One more update: having access to 3 colors of Mana in the megastage means we can use explosives to kill the Centaur and gain 3 life repeatedly, so back to [[Rust Tick]].
On a happier note, I have a tentative list! It's at 61 cards, so we need a cut, but as a starting point:
The Sanctum is there both to provide higher white Mana and as a manager buffer in the setup that won't take us infinite in the hyperstage. This gives us a gigastage, followed by 3 stages, followed by layers for each Panharmonicon on Keldon Overseer, for each Revel in Riches, for each Hive Mind copying White Sun's Zenith, for each Consecrated Sphinx trigger on Words of Wisdom, for each card in our opponent's library...
That gets us to f_{w^4+w3+4}(X), where X is the number of cards we can keep in the goldfish's library after we complete the setup, which I'm guessing is about 45. The setup seems difficult but definitely not impossible. The problem is finding that one card cut...
One issue with that list, and one I still haven't nearly solved, is that we don't have a way to tuck Reality Spasm into our library anymore. That Goblin Kites stage is some hot tech, though!
Ah, putting noncreatures into our library is a problem. I suppose we can replace Reality Spasm by Korozda Gorgon and Chasm Skulker, but that requires cutting a card. Looking at the rest of the deck, it looks like perhaps Mirrorworks can be cut; it was mainly there for Mirror of Fate, otherwise I don't see any other artifact for which it is absolutely necessary. Mimic Vat could be dicey, but I think we can get by with just one.
Edit: No, we have made use of +1/+1 counters already. If there were a way to just give humans flying without resorting to +1/+1 counters, that would be nice.
Edit: I guess we are using Mirrorworks in order to put a copy of Vedalken Orrery back onto the battlefield after it is eliminated; I don't see a way around that, so it stays in.
I suppose we can use another card to put Reality Spasm back into the library, one that can bring cards from the graveyard or exile back into the library, but not for too cheap. Riftsweeper should work; it doesn't get around Bone Harvest for Skyshroud Archer or Child of Alara, as we need to use Bone Harvest to bring Riftsweeper back into the library anyway. (Or use Woodland Bellower to get Bloodbond March triggers, but that is more expensive still.)
That means we have to cut a card; perhaps we can replace Consecrated Sphinx + Words of Wisdom with Wheel of Fortune or Memory Jar? We would be reduced to just one or two applications of F_{w^4 + w4}, but that would still be better than just three stages of course. The question is whether we can get started.
I've found that every time I try to set up, I wind up running out of mana and fizzling. I think we'll need some kind of easily extensible Mana generation that won't break the hyperstage, like Tolarian Academy. That probably takes another card slot, and I'm not even sure just Academy represents enough Mana.
We know that, with Horobi, Cowardice, and enough copies of Psychic Battle, targeting Su-Chi with anything is net-positive on Mana. At some point, we should figure out what that number is for this configuration. That way, we'll have a target to aim for in our setup.
In other news, I'm planning on posting the standard combo write-up this week. I've stuck with the Silent Gravestone version because it's the one I most trust to stay finite and because I've already written the relevant sections for it. I'd appreciate it if we could do one more check for infinite combos before I post!
Hmmm. When we play Mana Vault, each Mirrorworks trigger lets us generate another 1 colorless Mana, but there is also the question of how we make those Mirrorworks in the first place... You may be right.
In other news, there's one other possible cut: I only switched to Hive Mind so White Sun's Zenith would give our opponent lots of tokens. If we go back to Swarm Intelligence, though, we can have our lifegain sorcery in the gigastage gain a constant amount of life, keeping track of our progress by Mana turning into Swarm Intelligence copies. So, we can switch Righteous Fury for the famously-OP Archangel's Light, which fixes the graveyard->library problem with Reality Spasm.
Edit: a couple minor things: we can't use Wheel of Fortune, because it's a 3-mana sorcery and can be cast with Dark-Dwellers to trigger Volute. Either Memory Jar or Reforge the Soul will fix that. Let's say Reforge the Soul, because ironically enough it seems like Sorceries will be easier to recur at the setup level than artifacts (we can just leave it under a Spellweaver Helix and go to town with Spider Spawning and any way to generate black Mana we can find).
Then, the big question is tue same one this challenge always seems to end with: how many cards can we get ahead of the goldfish in the deck? Thankfully, the answer for this deck seems fairly high: we can get almost all of our enchantments by milling them with Broken Ambitions and bringing them back with Skull of Orm. By my count, that gets us Bloodbond March, March of the Machines, Opalescence, Spellweaver Volute, and possibly Psychic Battle and even Cowardice.
