Nah, Sage's Knowledge + Fork is better than Timetwister. Timetwister does allow you to reuse the whole deck, but all the non-Runic Repetition cards put together have less of an effect than a single Runic Repetition. So a single copy of Timetwister has less of an effect than 5 copies of Sage's Knowledge, since those could fetch Runic Repetition 5 times. This still means that 1 casting of Timetwister is more effective than 1 casting of Sage's Knowledge. But that 1 casting will increase our number of Thousand-Year Storms a gajillion times, so a second casting of Sage's Knowledge will get way more than 5 times as many copies as the original Timetwister had. More generally X+1 castings of Sage's Knowledge will beat X castings of Timetwister. So in the Timetwister deck, one copy of Praetor's Counsel can allow for one more casting of Timetwister, but in the Sage's Knowledge deck, one copy of Praetor's Counsel will allow for one more casting of Fork, gets a gajllion copies, and all of those copies can copy Praetor's Counsel and fetch Sage's Knowledge. So we get a gajlllion castings of Sage's Knowledge rather than one casting of Timetwister.
Whoops, I forgot what Reiterate was for a second. So yeah, go with Fork or one of its many variants. But we can't mix Fork or Doublecast with Timetwister and Praetor's Counsel, since Timetwister can allow us to redraw Doublecast, and Doublecast can fetch Timetwister by copying Praetor's Counsel. So we need to switch Timetwister for something that cannot fetch Fork, like Sage's Knowledge.
Edit: No, I guess Doublecast will not go infinite with Timetwister. But given the way it works, it doesn't add to the sequence. So we still would rather have Sage's Knowledge + Fork than Timetwister.
Interesting deck! I see you put a lot of time into it. I don't see any infinities offhand, so I will analyze what numbers it reaches.
At the bottom, we have to damage chain, so let's start with that. If A is the number of creatures that get created (buttressed by Doubling Season), B is the number of Intruder Alarms (so we get AB untaps) and C is the number of Rowan emblems on the battlefield (so we get C copies of each tap ability), then we can tap those creatures A(AB + A) times, for A(AB+A)(C+1) copies of the ability, so the damage chain yields roughly 2^[A(AB+A)(C+1)] counters. Then, if we have D copies of Angelic Chorus, our life increases by roughly D*2^[A(AB+A)(C+1)]. Note that we cannot increase our number of Rowan Kenriths at this level. We can incrase our number of Intruder Alarms and Angelic Choruses a little higher up, but it turns out that this is never a good idea; we get much more benenfit by copying Doubling Season, since each one has an exponential effect, and that effect applies to pretty much everything. So our life increases by roughly 2^(cX^2) where c is a constant, or exponential growth. This puts us at layer 1.
Next we have the other effect of adding more creatures. Most of the time when we add a token creature, we will want all the tokens to copy Doubling Season. If we have X Doubling Seasons in play, the effect of a token creation is to actually create 2^X more tokens, which will usually be or turn into Doubling Seasons, so each such ability takes X to 2^X. So this is layer 1 again.
Going through the rest of the combo:
Layer 1: With X Doubling Seasons in play, an ability to create a token can create 2^X Doubling Seasons.
Layer 2: With X Riku of Two Reflections in play, putting a nontoken creature into play will trigger token creation X+1 times, each time taking X to 2^X. The last creation can create more Rikus, so we wind up taking X Rikus to 2^^(X+1) Rikus. Casting any noncreature spell will trigger the flickering of Ojutai Exemplars, and therefore this layer. Casting White Sun's Zenith will achieve this layer in another way, since if we copy it the spell X times we will take X to 2^^(X+1) as well.
Layer 3: Resolving a copy of Cackling Counterpart will create us to create 2^X Teferi, Temporal Archmages, where X is the number of Doubling Seasons on the battlefield. If we have Y Rowan Kenriths on the battlefield, then we can create 2^X(Y+1) Teferi abilities, each of which can draw and recast White Sun's Zenith, so we can take X to 2^^^(2^X(Y+1)). We could also copy Rowan Kenrith and use his ability to increase Y, but unfortunately, the number of emblems doesn't increase anywhere near as fast as everything else, since unfortunately emblem creation isn't affected by Doubling Season. So the best we can do is to use the first 2^X-1 copies to copy Rowan Kenrith, and the last copy to copy Teferi Temporal Archmage. We can use the activate the 2^X-1 Rowan Kenriths' ultimate abilities, and each one doubles the number of emblems. So we wind up with Y*2^(2^X - 1) emblems, so we get 1+Y*2^(2^X-1) copies of Teferi's ability and we take X to 2^^^[Y*2^(2^X-1)].
