the stack, because the triggers resolve separately. I think Djinn of Wishes is safe, though: we can use it three times per cast, but can only recast it at sorcery speed, at which point all our critical spells will be in our library. It can get three of them out, but by my count there are five different Instants and sorceries we need on the stack to go off:
And that's even cutting Onward//Victory, which is possible but makes the combo harder to follow. Either way, that's more than can be drawn with the Djinn, so each time we do that we need to consume some other resource (draw 2).
re Vintage: I think the problem with switching from Smite the Monstrous to Gloomwidow's Feast is that we then get one more Explosives per megastage transition, but are draining two. So we will lose a Moggcatcher token each megastage transition, which I'm pretty sure is too much. On the other hand, if we remove one Explosives drain, then the megastage transition will be fine, but then the gigastage transition gets one more Explosives usage, because it bounces Goblin Dark-Dwellers as well. So we have to figure out how to drain another Explosives in the gigastage transition. Since that has proven to be difficult, the idea I had was to somehow bounce Goblin Dark-Dwellers in the megastage as well, so we would get the same number of extra Explosive uses in both n-stages, and the same draining strategy would work for both. (It is fine if the gigastage drains more, but that's not the way it is turning out.) My initial idea was to use Smite the Monstrous to bounce Goblin Dark-Dwellers; if that is out, I'm not sure how to replace it.
For the Soulshift creature, we can make it green and just include a second one: then, when the first dies, we can tutor the second one from library to battlefield, and shuffle the first one from graveyard to library. Harbinger of Spring is probably the safest. I think that works!
So, what's left for us to build on from there? Goblins are out, as are 1-drops, 2-drops, humans, flyers, creatures with power 4+, legends, and green or white creatures. We have 8 slots to work with, but that's really 7 because at least one will be taken up with whatever spell we use to draw everything. That actually closes off most of the stages I've seen: we can't use any paragons, because they pump each other into smite range. We can't use Ghosthelm Courier because it's a human, Tidewater Minion and all the possessed creatures for their power, etc.
Frightshroud Courier works, as does Streambed Arquitects, Giant Oyster/Rathi Assassin+ some "tap an untapped ___" effect, Dwarven Nomad, and any number of general creature targeters at the very top. I have no idea how to link any of those, though....
Edit: oh right, your list isn't on Battle Cry anymore. White creatures are safe!
Ooh, very nice solution to getting the Soulshift creature back!
For our stage creatures, we could do something like:
2 Psychic Battle
3 Cowardice
4 Horobi, Death's Wail
5 Bloodbond March
6 Cephalid Shrine
7 Mimic Vat
8 Omniscience
9 Vedalken Orrery
10 Mirror of Fate
11 March of the Machines
12 Perpetual Timepiece
13 Allay
14 Skull of Orm
15 Mirrorworks
16 Azorius Keyrune
17 Rust Tick
18 Meriok Scavenger
19 Molten Frame
20 Rebuild
21 Swarm Intelligence
22 Goblin Dark-Dwellers
23 Facevaulter
24 Moggcatcher
25 Panharmonicon
26 Engineered Explosives
27 Skyshroud Archer
28 Dearly Departed
29 Abzan Falconer
30 Verdant Succession
32 Roar of the Wurm
33 Smite the Monstrous
34 Child of Alara
35 Elvish Spirit Guide
36 Harbinger of Spring
37 Harbinger of Spring
38 Goryo's Vengeance
39 Wormfang Behemoth
40 Vile Redeemer
41 Spellweaver Helix
42 Rude Awakening
43 Worldpurge
44 Acorn Harvest
45 Acorn Harvest
46 Radiant Fountain
47 Glimmerpost
48 Titania, Protector of Argoth
49 Eye of Ugin
50 Breeding Pool
51 Flash
52 Academy Rector
54 Streambed Aquitects
55 Grand Architect
56 Giant Oyster
57 Dwarven Warriors
58 Disciple of Tevesh Szat
59 Reality Spasm
60 Words of Wisdom
More stages are probably possible, but as you can see we are out of deck space.
The next problem is... how do we get the combo off the ground? This deck will be much tougher than the megastage one, because of our need to do the whole Mimic Vat / Mirror of Fate / Perpetual Timepiece shebang just to generate one enchantment, as opposed to Dual Nature / Copy Enchantment where we could generate them very quickly. I see mana perhaps being a serious issue.