Hmm, is it F_{w^4+w4+2}(5) or F_{w^4+w4+2}(6)? For the latter, we need to get X up to F_{w^4+w4+1}(6) after the second Reforge the Soul, which means we need to cast five or six Reality Spasms at that time. (probably six) So if we can mill/fetch 30 cards, we can use the extra five draws to draw Reality Spasm another five times... it's probably pretty close.
Of course, what would make it easy would be if we could get a second Consecrated Sphinx on the board - for that, we would need a Mimic Vat, perhaps a Smite the Monstrous, and a few mana... would be tricky.
It would be nice if we can get going without the Tolarian Academy, then we could put in [c]Precursor Golem to go up to F_{w^4+w4+3}.
The reason for w4 and not 4w is that AB represents B copies of A put one after the other, so 4w is actually 4+4+4+4..., which is just w. w4 is then w+w+w+w, which is what we want.
(The reason AB is B copies of A rather than A copies of B, near as I can tell, is to make ordinal arithmetic nicer; for example, this way a^(b+c) comes out to a^b * a^c rather than a^c * a^b.)
Frankly, I"ll be shocked if we can get going without at least one Mana boost. I'm even worried the Academy won't be enough. More power to you if you can find a way, though!
Edit: actually, with Trashmaster as the hyperstage transition, we don't need the early boost at all. If we can get out Psychic Battle, Trashmaster, Mimic Vat, Cowardice, and some goblins in hand, that gets us four artifact bounces per goblin (because we can both sac the goblin and sac a vat token), plus tucking and searching out Changeling Sentinel with Dark-Dwellers' ability.
It's very wordy and quite rough, but I've outlined a start that works by turning all that Mana into 3 Bloodbond Marches and 3 Psychic Battles, and using some stack tricks to get two hasty Rust Ticks. That's enough to make a mini-stage worth a few hundred Mana, more than enough to get everything going.
That does involve sufficiently many searchable cards that two Reforge the Soul isn't enough to get everything, but f_{w^4+4w+3}(4) is still a huge improvement.
Concerning the standard deck, here are some corrections to the article:
When you describe the damage of the megastage deck (F_{w^3 + w13 + 3}) you superscript the ordinal and subscript the 3 in w^3, rather than the other way around.
Whenever you mention using Gideon's +1 ability, that should be a +2.
The result of repeatedly casting Cackling Counterpart of Precursor Golem is a lot more complicated. After the first Cackling Counterpart there will be 2 Precursor Golems and 6 vanilla Golems; after the second (assuming you target a vanilla Golem), there will be 8 and 38; after the third, if I calculate correctly, there will be 2048 and 25,858.
When you say "untapping Marwyn, the tapped elves", that is a bit confusing, since you can only untap one elf other than Marwyn. So it should probably be "the tapped elf".
This feels a bit like cheating, but I think I've found an easier way to kick-start that might even allow for Words of Wisdom.
If we switch from March of the Machines to Karn, Silver Golem, Karn+Mana Crypt+Salvaging Station is an infinite Mana combo: tap the crypt for 2, spend 1 to make it a creature, it dies, which untaps station, bring it back, repeat. If we swap Spirit Guide for Mox Emerald, we can also filter to green. None of that's an issue, as long as we still require a "target artifact" to make blue Mana (hello Silver Myr), we still can't go infinite, because you need too many Guildmage activations to keep going otherwise.
That makes the setup much, much easier: with no generic costs to worry about, making a second, even a third Sphinx via Reality Spasm on a vat isn't hard, as long as we can generate the UU needed to cast it in the first place. That doesn't seem very difficult.
Also, thanks for the revisions! Those are all good catches, I'll fix those when I get the chance.
Hmmm, I'm not sure. In order to keep the Rebuild infinite going, you have to generate at least one blue Mana to get Rebuild and any creatures under a vat back into the graveyard, and Bone Harvest. You also need a green/blue to feed into Engineered Explosives. If you go Mirrorworks, Panharmonicon and pay for the trigger, Keyrune/Signet/Eye, pay for all 3 triggers, that's a minimum of 4 Mana... Yeah, I think that does go infinite.
On the plus side, Karn seems unquestionably better here than March of the Machines, since it's much cheaper to recur artifacts with him vs. March, and therefore easier to set up. We don't have Dual Nature anymore, so we can afford to make things creatures as-needed.
That's a great catch with Precursor! It's certainly worth noting as a potential card cut later on.
One other interesting thing: because we can't refill the library as-needed, we also need to LOSE the clash on Broken Ambitions sometimes. That means we need a 0-drop card we can keep in the library, and also need a way to shuffle our library whenever we switch from Broken Ambitions to Psychic Battle. So we definitely need to swap Elvish Spirit Guide for Mox Emerald, whether or not we have the Keyrune: we can tuck it during the setup.