Layer 4: Each time we resolve a Precursor Golem trigger for Cackling Counterpart with X creatures in play, we get X-1 copies of Cackling Counterpart, and we can use Goblin Flectomancer to retarget the target to Mirage Mirror. So this takes X to 2^^^^(X-1).
Layer 5: Each time we cast Cackling Counterpart, we get X Precursor Golem triggers, where X is the number of Precursor Golems in play. So this takes X to 2^^^^^X.
Layer 6: Each time we cast Runic Repetition, we can copy it X times to retrieve Cackling Counterpart X+1 times, taking X to 2^^^^^^(X+1).
Layer 7: Each time we cast Timetwister we can copy it X times to redraw Runic Repetition X+1 times, taking X to 2^^^^^^^(X+1).
Layer 8: Each time we cast one the exile-for-retrieval spells, we can copy it X times to retrieve Timetwister X+1 times, taking X to 2^^^^^^^^(X+1).
There are various other effects and increases being done, but they are insignficant compared to this main sequence. Since we have 20 exile-for-retrieval spells, the final damage is between 2^^^^^^^^^22 and 2^^^^^^^^^23.
It would definitely help to add Thousand-Year Storm, since we get the Precursor Golem of effect of creating a bunch of spells, and each trigger copying the original spell a bunch of times. So that would change things to:
Layer 6: Each TYS trigger for Runic Repetition creates X copies of Runic Repetition.
Layer 7: Each time we cast Runic Repetition, we get X TYS triggers.
Layer 8: Each TYS trigger for Timetwister creates X copies of Timetweister.
Layer 9: Each time we cast Timetwister, we create X TYS triggers.
Layer 10: Each TYS trigger for Praetor's Counsel creates X copies of Praetor's Counsel.
Layer 11: Each time we cast Praetor's Counsel, we create X TYS Triggers.
So that ups the damage to between 2^^^^^^^^^^^^22 and 2^^^^^^^^^^^^23. (12 arrows)
As for improving the deck, it would be nice to extend the chain of retrieval cards. Here's a way to improve the chain by one: Replace Timetwister with Sage's Knowledge and Reiterate. Sage's Knowledge can fetch Runic Repetition. Reiterate can copy an exile-for-retrieval spell and fetch Sage's Knowledge. Then the exile-for-retrieval spells can fetch Reiterate. This adds two arrows, getting us to 2^^^^^^^^^^^^^^22 (14 arrows).
Interesingly, it seems that Alpha cards were unreservedly legal in the very early days of Magic - for example, the very first DCI tournament rules make no mention of Alpha cards.
To answer the intended question, in the 1998-1999 DCI tournament rules it states that decks must consist of all Alpha or all non-Alpha cards. In the 2000-2001 DCI tournament rules, it states that you could mix Alpha and non-Alpha cards provided you used opaque sleeves for your deck. I couldn't find the 1999-2000 tournament rules on the internet, but at least this narrows it down.
If that doesn't work, it wouldn't bother me at all for the writeup to be spread over multiple posts. You might also put spoiler tags around each section, so we can open and close them at our leisure.
Okay, so you can cast the first five spells, then cast Reiterate six times with buyback.
After casting the enchantments you cast Rite of Replication (unkicked, so that we can cast Reiterate the most times), which doesn't get copied. We can choose to target either Doubling Season or Thousand-Year Storm; I think it makes more sense to target Thousand-Year Storm, so that Radiate will make copies of Doubling Season, which we need to do in order to get exponential growth. Then you cast Radiate, which gets one copy from Thousand-Year Storm. That copy creates one copy if Rite of Replication, which targets Doubling Season. So we create two token copies of Doubling Season.
We don't want to resolve either Radiate or Rite, since we need to copy them. So we cast Reiterate next, with buyback. The Thousand-Year Storm triggers, creating two copies of Reiterate. So we get three Reiterates total, and each time we create a copy of Radiate. The next Reiterate will create three copies of Reiterate from Thousand-Year Storm, then 4,5,6,7. So we get a total of 3 + 4 + 5 + 6 + 7 + 8 = 33 Reiterates in total.