I was wondering whether you think you are "close" to the optimal solution. I realize that this is a very difficult question and also that any time you can add a layer, the damage gets astronomically larger. However, right now it seems like you are on the verge of getting f_{w^4+C}(N) damage where C is an ordinal smaller than w^4 and N is a rather large number. How far do you think the rabbit hole goes? f_{w^5}, f_{w^7}, f_{w^w}, f_{e_0}...? I'm would be surprised if it's the latter, but I'm very surprised by the progress you have already made.
Sorry to distract from your work. I was just curious.
It does seem to me we are somewhere close to the end. Even at w^4 there are many little issues to take care of, and they seem to grow worse as the number of stage types increases. For example, there is the problem of each successive transition collecting more benefits as you go up, and you have to pay all those out. So for example in our current deck, at the hyperstage (the w^2) level, we get Engineered Explosives back in our hand after the transition, which we promptly use to restart the hyperstage. At the megastage (w^3) level, we get a green mana, but we can't avoid being able to bounce Engineered Explosives back to our hand, so we have to be forced to pay that out as well as the green mana. Then, at the gigastage (w^4) level, we get 3 life, but we also get the green mana and the Explosives back that we have to pay out. At w^5 it gets even worse.
The other problem is we are running out of both different stage resources to use, and different ways to implement the stage. In the previous deck, we had the hyperstage use triggered instants, and have mana as the resource; in the megastage, we used triggered sorceries, with life as the resource. Stakfish was able to brilliantly come up with the idea for a new stage level that used instants with limited CMC (at most 3 or 4) and had creature tokens as the stage resource. But, I'm not sure how many more implementations we could come up with. And of course, we are running up against the 60-card limit. (Although it is interesting to wonder how far we could get with no card limit.) Right now, w^5 seems very remote, and w^6 looks pretty much impossible.
But, maybe I am thinking too much in my particular box. Stakfish had an interesting idea for an extensible structure that could chain different stage levels in succession; had the idea worked, w^20 might have been a possibility. But, there is a fundamental flaw in the idea that seems unfixable. However, that doesn't mean there isn't an idea that actually does work.
One thing that I have no idea how we would implement is a w^w structure. Our basic idea for the w^n structures is that you have groups of alternating abilities for the w-stage, then to go from w^n to w^(n+1) you add a beneficial ability to recreate the w^n stage, and a destructive ability that differentiates the w^n stages. These abilities have to be different of course, and it seems there are only finitely many distinct abilities in Magic, so how could you get w^w? One possibility is that we could create infinitely many different abilities that differ by some X component; that seems like a pretty big stretch though, that at each opportunity to make an X ability you would have exactly the right amount of resources to activate an ability with X but not X+1. But I can't say it is impossible, nor can I see there can't be some other completely different method that I haven't thought of.
Needless to say, I don't have much hope for e_0. There are some mathematical e_0 constructions; some involve trees, which I have no idea how we would implement in Magic. The most approachable construction is probably Beklemishev worms, which involve sequences of finite numbers. You could implement that with say the stack. But if you look at how the procedure is implemented, there seems like no way you could force a player to do that with the stack. (For example, you would have to resolve part of the stack, then have to remake the stack using the same abilities you just resolved...) So none of these seem likely.
Enough of my rambling, back to the deck!
With that said, I still have hopes for an extensible n-stage framework, which might get as high as f_{w^25}, but would probably require a complete rethinking of the deck. Even that approach, though is fundamentally flawed: we're building our deck recursively, through layers, stages, and eventually n-stages. That makes sense, because the number of permutations of a 60 card vintage-legal magic deck is impossible to evaluate and this lets us break the deck down into components like a hyperstage, a megastage, etc. That's the best strategy for maximizing the damage we can find, but I believe that even if we can get to the optimal recursive deck, we'll ultimately have found a local maximum.