Hmmm... if there turns out to not be an easy way to get a shuffle, we can exchange Broken Ambitions for Cephalid Shrine, since I don't see a need for milling now that we have Pull from Eternity. (Perhaps I'm missing something)
Well, we need milling for Archangel's Light I guess. Perhaps if we switched out Archangel's Light and go back to Righteous Fury? (Or maybe Divine Congregation is better) Then we could remove Swarm Intelligence, and add a card that will refill the library. Riftsweeper is a possibility, or some card that moves cards from the graveyard to the library directly.
Okay, here's a solution: every time we make a Bloodbond March copy, we kill it to put it on a vat. That triggers Verdant Succession, which lets us search, fail to find, and shuffle.
Also, assuming we can get a Goblin Welder on a vat before playing a Reality Spasm, we can also get artifacts out of the library by milling/reanimating, same as enchantments. That seems exceptionally easy with Trashmaster.
Ah, nice. The one issue is, can we move Mox Emerald in and out of the library as we need to? We can just mill the Mox and use Salvaging Station to bring it to the battlefield. Putting it back is a problem though. I guess we can just run the infinite combo long enough to generate all the green mana we need for the whole combo... sounds cheesy, but I guess it works!
Ah, nice. The one issue is, can we move Mox Emerald in and out of the library as we need to? We can just mill the Mox and use Salvaging Station to bring it to the battlefield. Putting it back is a problem though. I guess we can just run the infinite combo long enough to generate all the green mana we need for the whole combo... sounds cheesy, but I guess it works!
Play Karn/Station, use SSG to get the 1 Mana needed to start, get infinite colorless. Play Mirrorworks, Panharmonicon, and Mimic Vat, paying for all trigger each time. I believe that gets us 3 Panharmonicons and 5 Mimic Vats. Play Pull from Eternity, use the Swarm Intelligence copy to get back words, play Goblin Dark-Dwellers, draw another 8, leaving the original Pull on the stack.
Play Trashmaster/Welder/Silver Myr, and use the sacrifice ability to put Silver Myr and Welder on vats. Make a copy, sac Dark-Dwellers and put it on the Silver Myr vat and Silver Myr back in the graveyard, activate the hasty welder to swap Silver Myr with any random artifact, and sac the welder to kill the Myr again, putting it on a third vat. Make another copy, and now we have UU, arbitrary colorless, 5 cards in hand, and 2 open Mimic Vats, as well as three tapped ones imprinted with Silver Myr, Goblin Welder, and Goblin Dark-Dwellers imprinted. From here, we play Psychic Battle and Precursor Golem, which triggers Mirrorworks 4 times, each of which makes 8 more Golem tokens for 5 Precursors and 40 tokens, and Reality Spasm for 1. If we throw in Horobi and Izzet Guildmage, that lets us put Sphinx and Precursor on Mimic Vats, get up to about 60,000 Golems by the time the fifth trigger resolves, and use that to make a huge number of Sphixes, Welders, blue/black Mana, etc. We can even untap the Dark-Dwellers vat and use it to cast Words of Wisdom, and copy Words with Izzet Guildmage to trigger our thousands of Sphinxes.
By my count, we resolve Words of Wisdom 5 times, plus a 6th to trigger our Sphixes and redraw our Reality Spasm to start the real combo. That leaves the opponent with 47 cards in hand, enough for f_{w^4+w4+3}(48).
Also, a funny thing: with no more Battle Cry, white creatures are an acceptable stage resource again. I wonder if there are any improvements we can make that way.
Actually, here's a real problem: at the stage level, we can't produce enough U to keep going. We get one for each time we can put Myr on a Mimic Vat, but we need 2 U to both replace Rust Tick and replace the Myr in the graveyard with Pull from Eternity, so we can never actually profit on blue Mana. So I think this version might be kaput, unless we can get a way to get multiple cards out of exile with one Pull from Eternity.
Hmm, that's a shame. I guess if we want to keep the Pull from Eternity version, we should go back to not having an infinite colorless mana combo, and use Eye of Ramos or Simic Keyrune for blue mana.
Looking at the start with Reforge the Soul a little bit, I agree that getting access to the full deck plus 6 Reality Spasms after two Reforge the Souls doesn't seem to be happening. But we should have a gajillion Consecrated Sphinxes ready for the third Reforge the Soul, so getting past F_{w^4 + w4 + 3}(5) shouldn't be a problem.
It's probably not the most efficient, but it starts with the regular land+guide+Mox (we want Mox Emerald for startup value)+Vault+Eureka plan, into Reforge the Soul. That lets us play Rebuild, which bounces Mox Emerald and Mana Vault. We play Mox, Mirrorworks, then Panharmonicon. That triggers Mirrorworks twice, and we spend our remainng C and tap Emerald to pay for one. Mirrorworks now triggers 3 times. We play Mana Vault, and tap each one to pay for the next copy trigger, generating X+3=6 mana.
