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.
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!
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.
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".
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.)
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.
Woodland Bellower looks okay; do we have a way to get back our more expensive green creatures?
Another possibility for avoiding the problem is to get rid of Swarm Intelligence. Currently, we need Swarm Intelligence for Rend Spirit and Smite the Monstrous, as a single Righteous Fury will get us plenty of life. We can deal with the former by going back to Metallurgeon / Battle Cry, and we can deal with the latter by adding Copy Enchantment, so we can place a whole bunch of Spellweaver Volutes on Smite the Monstrous. Then, without Swarm Intelligence, only the original Worldfire can get us our gigastage mana.
Rather than Open the Vaults, we could perhaps just have a creature that fetches artifacts from the graveyard. Arcum Dagsson or Muzzio, Visionary Architect? For Muzzio, we would need a way to put artifacts into the library; for Arcum Dagsson, I'm worried that it might go infinite. Same for Daring Archaeologist. (If either is safe, they would eliminate the need for Salvaging Station.) What about Goblin Welder? We would need to use Moggcatcher to bring it back and sacrifice it for a hasted token. So if we were at all worried about getting to many Engineered Explosives in the gigastage, Goblin Welder would certainly solve that problem, as we have to use up a Moggcatcher token for all of our artifacts! (Well, I guess just the ones we need before the gigastage transition, but that's enough.)
Edit: Sorry, Daring Archaeologist is obviously no good since it brings an artifact back to our hand.
Edit: Stakfish, can you describe your alternative methods for making a hyperstage with Rust Tick? It mnight be useful here, as green mana is cheap in this deck.
Hmm, that seems very iffy, but maybe it could work.
Unfortunately, there is a problem with Essence Scatter: We have to cast our flashback spells at various levels of our stack, so we don't want to have them stuck on the stack. So we need some other way to deal with the Centaur Safeguard problem.
Ooh, good point about Ghitu Chronicler. Still, we can fetch every instant and sorcery in the deck, so if we can generate M red mana pre-combat, and have N instants/sorceries, then we can get 3 -> 3 -> (3 -> 3 -> NM+1 -> 3) -> 3 damage, which is still the best we currently have I think.
Hmm yes, we can give Acorn Harvest flashback. Although, recurring Dralnu, Lich Lord without Goryo's Vengeance is somewhat problematic; to do it in the megastage, it looks like we need to cast an Acorn Harvest (paying 3 life) without destroying Child of Alara, losing 3 life in the process. Then, we can use a Smite the Monstrous to destroy Dralnu, use a Bloodbond March trigger to bring it back, and then do the megastage transition again, this time giving Acorn Harvest flashback with Dralnu, and destroying Child of Alara, gaining our 3 life back. So we wind up even - but, that first Acorn Harvest without destroying Child of Alara will allow us to create a lot more Moggcatcher tokens, so we wind up ahead I think. So yeah, it doesn't work.
The idea behind the second Skyshroud Archer was that we would have both on the battlefield, and we would destroy them both, create two tokens, and put Verdant Succession triggers on the stack. Then we would use a Moggcatcher token to create a hyperstage, getting Engineered Explosives back, and destroy Nantuko Tracer, with the Mimic Vat token allowing us to shuffle the Nantuko Tracer and both of the Skyshroud Archers back into the library, and then the Verdant Succession triggers would bring them back. I think that works, although I may be missing something.
Hmm, nice call with Natural Order. Can we make it so we don't need to bring any cards back to our hand? The first problem is all the artifacts that we have. A card that can bring back artifacts from the graveyard or library would be in order here; Arcum Dagsson+ Goryo's Vengeance would work great, except that we can't have that combo along with Izzet Guildmage. So we have to sacrifice one or the other.
If Run Aground works to gain life, that would be great! Being able to target artifacts seems okay; we don't need Abandoned Sarcophagus, we can use Ghitu Chronicler to cast many more instants than we could with the Sarcophagus. If we can put four The First Eruptions onto the battlefield the turn before, we can use the eight red mana to get to something like 3 -> 3 -> 9 -> 4, and then we attack with Radha to get to 3 -> 3 -> (3 -> 3 -> 9 -> 4) -> 4.
Yeah, for most of our decks it doesn't much matter if we get extra turns; maybe we could turn that into an extra layer or two. The biggest difference would be if it is tough to get started; for example, in our current gigastage deck, we have precious few card slots, and getting started might cost us a few stages.
Edit: Thinking about the theoretical optimal solution, I imagine the circumstances are the same; being restricted to a few turns just means that you will have to devote a very few number of cards to getting set up. In some unusual circumstances this might cause you to just miss being able to complete a powerful construction, but I think it is more likely that the loss will be minor, e.g. you might miss a few stages.
Of course, this is calling a function "slow-growing" if it gets close to its natural place in the fast-growing hierarchy at low values, but that is not really what slow-growing means. For example, TREE(n) grows at the level of theta(Omega^omega * omega) in the fast-growing hierarchy, and already TREE(3) is roughly in that neighborhood, but people don't normally call TREE(n) slow-growing!