I believe that the true optimal solution is probably not made of a coherently organized deck like ours is, possibly avoiding shortcuts like Omniscience altogether. I think the optimal deck probably implements a non-universal Turing machine whose ruleset implements some fast-growing function in the f_{e_0} range, and that we will never find it, because the individual cards have to be so precise, assuming that we are even aware of that fast-growing function's existence. It may be a function we have not yet discovered. It may require algorithmic knowledge we don't yet have (for instance, a machine that is Turing-equivalent to one which solves the problem, but can be implemented safely in magic while the "primary" machine cannot). I have no doubt that it exists.
My best guess is that this hypothetical deck implements a derivative function to the Kirby-Paris Hydra or the Beklemishev Worm. Both are functions that seem simple enough to implement in a Turing machine that something equivalent to that machine could be constructed in magic, and both get to f_{e_o}, but even if we focused all our efforts on the problem, I doubt we'd ever find it.
For practical purposes, I think we're getting there. We may never find the exact optimum, but I don't think we'll get higher than f_{w^4}, even if we find n-stages that can be added to this structure: there won't be enough space in 60 cards for them. I think we'll top out at f_{w^4+w*m+k}, for some m between 5 and 10, and that's where the challenge will stabilize, until someone figures out an extensible n-stage framework, someone finds a way to cut a large number of cards from the current n-stages, or Wizards prints multiple double-purpose cards that let us combine slots we currently need multiples for.
Turning back to the vintage deck, to what extent is Reality Spasm necessary? If we could replace that with some mechanism to generate a lot of colorless, that would go a long way towards helping the startup. Mana Echoes, Thran Dynamo, even Mana Reflection. Especially with Mirrorworks also costing mana, I don't see how we get started at all with that exact configuration.
Edit: Deedlit, what you were saying about your old work on the Ackermann combo is fascinating to me. It's a shame Wizards closed down their forums, I would have liked to read the old discussions. I've gotten a kick out of seeing the progress made in this thread, but it's a story that begins in the middle.
One problem with generating mana is that we have to watch for infinities in each of the stage transitions. For the hyperstage transition, for instance, we have to watch out for being able to restart the hyperstage without any Metallurgeon tokens, just with mana and the artifacts that we can use, and generate more mana each time we repeat. Mana Echoes and Mana Reflection look very dangerous to me, since if the deck works we would be able to have gajillions of those, and could get a lot of mana without any Metallurgeons. Thran Dynamo / Mana Vault may be safe, since then there is a limit to how much mana we can get starting from no artifacts.
When the Wizards forum was going to be shut down, I took the time to save all the pages on archive.org: http://web.archive.org/web/20150917153327/http://community.wizards.com/forum/cards-and-combos/threads/2230096
Have fun!
Edit: Hmm, since you have switched to Thorn Lieutenant, is Shrewd Negotation really necessary? You just need to copy Onward // Victory prior to copying Switcheroo to give Raging Swordtooth to the opponent.
Though actually, if we could just keep a Merfolk alive through the Swordtooth triggers, we could donate that instead of an artifact, and Slinn Voda wouldn't bounce it. We only need one card on our opponent's board to persist at a time: we can exchange that one with Lieutenant, exchange Lieutenant with Swordtooth, making a token, then exchange the token with this Merfolk.
If we replaced Merfolk Branchwalker from the setup with Jadelight Ranger, a ranger brought back right before or after a Swordtooth entering would always have more than twice as much toughness as Swordtooth triggers. Then, whenever we get back Teshar, we can also bring back the Jadelight Ranger? Though I'm not sure how we'd get back the Ranger then. Damn.
Well, there's a huge amount of free slots we can't productively fill in the deck, since the Ackermann stage shuts out all creatures, four colors of Mana, our life total, Instants and Sorceries, artifacts, and enchantments, so we can just add a Jungle Delver or a Deeproot Champion and call it a day. That makes the stack setup a lot easier. I wonder if it allows us to use The Antiquities War to double up on Gravestones again....
Edit: I just realized: for the vintage combo, we no longer have a way to recur Moriok Scavenger during the hyperstage, so that construction is out. That should be fine though, because I don't see us ever using Explosives for 2.
We also need Orzhov Keyrune, not Azorius, because Facevaulter requires black Mana.