We play Consecrated Sphinx, Spider Spawning, and finally Spellweaver Helix, which triggers several times. We imprint Spawning, Eureka, and Reforge the Soul on it. Our last card is the second Spawning, we play it, and cast the other Sorceries, drawing 21 cards.
From there, we just have to make as many Mirrorworks and Panharmonicon copies as possible. We can produce Mana by killing Mana Vault and reanimating it with Salvaging Station, getting Mirrorworks triggers. If we start by doing that with Goblin Kites and Goblin Trashmaster, that gets us up to 12 Mana.
From there, I play Orrery, Karn and animate Mirrorworks, Mimic Vat (pay for 2 triggers) and Goblin Welder. Sac Goblin Welder to destroy Mirrorworks and put it on a Vat, and put Welder on another vat. Make copies of both, Pull the Mirrorworks back, and use Welder to exchange Mirrorworks for a Mana Vault (the untap trigger on Salvaging Station means we can get it back immediately, and go back from 5 Mana to a lot more). The token makes for 3 Mirrorworks triggers on the nontoken. We pay for 2, and then reanimate Vault like I said. Now, Mirrorworks triggers 3X+3, where X is 1+Panharmonicons (or 3), so that takes us from 1 Mana left to 13, enough to pay for the last Mirrorworks trigger and get a fourth one. We now get 12 triggers per Artifact ETB.
I imagine we can do better with some Cowardice/Panharmonicon shenanigans, but I just went with the naive version: sac the welder token to kill and reanimate Vault again to get up to 25 Mana, play Precursor Golem, get 12 triggers, pay for 11 of them, get a total of 12 Precursors and 72 golem tokens, play Simic Keyrune and pay for one trigger to get UU, play Psychic Battle, and spend all remaining Mana to target a Golem with Reality Spasm. That gives us 12 copy-for-each triggers, the first of which will make 84 copies. The rest can dump Golem on a Mimic Vat and make tokens, bounce Panharmonicon or Mirrorworks, whatever maximizes the Golem count on the next trigger. We should easily be able to clear 10,000 Golems by the twelfth trigger. We can use the last one to make some black Mana, some enchantment copies, etc., And use the rest to make many thousands of Consecrated Sphinx tokens. From there, actually going off should be trivial.
The other bit of good news here is that, since we no longer need to worry about milling cards to get them without drawing, we no longer need Broken Ambitions: our other spells we need in the graveyard can all be cast, besides Dearly Departed, and the Departed is easy enough to kill. So we no longer have to worry about keeping a 0-cmc card in the library, which is nice.
Actually, I just assumed Goblin Kites was a goblin creature. Oops... Replace Kites with Dark-Dwellers, then, or sac the Trashmaster to itself.
I think we can still get the light going by storing our Instants/Sorceries in exile, our artifacts and enchantments in hand/Battlefield, etc., And letting the Light shuffle back Goblins and green 3- drops like Centaur Safeguards, stuff like that.
It's not particularly important, however, since we can just tuck Mox and use Keyrune to make green.
Shoot, I think there is a problem in the hyperstage. We don't need to create a Rust Tick token prior to tapping Moggcatcher to start the hyperstage transition, because Goblin Trashmaster can destroy our nontoken Rust Tick and create hasted tokens during the hyperstage. These hasted tokens can help us get any mana we need for the hyperstage transition. We can get the Mirrorworks trigger for Vedalken Orrery by playing both artifacts after Rebuild returns them to our hand. So, I'm not seeing anything that makes us use a Psychic Battle trigger from Rust Tick.