The first Radiate will target the three Doubling Seasons, getting us to 3 + 2^3 = 11, then 11 + 2^11 = 2059, then 2059 + 2^2059 ~ 6.6 * 10^619 Doubling Seasons.
The second Radiate will target all 6.6 * 10^619 Doubling Seasons, so we get more than 10 ^^ (10^619) > 10 ^^^ 2 Doubling Seasons in the end. Then the third will get more than 10 ^^ 10 ^^ (10^619) > 10 ^^^ 3 Doubling Seasons, and so on, until the 33rd generates more then 10 ^^^ 33 Doubling Seasons.
That's more than the old 7, but less than the improvements that I mentioned. The major problem is that we aren't getting more Thousand-Year Storms. So I guess it makes sense to use one of the Reiterates to copy Rite of Replication rather than Radiate, so that we can get both enchantments going.
So, let's go back to the first casting of Reiterate. We don't need more Thousand-Year Storms until the next Reiterate is cast, so let's use the first two Reiterate copies to copy Radiate; that takes us to more than 10 ^^ (10^619) Doubling Seasons. Then we use the original Reiterate to copy Thousand-Year Storm, which creates 10 ^^ (10^619) Thousand-Year Storms.
We then cast Reiterate the second time, and now we get more than 10 ^^ (10^619) copies. Each one takes X Doubling Seasons to 10 ^^ X Doubling Seasons, so by the end we will have more than 10 ^^^ (10 ^^ (10^619)) Doubling Seasons and Thousand-Year Storms. The same thing happens for each of the next Reiterate castings, so by then end we will have between 10 ^^^^ 6 and 10 ^^^^ 7 creatures. More than before, but still less than the last deck I mentioned. (Of course, you do have a seventh card to make things bigger!)
Edit: So, the following seems like a good entry into the seven card challenge:
So, we start off by casting the enchantments. We cast Cackling Counterpart, targeting Doubling Season. We cast Radiate, getting one copy via Thousand-Year Storm. The copy copies Rite of Replication, targeting Thousand-Year Storm, creating two Thousand-Year Storm copies.
Next, we cast Increasing Vengeance, which triggers all three Thousand-Year Storms, each one creating two copies of Increasing Vengeance. So we will get seven copies of Increasing Vengeance in total. We can use the first to copy Cackling Counterpart, getting two more Doubling Seasons. Then the next six copies can copy Radiate. The first targets two Doubling Seasons and three Thousand-Year Storms, getting 11 and then 2059 Doubling Seasons, and then 3 + 3*2^2059 Thousand-Year Storms. The second Radiate will create more than 10 ^^ 2059 Doubling Seasons and Thousand-Year Storms, the third 10 ^^ 10 ^^ 2059, and by the sixth we have more than 10 ^^^ 6 Doubling Seasons, and Thousand-Year Storms.
Finally we cast Mystic Retrieval, which will trigger more than 10 ^^^ 6 Thousand-Year Storms. We can use the first trigger to fetch the three spells from the graveyard, and then cast the three again. Keep the Cackling Counterpart and Radiate on the stack, and only resolve the Increasing Vengeance card. Then each copy of Mystic Retrieval can fetch Increasing Vengeance. Each Increasing Vengeance spell can copy Radiate, which takes X to 10 ^^ X; casting Increasing Vengeance triggers many copies, so we take X to 10 ^^^ X; casting Mystic Retrieval triggers many copies, each of which can fetch Increasing Vengeance, so we take X to 10 ^^^^ X. So we wind up with more than 10 ^^^^ (10 ^^^ 6) creatures after Mystic Retrieval fully resolves. Then we can flashback it from the graveyard, so we get a final number of more than 10 ^^^^ (10 ^^^^ (10 ^^^ 6)).
Eh, that's okay I guess. I'm pretty sure we can do more.