Edit: much bigger problem: I think I've found an infinite. If we use a Smite the Monstrous to bounce Child instead of killing it, we can use the next one down to bounce and replay Dark-Dwellers, put all the prep we need to rebuild the megastage on the stack, then use a Dwellers trigger to flash in... Wait, never mind. We can't stack a Flash cast above the artifact triggers we need to rebuild, and we can't make those triggers after Child of Alara. So I think that's actually safe.
That has turned me into a cut, though: we no longer need Goryo's Vengeance. We can't pull off any shenanigans between Volute triggers and therefore underneath the death trigger, but we can still do stuff after Child dies, but before the trigger resolves. If we kill the Child, and put a Verdant Succession trigger under its own death trigger, with the destroy-all-nonlands still on the stack we can respond by shuffling Child back in with Timepiece. Then the death trigger resolves like usual, and the Verdant Succession trigger below it still has something to search for.
One more edit: I'm still unclear on how generating too much generic/white/lower resources of Mana can go infinite. We can use it to generate more copies of enchantments, or pay for Mirrorworks triggers, but the only resources we can't let ourselves generate are untapped hasty tokens of our various creatures, green Mana, and life, and I don't see any way lots of colorless Mana would change the amounts of those that we have. What am I missing?
Vintage: Nice removal of Goryo's Vengeance. If we can use colorless mana to generate more and more copies of enchantments, then if we have an infinite mana loop, we can leave the loop with as many enchantments as we want. Then, for example, we can resolve a layer of Bloodbond March triggers for Metallurgeon to create a gajillion tapped hasted Metallurgeons, cast Battle Cry to untap them, and attack.
For Vintage, a few other observations:
1. Right now, we have no way to generate specifically colorless in the deck, and need it for Vile Redeemer, so we'll need to adjust the low-level mana rocks a bit.
2. When I switched over to the Skull of Orm base level, I didn't realize that we can't make token copies of Spellweaver Volute with Mimic Vat under any circumstances, so we need to add a Copy Enchantment.
3. Facevaulter has cmc 1 and dies when we pop Engineered Explosives, so we need to replace it with a 2-drop goblin-sacrificer. Goblin Turncoat seems like the best option.
4. We no longer need to kill Llanowar Scout, so we can save a card (and fix the issue I mentioned with Moriok Scavenger) by going back to Metallurgeon/Battle Cry. We have no white creature stages, so that's fine.
5. I think that if we can squeeze a Consecrated Sphinx in there, we can start okay.
Start with Breeding pool (18 life), exile Elvish Spirit Guide, Flash in Academy Rector, get Omniscience. 3 cards in hand.
Play Swarm Intelligence, Copy Enchantment as Swarm Intelligence, and Words of Wisdom, which gets copied twice. We draw 6, our opponent draws 3.
Play Vedalken Orrery, Mirrorworks, Panharmonicon, and in response to the Mirrorworks trigger on Panharmonicon, Perpetual Timepiece. In response to the Mirrorworks trigger, activate the nontoken timepiece to mill both Radiant Fountain and Glimmerpost, then play Titania, Protector of Argoth, getting back both lands and bringing us up to 21 life. Tap both to pay for Mirrorworks, getting a second Timepiece. Last card is Goblin Dark-Dwellers, which gives us 3 copies of Words of Wisdom again.
The first copy can draw Consecrated Sphinx and Rude Awakening. We play both, and tap our lands for 3 mana again.
The next copy draws 4 total cards. We use the token Timepiece to mill Acorn Harvest and Dearly Departed, then play Spellweaver Helix, exiling Rude Awakening and Acorn Harvest. We play the second Acorn Harvest from our hand, and get 3 Rude Awakenings to get us to 12 mana. We can spend 2 of that and the 3 life we gained earlier to flashback Acorn Harvest, which gets us to 22 mana floating. I think that's probably enough to get the combo going.