The original Metallurgeon / Battle Cry seems to be okay in this regard, since we need hasted Rust Ticks to untap for Battle Cry. I think another possibility is Muzzio, Visionary Architect, which just fetches artifacts from the library, so we need a hasted Rust Tick to destroy the nontoken Rust Ticks that we fetch. But the combos that can target artifacts don't seem to work. (unless there is something we have to do in the hyperstage transition that I am missing)
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On a happier note, I have a tentative list! It's at 61 cards, so we need a cut, but as a starting point:
2 Psychic Battle
3 Cowardice
4 Horobi, Death’s Wail
5 Bloodbond March
6 Broken Ambitions
7 Mimic Vat
8 Omniscience
9 Vedalken Orrery
10 Pull from Eternity
11 March of the Machines
12 Izzet Guildmage
13 Kiora’s Dismissal
14 Skull of Orm
15 Mirrorworks
16 Su-Chi
17 Eye of Ramos
18 Rust Tick
19 Pteron Ghost
20 Rend Spirit
21 Rebuild
22 Goblin Dark-Dwellers
23 Moggcatcher
24 Goblin Turncoat
25 Panharmonicon
26 Engineered Explosives
27 Salvaging Station
28 Skyshroud Archer
29 Verdant Succession
30 Dearly Departed
31 Abzan Falconer
32 Bone Harvest
34 Spellweaver Volute
35 Acorn Harvest
36 Smite the Monstrous
37 Child of Alara
38 Centaur Safeguard
39 Centaur Safeguard
40 Spellweaver Helix
41 Righteous Fury
42 Worldfire
43 Spider Spawning
44 Spider Spawning
45 Titania, Protector of Argoth
46 Woodland Bellower
47 Goblin Welder
48 Bayou
49 Eureka
50 Mana Crypt
51 Elvish Spirit Guide
52 Polluted Dead
53 Frightshroud Courier
55 Barrage Ogre
56 Serra’s Sanctum
57 Keldon Overseer
58 Revel in Riches
59 White Sun’s Zenith
60 Consecrated Sphinx
61 Words of Wisdom
The Sanctum is there both to provide higher white Mana and as a manager buffer in the setup that won't take us infinite in the hyperstage. This gives us a gigastage, followed by 3 stages, followed by layers for each Panharmonicon on Keldon Overseer, for each Revel in Riches, for each Hive Mind copying White Sun's Zenith, for each Consecrated Sphinx trigger on Words of Wisdom, for each card in our opponent's library...
That gets us to f_{w^4+w3+4}(X), where X is the number of cards we can keep in the goldfish's library after we complete the setup, which I'm guessing is about 45. The setup seems difficult but definitely not impossible. The problem is finding that one card cut...
2 Psychic Battle
3 Cowardice
4 Horobi, Death’s Wail
5 Bloodbond March
6 Broken Ambitions
7 Mimic Vat
8 Omniscience
9 Vedalken Orrery
10 Pull from Eternity
11 March of the Machines
12 Izzet Guildmage
13 Kiora’s Dismissal
14 Skull of Orm
15 Mirrorworks
16 Su-Chi
17 Eye of Ramos
18 Rust Tick
19 Pteron Ghost
20 Rend Spirit
21 Rebuild
22 Goblin Dark-Dwellers
23 Moggcatcher
24 Goblin Turncoat
25 Panharmonicon
26 Engineered Explosives
27 Salvaging Station
28 Skyshroud Archer
29 Verdant Succession
30 Dearly Departed
31 Abzan Falconer
32 Bone Harvest
34 Spellweaver Volute
35 Acorn Harvest
36 Smite the Monstrous
37 Child of Alara
38 Centaur Safeguard
39 Centaur Safeguard
40 Spellweaver Helix
41 Righteous Fury
42 Worldfire
43 Spider Spawning
44 Spider Spawning
45 Titania, Protector of Argoth
46 Woodland Bellower
47 Goblin Welder
48 Bayou
49 Eureka
50 Mana Crypt
51 Elvish Spirit Guide
52 Polluted Dead
53 Frightshroud Courier
55 Goblin Kites
56 Old Man of the Sea
57 Xathrid Gorgon
58 Reality Spasm
59 Consecrated Sphinx
60 Words of Wisdom
However, I haven't checked if we can get started with this.
Edit: No, we have made use of +1/+1 counters already. If there were a way to just give humans flying without resorting to +1/+1 counters, that would be nice.
Edit: I guess we are using Mirrorworks in order to put a copy of Vedalken Orrery back onto the battlefield after it is eliminated; I don't see a way around that, so it stays in.
I suppose we can use another card to put Reality Spasm back into the library, one that can bring cards from the graveyard or exile back into the library, but not for too cheap. Riftsweeper should work; it doesn't get around Bone Harvest for Skyshroud Archer or Child of Alara, as we need to use Bone Harvest to bring Riftsweeper back into the library anyway. (Or use Woodland Bellower to get Bloodbond March triggers, but that is more expensive still.)
That means we have to cut a card; perhaps we can replace Consecrated Sphinx + Words of Wisdom with Wheel of Fortune or Memory Jar? We would be reduced to just one or two applications of F_{w^4 + w4}, but that would still be better than just three stages of course. The question is whether we can get started.
If not, we could remove the Goblin Kites stage and perhaps go with Reality Spasm + Precursor Golem + Consecrated Sphinx + Words of Wisdom.
We know that, with Horobi, Cowardice, and enough copies of Psychic Battle, targeting Su-Chi with anything is net-positive on Mana. At some point, we should figure out what that number is for this configuration. That way, we'll have a target to aim for in our setup.