Edit: Wait, I'm being stupid. Each Increasing Vegeance Spell can copy Mystic Retrieval, which can fetch Radiate from the graveyard. A single Radiate spell takes X to 10 ^^ X; a casting of Radiate creates many Thousand-Year Storm copies, which takes X to 10 ^^^ X. Each Increasing Vengeance can fetch a Radiate to recast it. Each Increasing Vengeance cast creates many copies, so it takes X to 10 ^^^^ X. Each Thousand-Year Storm trigger for Mystic Retrieval creates many copies of Mystic Retrieval, each of which can fetch Increasing Vengeance, so it takes X to 10 ^^^^^ X. And finally, casting Mystic Retrieval creates many Thousand-Year Storm triggers, so it takes X to 10 ^^^^^^ X.
So we wind up with more than 10 ^^^^^^ (10 ^^^^^^ (10 ^^^ 6) creatures. Much better.
So, getting back to the gigastage deck, I was examining again the idea of switching out Mimic Vat for Soul Foundry. Unfortunately Soul Foundry does not grant its created tokens with haste, so the idea is to add Brudiclad to give tokens haste. This in turn runs afoul of artifacts because of Mirrorworks, so we want to remove that. That means that we will have trouble using Mirror of Fate, so we go back to Pull from Eternity and Izzet Guildmage.
With those changes in hand, we bring in Godtoucher to bounce Boggart Mob. Godtoucher can also target Child of Alara, but now we can only bounce it back to our hand, which does not trigger the destruction of permanents. So I think this is safe.
One minor correction: You can't copy Opalescence with Riku, but I imagine that is just a typo.
The number you get after four Parallel Evolutions is a lot more than 35 million digits long:
After the first Parallel Evolution, you copy the two Parallel lives tokens, and each copy gets multiplied by 2^3 (since you have 3 Parallel Lives total) so you wind up with 2 + 2*2^3 = 18 Parallel Lives tokens.
After the second, you get 18 + 18 * 2^19 = 9437202 tokens total.
After the third, you get 9437202 + 9437202 * 2^9437203 ~ 1.42 * 10^2,840,888
After the fourth, you apply the formula one more time to get more than 10^10^2,840,887 Parallel Lives tokens.
Next comes the Rites and Radiates. each copy of Rite will add another "10^" to the exponential tower (the highest exponent will drop by an infinitesimal amount each time, but it will never dip below 2,840,887), and the first Radiate creates more than 10^10^2,840,887 Rites, so you will wind up with more than an exponential tower of 10^10^2,840,887 10's. In Knuth arrow notation, this is written 10^^(10^10^2,840,887). (More generally, a^^b is defined as a^(a^(...(a^a)...)) with b copies of a.) Similarly, the second Radiate will create more than 10^^(10^10^2,840,887) copies of Rite, so we will get more than 10^^10^^(10^10^2,840,887) token copies in the end. Since we wind up with six copies of Radiate in total, the final number will be more than 10^^10^^10^^10^^10^^10^^(10^10^2,840,887) token copies. We can simplify this with triple arrow notation: a^^^b = a^^(a^^...(a^^a)...) with b copies of a. So the final number is more than 10^^^7 but less than 10^^^8.
If I may make a suggestion, we could improve the number by replacing Riku of Two Reflections with a spell copier that we can have multiples of, like Swarm Intelligence. This would mean we don't get copies of Parallel Lives to start with, so casting Parallel Lives early on doesn't help. So perhaps we could replace that with [c]Clone Legion. We'll get only two copies of Clone Legion; the first will take us to 27 Parallel Lives and Swarm Intelligences, the second to 3623878683 each. That is a lot less than before, but now look what happens with the Radiates; The first cast gets copied 3623878683 times so we wind up with more than 10^^^3623878684 Parallel Lives and Swarm Intelligences. We then cast Radiate two more times, so we wind up with more than 10^^^10^^^10^^^3623878684 creatures, or between 10^^^^4 and 10^^^^5 creatures in the end.
Another improvement: Replace Mystic Retrieval with Bound // Determined. We can only cast Bound once, but since it is an instant, we can cast it while Rite of Replication is still on the stack. So we cast it after all the Radiates resolve, but while the original Rite is still on the stack. I just noticed that we can resolve the copies of the original Rite before we start resolving Radiates; so after the Rites resolve we will have more than 10^^3623878684 Swarm Intelligences, then the Radiates resolve and we go to 10^^^10^^3623878684 copies. Then we cast Bound, getting more than 10^^^(10^^3623878684) copies, and each one can fetch Radiate and recast it, increasing the final number to 10^^^^(10^^^(10^^3623878684)) creatures.