Edit: Speaking of the Vintage challenge, I was thinking about it for a bit. I had the happy thought that perhaps Centaur Safeguard could be retrieved after gigastage transition by resolving down the the next megastage transition and getting to a Verdant Succession trigger; this would not only allow us to take care of retrieving Centaur Safeguard without any further cards, it would also take care of any extra uses of Engineered Explosives we would have from the gigastage transition. Alas, if the only way to kill Centaur Safeguard is by destroying Child of Alara, then the Verdant Succession triggers are going to go above any trigger for putting a Vedalken Orrery token on the battlefield, so we won't be able to restart the megastage without resolving all the Succession triggers. So it doesn't look like that perfect scenario can happen.
So, I didn't think of a nonklutzy way to bring back Centaur Safeguard, so I just added River's Rebuke to the deck; I don't think that will eliminate future creature stages, because when you resolve it it eliminates all creature tokens. We still need to handle extra Engineered Explosives, so I went back to Dralnu, Lich Lord. For the Skyshroud Archer problem, I figured we could add another Skyshroud Archer; then we could use one Explosives to destroy the Archers, and another to destroy Nantuko Tracer, getting two Archers for two Explosives, which should work out correctly. So we have:
Ah, I see; so it's not quite done yet. We don't want to be able to draw our instants or sorceries, so the only card I see that we can use is Dukhara Scavenger. Would it be possible to do Thorn Lieutenant / Switcharoo / Seal Away shenanigans with the Scavenger?
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.
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.
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.
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.)
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.
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.
Another possibility for avoiding the problem is to get rid of Swarm Intelligence. Currently, we need Swarm Intelligence for Rend Spirit and Smite the Monstrous, as a single Righteous Fury will get us plenty of life. We can deal with the former by going back to Metallurgeon / Battle Cry, and we can deal with the latter by adding Copy Enchantment, so we can place a whole bunch of Spellweaver Volutes on Smite the Monstrous. Then, without Swarm Intelligence, only the original Worldfire can get us our gigastage mana.
Rather than Open the Vaults, we could perhaps just have a creature that fetches artifacts from the graveyard. Arcum Dagsson or Muzzio, Visionary Architect? For Muzzio, we would need a way to put artifacts into the library; for Arcum Dagsson, I'm worried that it might go infinite. Same for Daring Archaeologist. (If either is safe, they would eliminate the need for Salvaging Station.) What about Goblin Welder? We would need to use Moggcatcher to bring it back and sacrifice it for a hasted token. So if we were at all worried about getting to many Engineered Explosives in the gigastage, Goblin Welder would certainly solve that problem, as we have to use up a Moggcatcher token for all of our artifacts! (Well, I guess just the ones we need before the gigastage transition, but that's enough.)
Edit: Sorry, Daring Archaeologist is obviously no good since it brings an artifact back to our hand.
Edit: Stakfish, can you describe your alternative methods for making a hyperstage with Rust Tick? It mnight be useful here, as green mana is cheap in this deck.
Unfortunately, there is a problem with Essence Scatter: We have to cast our flashback spells at various levels of our stack, so we don't want to have them stuck on the stack. So we need some other way to deal with the Centaur Safeguard problem.
Broken Ambitions and Kiora's Dismissal look good.
Hmm yes, we can give Acorn Harvest flashback. Although, recurring Dralnu, Lich Lord without Goryo's Vengeance is somewhat problematic; to do it in the megastage, it looks like we need to cast an Acorn Harvest (paying 3 life) without destroying Child of Alara, losing 3 life in the process. Then, we can use a Smite the Monstrous to destroy Dralnu, use a Bloodbond March trigger to bring it back, and then do the megastage transition again, this time giving Acorn Harvest flashback with Dralnu, and destroying Child of Alara, gaining our 3 life back. So we wind up even - but, that first Acorn Harvest without destroying Child of Alara will allow us to create a lot more Moggcatcher tokens, so we wind up ahead I think. So yeah, it doesn't work.
Oh right, Salvaging Station.
The idea behind the second Skyshroud Archer was that we would have both on the battlefield, and we would destroy them both, create two tokens, and put Verdant Succession triggers on the stack. Then we would use a Moggcatcher token to create a hyperstage, getting Engineered Explosives back, and destroy Nantuko Tracer, with the Mimic Vat token allowing us to shuffle the Nantuko Tracer and both of the Skyshroud Archers back into the library, and then the Verdant Succession triggers would bring them back. I think that works, although I may be missing something.
Reclaim is a good idea, but to prevent us from using Mirror of Fate to put our cards back in the library, we added Izzet Guildmage and Pull from Eternity, so we can also cast Reclaim cheaply.