In other news, I'm planning on posting the standard combo write-up this week. I've stuck with the Silent Gravestone version because it's the one I most trust to stay finite and because I've already written the relevant sections for it. I'd appreciate it if we could do one more check for infinite combos before I post!
http://alex.shankland.org/index.php/2018/06/13/how-to-deal-more-than-grahams-number-damage-in-magic-the-gathering/
In other news, there's one other possible cut: I only switched to Hive Mind so White Sun's Zenith would give our opponent lots of tokens. If we go back to Swarm Intelligence, though, we can have our lifegain sorcery in the gigastage gain a constant amount of life, keeping track of our progress by Mana turning into Swarm Intelligence copies. So, we can switch Righteous Fury for the famously-OP Archangel's Light, which fixes the graveyard->library problem with Reality Spasm.
Edit: a couple minor things: we can't use Wheel of Fortune, because it's a 3-mana sorcery and can be cast with Dark-Dwellers to trigger Volute. Either Memory Jar or Reforge the Soul will fix that. Let's say Reforge the Soul, because ironically enough it seems like Sorceries will be easier to recur at the setup level than artifacts (we can just leave it under a Spellweaver Helix and go to town with Spider Spawning and any way to generate black Mana we can find).
Then, the big question is tue same one this challenge always seems to end with: how many cards can we get ahead of the goldfish in the deck? Thankfully, the answer for this deck seems fairly high: we can get almost all of our enchantments by milling them with Broken Ambitions and bringing them back with Skull of Orm. By my count, that gets us Bloodbond March, March of the Machines, Opalescence, Spellweaver Volute, and possibly Psychic Battle and even Cowardice.
We can get Centaur Safeguards and Skyshroud Archer via Woodland Bellower, and nearly all of our Instants, besides Broken Ambitions itself, via milling. From what I can see, we need one Reforge the Soul immediately after Eureka, but after that, we can play a Consecrated Sphinx, and the next 21 cards should be all we need to go off: that will give us 35 cards, and I count 25 cards (liberally) that we can fetch out of the library by other means. Leaving one slot spare for Tolarian Academy to kickstart the setup, that leaves just enough room for your extra stage, Reality Spasm, and Consecrated Sphinx. I think that represents a layer for each Swarm Intelligence on Reality Spasm, untapping a Mimic Vat, and a layer for each Consecrated Sphinx. We'll have the ability to play Reforge the Soul 5 more times, so... F_{w^4+w4+2}(5)? F_{w^4+w4+2}(6)? I'm not sure which.
Also, why is it w4 and not 4w? I picked that up from your write-up but I don't get the rationale.
Hmm, is it F_{w^4+w4+2}(5) or F_{w^4+w4+2}(6)? For the latter, we need to get X up to F_{w^4+w4+1}(6) after the second Reforge the Soul, which means we need to cast five or six Reality Spasms at that time. (probably six) So if we can mill/fetch 30 cards, we can use the extra five draws to draw Reality Spasm another five times... it's probably pretty close.
Of course, what would make it easy would be if we could get a second Consecrated Sphinx on the board - for that, we would need a Mimic Vat, perhaps a Smite the Monstrous, and a few mana... would be tricky.
It would be nice if we can get going without the Tolarian Academy, then we could put in [c]Precursor Golem to go up to F_{w^4+w4+3}.
The reason for w4 and not 4w is that AB represents B copies of A put one after the other, so 4w is actually 4+4+4+4..., which is just w. w4 is then w+w+w+w, which is what we want.
(The reason AB is B copies of A rather than A copies of B, near as I can tell, is to make ordinal arithmetic nicer; for example, this way a^(b+c) comes out to a^b * a^c rather than a^c * a^b.)
One new problem: Simian Spirit Guide is a spirit and can be bounced in the hyperstage. Out with Hearth Kami, Rend Spirit, and Goblin Turncoat, in with Sarkhan's Triumph, Goblin Trashmaster, and the (nongreen) French vanilla Changeling of our choice. That also means we need a Simic Mana rock, instead of monoblue.
Frankly, I"ll be shocked if we can get going without at least one Mana boost. I'm even worried the Academy won't be enough. More power to you if you can find a way, though!
Edit: actually, with Trashmaster as the hyperstage transition, we don't need the early boost at all. If we can get out Psychic Battle, Trashmaster, Mimic Vat, Cowardice, and some goblins in hand, that gets us four artifact bounces per goblin (because we can both sac the goblin and sac a vat token), plus tucking and searching out Changeling Sentinel with Dark-Dwellers' ability.
It's very wordy and quite rough, but I've outlined a start that works by turning all that Mana into 3 Bloodbond Marches and 3 Psychic Battles, and using some stack tricks to get two hasty Rust Ticks. That's enough to make a mini-stage worth a few hundred Mana, more than enough to get everything going.
That does involve sufficiently many searchable cards that two Reforge the Soul isn't enough to get everything, but f_{w^4+4w+3}(4) is still a huge improvement.