I have updated the OP with links to a few of the explanatory posts.
EDIT: I've been wondering whether the "target artifact" combos like Arcum Dagsson or Goblin Trashmaster could be made to work after all. They would certainly go infinite in the old version of the combo, but in this new version we have a lot of new moving parts, and maybe those can't be refreshed as easily as the artifacts. But I'm not sure.
If it still goes infinite, then I think we can still make it work by getting rid of Mirrorworks. Just replace Mirror of Fate and Mirrorworks with Izzet Guildmage and Pull from Eternity, I think. That resolves the problem of requiring more Psychic Battle triggers of our stage artifact, since we will need lots of mana for the stage transition and one copy of Mana Vault and a Keyrune won't be enough. However, we may have the Muzzio problem of being able to bring a copy of whatever we are using to target/sacrifice an artifact below the hyperstage transition, and getting an extra use out of them. Stakfish, do you know an artifact combo that would be safe from this happening?
EDIT: It just occurred to me that getting rid of Mirrorworks can also solve the problem we had with the Soul Foundry deck. That deck also used Muzzio, so we have to switch that out as well, but perhaps we can squeeze out an extra stage or two with that deck.
Very nice summary. I will definitely link it in the OP when I have time, right now I am abroad without a lot of time, and unsure of how to copy post links on mobile. But putting important posts in the OP is something I definitely should have been doing a long time ago.
Skull of Orm looks like a good cut! As for fetching Academy Rector, we can just use Misery Charm, can we not? That will allow the fifth stage back in, getting us to w^4 + w5 + 4.
Concerning the problem FireRogue517 mentioned, there was indeed an error with the early setup, which came about because I was greedy and tried to squeeze in another couple of copies of Dual Nature by having Copy Enchantment copy Dual Nature at strategic points. The additional Dual Nature means one more Dual Nature token is created, but when we bounce the Copy Enchantment (which is copying Dual Nature) card, we trigger the Dual Nature destruction abilities. I believe there was one point where doing this is okay, but there was another point where it was not. That problem could be easily fixed by simply not having Copy Enchantment copy Dual Nature at the wrong time; this will cost us a Dual Nature token, but not a big deal. Of course, the gigastage deck doesn't currently have Dual Nature in it, although perhaps we should keep an eye for its possible inclusion whenever it may become legal, since it may be more efficient.
Oh phooey... I guess being able to save Muzzio through the hyper stage transition is what causes the infinite. So there's currently no other setup for the main stage other than Metallurgeon?
If not, then it looks like the best we can do is to remove the Merfolk stage and end with World at War, getting to w^4+w4+4. I'm not seeing any cuts, although the need for both Salvaging Station and Goblin Welder is driving me nuts.
Heh, the ultracombo article. When I started writing it, I had every intention of finishing it. But, as the gigastage deck started coalescing, I lost the motivation to complete a description of what would (perhaps) become an outdated deck. Maybe if this deck never quite comes together, I will write up the older version.
As for this version, we haven't started writing up anything for it, as it is still in the development stages. I can summarize it briefly I suppose.
The setup is a little different from the previous version. Like before, the basic mechanic is to generate more and more Psychic Battle and Bloodbond March enchantments, and the number of these enchantments are basically our "current number" throughout the combo. However, in this version we removed Dual Nature, since it causes certain problems (currently the problem is with Child of Alara I think). So to get lots of copies of enchantments, we destroy them (using Psychatog in the above deck), and then imprint them on a Mimic Vat to generate a token copy. Then we use Skull of Orm to bring the enchantment back so we can sacrifice it again. This is less efficient then Dual Nature, but that won't affect the final estimate.
The primary stage is similar to the previous version - we have an artifact creature (either Metallurgeon or Rust Tick) that functions as our stage resource, and the stage is built up with alternating Psychic Battle and Bloodbond March groups of triggers.
The hyperstage is the biggest difference in the new version. In the previous version, we had Spellweaver Volute in the hyperstage andSpellweaver Helix in the megastage - in this version, those two cards are bumped up to the megastage and gigastage respectively, and the hyperstage is what's new here. The new spell triggerer is Goblin Dark-Dwellers, which can cast an instant or sorcery of CMC 3 or less when it enters the battlefield. With Panharmonicon, it can cast multiple spells. So we use Battle Cry, or Goryo's Vengeance couple with Muzzio, Visionary Architect, to get lots of Metallurgeons or Rust Ticks; we use Rebuild to regain our hyperstage resource, but at the cost of losing all our stage creatures.