Hmm, nice call with Natural Order. Can we make it so we don't need to bring any cards back to our hand? The first problem is all the artifacts that we have. A card that can bring back artifacts from the graveyard or library would be in order here; Arcum Dagsson+ Goryo's Vengeance would work great, except that we can't have that combo along with Izzet Guildmage. So we have to sacrifice one or the other.
Other than that... we have Horobi, Death's Wail and Yuki-Onna. Switching to Goryo's Vengeance would take care of those as well. Then we have Moggcatcher and Abzan Falconer. Can we set up Bloodbond March triggers in preparation for the gigastage transition? I don't see why not.
So it looks like this almost works, except that we can't have both the Arcum Dagsson / Goryo's Vengeance combo to make sure that we can bring back everything, and also the Izzet Guildmage / Pull from Eternity combo so that we can use Nantuko Tracer or Reclaim to soak up our uses of Engineered Explosives in the megastage transition. I think we're close, though.
If Run Aground works to gain life, that would be great! Being able to target artifacts seems okay; we don't need Abandoned Sarcophagus, we can use Ghitu Chronicler to cast many more instants than we could with the Sarcophagus. If we can put four The First Eruptions onto the battlefield the turn before, we can use the eight red mana to get to something like 3 -> 3 -> 9 -> 4, and then we attack with Radha to get to 3 -> 3 -> (3 -> 3 -> 9 -> 4) -> 4.
Yeah, for most of our decks it doesn't much matter if we get extra turns; maybe we could turn that into an extra layer or two. The biggest difference would be if it is tough to get started; for example, in our current gigastage deck, we have precious few card slots, and getting started might cost us a few stages.
Edit: Thinking about the theoretical optimal solution, I imagine the circumstances are the same; being restricted to a few turns just means that you will have to devote a very few number of cards to getting set up. In some unusual circumstances this might cause you to just miss being able to complete a powerful construction, but I think it is more likely that the loss will be minor, e.g. you might miss a few stages.
Of course, this is calling a function "slow-growing" if it gets close to its natural place in the fast-growing hierarchy at low values, but that is not really what slow-growing means. For example, TREE(n) grows at the level of theta(Omega^omega * omega) in the fast-growing hierarchy, and already TREE(3) is roughly in that neighborhood, but people don't normally call TREE(n) slow-growing!
Edit: Speaking of the Vintage challenge, I was thinking about it for a bit. I had the happy thought that perhaps Centaur Safeguard could be retrieved after gigastage transition by resolving down the the next megastage transition and getting to a Verdant Succession trigger; this would not only allow us to take care of retrieving Centaur Safeguard without any further cards, it would also take care of any extra uses of Engineered Explosives we would have from the gigastage transition. Alas, if the only way to kill Centaur Safeguard is by destroying Child of Alara, then the Verdant Succession triggers are going to go above any trigger for putting a Vedalken Orrery token on the battlefield, so we won't be able to restart the megastage without resolving all the Succession triggers. So it doesn't look like that perfect scenario can happen.
So, I didn't think of a nonklutzy way to bring back Centaur Safeguard, so I just added River's Rebuke to the deck; I don't think that will eliminate future creature stages, because when you resolve it it eliminates all creature tokens. We still need to handle extra Engineered Explosives, so I went back to Dralnu, Lich Lord. For the Skyshroud Archer problem, I figured we could add another Skyshroud Archer; then we could use one Explosives to destroy the Archers, and another to destroy Nantuko Tracer, getting two Archers for two Explosives, which should work out correctly. So we have:
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 Pull from Eternity
11 Izzet Guildmage
12 March of the Machines
13 Shriekhorn
14 Nantuko Tracer
15 Allay
16 Skull of Orm
17 Mirrorworks
18 Simic Keyrune
19 Mana Vault
20 Rust Tick
21 Yuki-Onna
22 Rend Spirit
23 Rebuild
24 Swarm Intelligence
25 Goblin Dark-Dwellers
26 Goblin Turncoat
27 Moggcatcher
28 Panharmonicon
29 Engineered Explosives
30 Skyshroud Archer
31 Skyshroud Archer
32 Dearly Departed
33 Abzan Falconer
34 Verdant Succession
36 Acorn Harvest
37 Smite the Monstrous
38 Child of Alara
39 Centaur Safeguard
40 Centaur Safeguard
41 Spellweaver Helix
42 Righteous Fury
43 Worldfire
44 River's Rebuke
45 Bump in the Night
46 Bump in the Night
47 Dralnu, Lich Lord
48 Lilliana's Mastery
49 Plateau
50 Titania, Protector of Argoth
51 Replenish
52 Judge of Currents
53 Streambed Aquitects
54 Giant Oyster
55 Grand Architect
57 Minion of Tevesh Szat
58 Reality Spasm
59 Consecrated Sphinx
60 Words of Wisdom
Edit: A few comments on the article, mostly nit-picky:
The problematic Legion Lieutenant is still on the list.
Early on you say you deal the damage on turn 7, but it looks like it happens on turn 6 this time.
At one point you say "uncounterable number"