So, the combo overall looks like:
2 Psychic Battle
3 Cowardice
4 Horobi, Death's Wail
5 Bloodbond March
6 Cephalid Shrine
7 Mimic Vat
8 Omniscience
9 Vedalken Orrery
10 Mirror of Fate
11 March of the Machines
12 Perpetual Timepiece
13 Allay
14 Skull of Orm
15 Mirrorworks
16 Tooth of Ramos
17 Copy Enchantment
18 Metallurgeon
19 Battle Cry
20 Rebuild
21 Swarm Intelligence
22 Goblin Dark-Dwellers
23 Moggcatcher
24 Goblin Turncoat
25 Panharmonicon
26 Engineered Explosives
27 Skyshroud Archer
28 Dearly Departed
29 Abzan Falconer
30 Verdant Succession
32 Roar of the Wurm
33 Smite the Monstrous
34 Child of Alara
35 Elvish Spirit Guide
36 Harbinger of Spring
37 Harbinger of Spring
38 Wormfang Behemoth
39 Vile Redeemer
40 Spellweaver Helix
41 Rude Awakening
42 Worldpurge
43 Acorn Harvest
44 Acorn Harvest
45 Radiant Fountain
46 Glimmerpost
47 Titania, Protector of Argoth
48 Eye of Ugin
49 Breeding Pool
50 Flash
51 Academy Rector
53 Streambed Aquitects
54 Grand Architect
55 Giant Oyster
56 Dwarven Warriors
57 Disciple of Tevesh Szat
58 Reality Spasm
59 Consecrated Sphinx
60 Words of Wisdom
Radiant Fountain and Glimmerpost can make colorless mana for Vile Redeemer, so I think that's okay.
Edit: I have a card cut: Since black mana is not available, we can replace Radiant Fountain and Glimmerpost with Stormscape Battlemage, and save a card. We still need Breeding Pool to generate green mana, and therefore Titania, Protector of Argoth, unless we could find a replacement for Rude Awakening.
With the extra card slot, we could add Precursor Golem for another layer; however, this does make it harder to generate mana via Rude Awakening, so I don't know if we can still get started.
One interesting observation: we're actually very close to having Dual Nature be safe again. In the past, it was dangerous with Spell Queller and champion creatures, but it's safe with the new Dark-Dwellers version of the hyperstage. The only problem is that, since Child of Alara is legendary, we can bounce it instead of killing it, do our preprocessing, replay it, and then use the Dual Nature triggers to kill the nontoken via legend rule. I don't think that problem is solvable, but it's certainly worth keeping in mind.
For the startup, we can also generate some Mana by playing March of the Machines, then running Engineered Explosives and Tooth of Ramos into it. If we can cast Vile Redeemer a couple times, those artifacts dying will power it up to make a lot of spawn tokens.
Actually, one major problem: I don't see how we can actually generate a net positive Mana at all. We generate Mana by bouncing Tooth of Ramos with Metallurgeon, but to bounce it, we need March out, and to tap it, we need to remove March. That means we need to spend 3 for each Psychic Battle trigger, on Allay, but we only generate 1 Mana from that line. We might have to bite the bullet and go back to Su-chi and a priest or something...
For Starlit Sanctum, what if we replaced Rude Awakening with Planar Cleansing? No, that would get rid of Vedalken Orrery so we couldn't get more than one green mana.
Yeah, it does look like Tooth of Ramos is not enough. So we may need to eliminate the Giant Oyster stage, which will give us two more cards, with which we could really get going quickly probably. Although we'll probably want to use one of them on Precursor Golem, so I guess we'll just get that plus a card that generates a lot of easy mana.
It occurs to me that we don't have to worry about generating too much colorless/white mana, provided we can require something else to use a Metallurgeon PB targeting. I thought about Millikin, but that seems to require using it just to mill itself, so I don't think that works. But Gold Myr should be fine, as we can't cast Engineered Explosives without the white mana. So that opens up arbitrary cards that can generate lots of colorless mana.
Now, if we had a spell that tutored Dwellers from the library or graveyard directly...
On a slightly lighter note, I think I've found a way to invert the mana-life ordering of the higher stages that shaves multiple cards:
33 Acorn Harvest
34 Smite the Monstrous
35 Child of Alara
36 Centaur Safeguard
37 Centaur Safeguard
39 Spellweaver Helix
40 Righteous Fury
41 Worldfire
42 Spider Spawning
43 Spider Spawning
44 Titania, Protector of Argoth
45 Natural Order
Boggart Birth Rite can tutor Goblins, but of course that's a sorcery so it's out of bounds. Goblin Matron and Goblin Ringleader can on ETB as well, but I suppose those are not what you are looking for.