The hyperstage mechanism is actually pretty complicated, as we couldn't get anything simple to work out. So, Rebuild has to replenish our hyperstage resource (which will turn out to be Moggcatcher tokens). It does this by bringing Engineered Explosives back to our hand, so that it can be cast again with a counter, and sacrificed. (Note that if we just bring Engineered Explosives back from the graveyard to the battlefield, it won't get a counter and we can't use it properly - this is quite handy.) Sacrificing Explosives will destroy Scattershot Archer, which can then be imprinted on a Mimic Vat to create a hasted copy. (Engineered Explosives can be brought back from the graveyard using Muzzio or Salvaging Station, while Scattershot Archer can be brought back using Verdant Succession.) The Scattershot Archer can then be activated to destroy Moggcatcher and Strongarm Thug, with the aid of Dearly Departed, Abzan Falconer, and Gratuitous Violence. The Moggcatcher can be imprinted to create a Moggcatcher token, and our hyperstage resource is renewed; the Strongarm Thug can be imprinted to create a token copy, which along with Panharmonicon can bring back both the Moggcatcher and Strongarm Thug to our hand. The Moggcatcher can then be tapped to bring Goblin Dark-Dwellers from the graveyard, which triggers the casting of our cheap instants, including Rebuild to bring Engineered Explosives back to our hand again.
The megastage, like the hyperstage of our previous version, uses Spellweaver Volute as our spell triggerer, and it triggers on the casting of an sorcery. So we use Acorn Harvest as our sorcery (which has CMC greater than 3, so no conflict with Goblin Dark-Dweller), which we can flashback for mana (which we can get cheaply) and 3 life. So life becomes are megastage resource. The instant that renews our hyperstage resource, and the instant that renews our megstage resource but eliminates all of our hyperstage resource, are combined into one: Smite the Monstrous. When we cast Acorn Harvest, we get many copies of Smite the Monstrous via Thousand-Year Storm. Most of the copies can be used to bounce Goblin Dark-Dwellers, which can then be replayed to cast our cheap instants and build up the hyperstage. (Note: we can't rebuild the hyperstage all at once, rather it gradually gets built as it resolves, with the lower Smite the Monstrous triggers standing as placeholders for hyperstage layers until the eventually get resolved and replaced by Battle Cry/Rebuild.) A few more Smite the Monstrous can be used to bounce large creatures that we need to return to our hand. The last Smite the Monstrous is used to destroy Child of Alara, which destroys all nonland permanents, and therefore all of our hyperstage resources (Moggcatcher tokens), but also destroys Centaur Safeguard, which allows us to gain 3 life, renewing our megastage resource. Note that we can't use multiple copies of Smite the Monstrous to destroy Child of Alara and therefore Centaur Safeguard multiple times, because - hmm, I'm forgetting the exact way that works. Stakfish, could you go over that again?
The gigastage uses Spellweaver Helix to cast sorceries, which can only be triggered by a sorcery we have more than one of, so no flashbacking Acorn Harvest to trigger it. So only Spider Spawning can trigger it, which requires a black mana to cast it via flashback. The we get a Frightshroud Courier stage that produces black mana, and a few stages after that; the last deck gets us to around F_{w^4 + w5 + 4}, assuming it works properly.
As for the challenges the deck presents, there have been many. Having four different stage types in the main combo leads to a lot of possible conflict with one level potentially affecting another in undesirable ways. And the newest hyperstage is partcularly complicated; the more complicated, the greater the chance of a flaw or a hidden infinite. Probably the greatest lingering problem is the potential infinite that can come from the megastage and gigastage transitions potentially renewing lower level resources improperly. For example, the way the megastage transition is supposed to work is that we spend three life, go to a higher hyperstage, and gain three life back; but to do so we need to use up a minimal amount of resources from below the megastage transition so as not to go infinite. (Basically, we need to use a few Metallurgeon/Rust Tick triggers to set up the megastage transition). Unfortunately, passing to the higher hyperstage allows us to bring Engineered Explosives and Goblin Dark-Dwellers back to our hand, and keep them in our hand when we pass back down to the lower hyperstage. This allows us to add a couple Moggcatcher tokens, which more then compensates for a couple of lost Metallurgeons, so that goes infinite. So we need to find a way for the megastage transition to use up those extra uses of Engineered Explosives that we gain, and that has been plaguing us for a while. We may have solved it with the necessity to resolve Verdant Succession to gain certain green creatures back, but I'm not totally sure there isn't a hidden infinite somewhere.