Concerning the standard deck, here are some corrections to the article:
When you describe the damage of the megastage deck (F_{w^3 + w13 + 3}) you superscript the ordinal and subscript the 3 in w^3, rather than the other way around.
Whenever you mention using Gideon's +1 ability, that should be a +2.
The result of repeatedly casting Cackling Counterpart of Precursor Golem is a lot more complicated. After the first Cackling Counterpart there will be 2 Precursor Golems and 6 vanilla Golems; after the second (assuming you target a vanilla Golem), there will be 8 and 38; after the third, if I calculate correctly, there will be 2048 and 25,858.
When you say "untapping Marwyn, the tapped elves", that is a bit confusing, since you can only untap one elf other than Marwyn. So it should probably be "the tapped elf".
One of the links for Planar Bridge is broken.
After then second Sparring Mummy comes into play, you say you have 5G and R, but I believe you have 6G and R. Doesn't seem to affect anything though.
You mention fetching the three Yargles, but the list has only two.
The list of remaining cards after all the fetching and milling does not include Seal Away.
It looks like you want to link here on the last line of the article, but there is no link.
So mainly just a bunch of minor errors. I didn't see any major problems, although I'm not that good at finding them.
If we switch from March of the Machines to Karn, Silver Golem, Karn+Mana Crypt+Salvaging Station is an infinite Mana combo: tap the crypt for 2, spend 1 to make it a creature, it dies, which untaps station, bring it back, repeat. If we swap Spirit Guide for Mox Emerald, we can also filter to green. None of that's an issue, as long as we still require a "target artifact" to make blue Mana (hello Silver Myr), we still can't go infinite, because you need too many Guildmage activations to keep going otherwise.
That makes the setup much, much easier: with no generic costs to worry about, making a second, even a third Sphinx via Reality Spasm on a vat isn't hard, as long as we can generate the UU needed to cast it in the first place. That doesn't seem very difficult.
Also, thanks for the revisions! Those are all good catches, I'll fix those when I get the chance.
Putting Mana Crypt back in nicely allows for Consecrated Sphinx + Words of Wisdom, I think that has a pretty good chance of being enough.
Does Eye of Ramos or Simic Keyrune go infinite with this setup?
It occurs to me that having a potentially large Changeling in the deck allows us to use Precursor Golem with Smite the Monstrous, which might pave the way to eliminate Swarm Intelligence. However, we just switched Righteous Fury with Archangel's Light, and we need Swarm Intelligence for that.
On the plus side, Karn seems unquestionably better here than March of the Machines, since it's much cheaper to recur artifacts with him vs. March, and therefore easier to set up. We don't have Dual Nature anymore, so we can afford to make things creatures as-needed.
That's a great catch with Precursor! It's certainly worth noting as a potential card cut later on.
One other interesting thing: because we can't refill the library as-needed, we also need to LOSE the clash on Broken Ambitions sometimes. That means we need a 0-drop card we can keep in the library, and also need a way to shuffle our library whenever we switch from Broken Ambitions to Psychic Battle. So we definitely need to swap Elvish Spirit Guide for Mox Emerald, whether or not we have the Keyrune: we can tuck it during the setup.
I'm not sure how to get the shuffle, though...
Well, we need milling for Archangel's Light I guess. Perhaps if we switched out Archangel's Light and go back to Righteous Fury? (Or maybe Divine Congregation is better) Then we could remove Swarm Intelligence, and add a card that will refill the library. Riftsweeper is a possibility, or some card that moves cards from the graveyard to the library directly.
Also, assuming we can get a Goblin Welder on a vat before playing a Reality Spasm, we can also get artifacts out of the library by milling/reanimating, same as enchantments. That seems exceptionally easy with Trashmaster.
Play Swarm Intelligence, Spellweaver Volute on Words, Orrery, and Spider Spawning. Cast Words and exile it, draw 8 cards.
Play Karn/Station, use SSG to get the 1 Mana needed to start, get infinite colorless. Play Mirrorworks, Panharmonicon, and Mimic Vat, paying for all trigger each time. I believe that gets us 3 Panharmonicons and 5 Mimic Vats. Play Pull from Eternity, use the Swarm Intelligence copy to get back words, play Goblin Dark-Dwellers, draw another 8, leaving the original Pull on the stack.