One idea that we used in the past was to make the gigastage transition require going down to the next megastage transition with Knight of the Reliquary. If we can no longer use that, perhaps we could use Dralnu, Lich Lord to give a spell flashback in the gigastage. Of course, we would have to be careful with our spell colors, making sure we can't cast a spell too cheaply.
Edit: So here's an idea for a deck using Dralnu, Lich Lord and Worldpurge:
2 Psychic Battle
3 Cowardice
4 Horobi, Death's Wail
5 Bloodbond March
6 Cephalid Shrine
7 Mimic Vat
8 Omniscience
9 Vedalken Orrery
10 Mirror of Fate
11 March of the Machines
12 Perpetual Timepiece
13 Allay
14 Skull of Orm
15 Mirrorworks
16 Horn of Ramos
17 Rust Tick
18 Arcum Dagsson
19 Arcum Dagsson
20 Goryo's Vengeance
21 Rebuild
22 Swarm Intelligence
23 Goblin Dark-Dwellers
24 Goblin Turncoat
25 Moggcatcher
26 Panharmonicon
27 Engineered Explosives
28 Skyshroud Archer
29 Dearly Departed
30 Abzan Falconer
31 Verdant Succession
33 Scorching Missle
34 Gloomwidow's Feast
35 Brace for Impact
36 Child of Alara
37 Simian Spirit Guide
38 Harbinger of Spring
39 Harbinger of Spring
40 Wormfang Behemoth
41 Vile Redeemer
42 Spellweaver Helix
43 Devouring Greed
44 Worldpurge
45 Fumarole
46 Fumarole
47 Dralnu, Lich Lord
48 Radiant Fountain
49 Glimmerpost
50 Swamp
51 Titania, Protector of Argoth
52 Eye of Ugin
54 Flash
55 Academy Rector
56 Judge of Currents
57 Streambed Aquitects
58 Reality Spasm
59 Consecrated Sphinx
60 Words of Wisdom
It really uses a lot of card slots, although we can probably cut some cards down. Perhaps switching Worldpurge out with Worldfire can save a few slots, but we still have to get that working. The first question is whether this can work.
Edit: We do need a copy of Copy Enchantment and a more productive source of mana, but there are a couple of card cuts we can make as well: Swamp and Breeding Pool can be replaced by Underground Sea, and the pair of Harbinger of Springs can be replaced by a single Kodama of the Center Tree.
Edit: Something I just noticed: For the Standard version, not only do we need to cast Raff Capashen, we also need to cast Samut, Voice of Dissent after Slinn Voda brings everything back to our hand. Does this give Firesong and Sunspeaker any possibility?
Right now, we need to have a way to get Child of Alara from the graveyard to the library, while its death trigger is on the stack, or we fizzle. This way, we have to crack explosives for 3 while the trigger is on the stack, so that we can kill the Shaman, exile it with Vat, bring it back by sacrificing a mirror, and search it out again with Verdant Succession, shuffling Child back into the library. That means Explosives can't be in our hand when we resolve it, so that's one explosives down. We can play or tutor for Dark-Dwellers to save it afterwards, but then we won't have any way to bounce it before the trigger resolves, so even if that doesn't fizzle us outright, it will cost a later explosives to get back explosives.
Then, once the Alara trigger resolves, we need a way to get back Loaming Shaman. If we don't include a second copy, the only way to do that will be to, before starting the megastage transition, spend an explosives on 3 to blow up the Shaman and put a lot of Succession triggers on the stack, under our sorcery cast and some other busywork. That accounts for the second (first?) Explosives, so of the two we could originally store, we have to spend one to keep the Child of Alara around later, and one to keep the Shaman around after the megastage transition resolves.
Edit: Shaman has the same cmc as Safeguards, but Nantuko Tracer should work fine. Then we switch to Rust Tick. One other nice thing: since Smite the Monstrous is safe, we don't need to copy Volute anymore and can cut Copy Enchantment, because we only ever need to enchant one instant at a time.
The problem with Rust Tick is that the combos I know of use Goryo's Vengeance, which can bring back Child of Alara without any problem. Perhaps there is another Rust Tick setup?
Another issue is that we can get around it by exiling Child to a Vat and then shuffling it with Mirror of Fate. That seems fixable, though.