Play Trashmaster/Welder/Silver Myr, and use the sacrifice ability to put Silver Myr and Welder on vats. Make a copy, sac Dark-Dwellers and put it on the Silver Myr vat and Silver Myr back in the graveyard, activate the hasty welder to swap Silver Myr with any random artifact, and sac the welder to kill the Myr again, putting it on a third vat. Make another copy, and now we have UU, arbitrary colorless, 5 cards in hand, and 2 open Mimic Vats, as well as three tapped ones imprinted with Silver Myr, Goblin Welder, and Goblin Dark-Dwellers imprinted. From here, we play Psychic Battle and Precursor Golem, which triggers Mirrorworks 4 times, each of which makes 8 more Golem tokens for 5 Precursors and 40 tokens, and Reality Spasm for 1. If we throw in Horobi and Izzet Guildmage, that lets us put Sphinx and Precursor on Mimic Vats, get up to about 60,000 Golems by the time the fifth trigger resolves, and use that to make a huge number of Sphixes, Welders, blue/black Mana, etc. We can even untap the Dark-Dwellers vat and use it to cast Words of Wisdom, and copy Words with Izzet Guildmage to trigger our thousands of Sphinxes.
By my count, we resolve Words of Wisdom 5 times, plus a 6th to trigger our Sphixes and redraw our Reality Spasm to start the real combo. That leaves the opponent with 47 cards in hand, enough for f_{w^4+w4+3}(48).
Also, a funny thing: with no more Battle Cry, white creatures are an acceptable stage resource again. I wonder if there are any improvements we can make that way.
Could you describe your start in more detail?
We play Consecrated Sphinx, Spider Spawning, and finally Spellweaver Helix, which triggers several times. We imprint Spawning, Eureka, and Reforge the Soul on it. Our last card is the second Spawning, we play it, and cast the other Sorceries, drawing 21 cards.
From there, we just have to make as many Mirrorworks and Panharmonicon copies as possible. We can produce Mana by killing Mana Vault and reanimating it with Salvaging Station, getting Mirrorworks triggers. If we start by doing that with Goblin Kites and Goblin Trashmaster, that gets us up to 12 Mana.
From there, I play Orrery, Karn and animate Mirrorworks, Mimic Vat (pay for 2 triggers) and Goblin Welder. Sac Goblin Welder to destroy Mirrorworks and put it on a Vat, and put Welder on another vat. Make copies of both, Pull the Mirrorworks back, and use Welder to exchange Mirrorworks for a Mana Vault (the untap trigger on Salvaging Station means we can get it back immediately, and go back from 5 Mana to a lot more). The token makes for 3 Mirrorworks triggers on the nontoken. We pay for 2, and then reanimate Vault like I said. Now, Mirrorworks triggers 3X+3, where X is 1+Panharmonicons (or 3), so that takes us from 1 Mana left to 13, enough to pay for the last Mirrorworks trigger and get a fourth one. We now get 12 triggers per Artifact ETB.
I imagine we can do better with some Cowardice/Panharmonicon shenanigans, but I just went with the naive version: sac the welder token to kill and reanimate Vault again to get up to 25 Mana, play Precursor Golem, get 12 triggers, pay for 11 of them, get a total of 12 Precursors and 72 golem tokens, play Simic Keyrune and pay for one trigger to get UU, play Psychic Battle, and spend all remaining Mana to target a Golem with Reality Spasm. That gives us 12 copy-for-each triggers, the first of which will make 84 copies. The rest can dump Golem on a Mimic Vat and make tokens, bounce Panharmonicon or Mirrorworks, whatever maximizes the Golem count on the next trigger. We should easily be able to clear 10,000 Golems by the twelfth trigger. We can use the last one to make some black Mana, some enchantment copies, etc., And use the rest to make many thousands of Consecrated Sphinx tokens. From there, actually going off should be trivial.
The other bit of good news here is that, since we no longer need to worry about milling cards to get them without drawing, we no longer need Broken Ambitions: our other spells we need in the graveyard can all be cast, besides Dearly Departed, and the Departed is easy enough to kill. So we no longer have to worry about keeping a 0-cmc card in the library, which is nice.
I think Spellweaver Helix makes us exile sorceries in pairs, so we can't exile exactly 3. We don't need Eureka though, so that should be fine.
How are we killing Mana Vault with Goblin Kites and Goblin Trashmaster - do you just mean sacrificing the Trashmaster to itself?
I think we still need Broken Ambitions to put cards in the graveyard after Archangel's Light.
I think we can still get the light going by storing our Instants/Sorceries in exile, our artifacts and enchantments in hand/Battlefield, etc., And letting the Light shuffle back Goblins and green 3- drops like Centaur Safeguards, stuff like that.
It's not particularly important, however, since we can just tuck Mox and use Keyrune to make green.
The original Metallurgeon / Battle Cry seems to be okay in this regard, since we need hasted Rust Ticks to untap for Battle Cry. I think another possibility is Muzzio, Visionary Architect, which just fetches artifacts from the library, so we need a hasted Rust Tick to destroy the nontoken Rust Ticks that we fetch. But the combos that can target artifacts don't seem to work. (unless there is something we have to do in the hyperstage transition that I am missing)