Another problem is that Iijil has left, and he was the best one at finding problems and infinities in our prospective decks.
Stakfish, feel free to point out any further details that are worth mentioning.
Yeah, I also would prefer to use Psychic Battle over Grip of Chaos, so that the probabilities remain "reasonable". I had a discussion with Iijil over this, and he was of the opinion that we should just present the deck that had the maximum possible damage. I guess that makes sense. I think, if our deck winds up favoring Grip of Chaos, we would write up the article about that deck, but include an addendum on how things change if we want to replace Grip of Chaos with Psychic Battle.
So long as Grip of Chaos is not in the deck, there is really nothing that requires chance other than card drawing, and with Mirror of Fate we can arrange the library however we want to. So the probability of getting the combo off only includes the cards we draw up until we start using Mirror of Fate, so really it can be no worse (and in fact will be much better than) 1 in 60!, which I would consider firmly in "imaginable numbers".
About your question: Yes, any possible infinite damage combo is prohibited; I'm not sure what you mean by "optimal luck" never reaching that state, could you clarify?
I played around certain times with combos that would be finite if the opponent always acted in a way to keep the maximum damage down, rather than cooperate to go infinite; but Iijil and Stakfish seem to like the rules as they are.
Yes, the fast-growing hierarchy is consistent with the definition of ordinals. Now, F_w (64) is not greater than F_m (64) for all finite ordinals m, and cannot possibly be, since F_m (64) can be arbitrarily large. But, F_w (n), as a function of n, is faster growing than F_m (n) for any fixed m; we have F_w (n) = F_n (n) < F_m (n) for n < m, but once n passes m we have F_w (n) = F_n (n) > F_m (n). So past a certain point F_w (n) will always be greater. More generally, if we have two ordinals a < b < epsilon_0, then F_b will be a faster growing function than F_a. So the ordinals match up orderwise with their corresponding functions.
This might strike one as an academic point, since F_w (n) = F_n (n) doesn't seem like a major leap forward from the finite ordinals. But really it is. To see the big difference, look at what happens with F_{w+1} (n). For example, take F_{w+1} (4). This is equal to F_w (F_w (F_w (F_w (4)))). First we have F_w (4) = F_4 (4) ~ an exponential stack of 10's of height an exponential stack of 10's of height an exponential stack of 10's of height about 10^10^10^21. Big, but not that big given F_m on finite ordinals. But then we go to F_w (F_w (4)) = F_{F_4 (4)) (F_4 (4)) - now, that big number we just described goes into both the variable and the subscript of F, so now we are at an extremely high level in the finite hierarchy. Then we plug the extremely big result of that into the next F_w, so the extremely big result goes into both the variable and the subscript again. Then we plug that value into F_w one final time. Clearly, this is much bigger than an F_m (n) that we can easily describe without this iteration process. That's the power of diagonalization at limit ordinals.
Edit: No, I guess Doublecast will not go infinite with Timetwister. But given the way it works, it doesn't add to the sequence. So we still would rather have Sage's Knowledge + Fork than Timetwister.
At the bottom, we have to damage chain, so let's start with that. If A is the number of creatures that get created (buttressed by Doubling Season), B is the number of Intruder Alarms (so we get AB untaps) and C is the number of Rowan emblems on the battlefield (so we get C copies of each tap ability), then we can tap those creatures A(AB + A) times, for A(AB+A)(C+1) copies of the ability, so the damage chain yields roughly 2^[A(AB+A)(C+1)] counters. Then, if we have D copies of Angelic Chorus, our life increases by roughly D*2^[A(AB+A)(C+1)]. Note that we cannot increase our number of Rowan Kenriths at this level. We can incrase our number of Intruder Alarms and Angelic Choruses a little higher up, but it turns out that this is never a good idea; we get much more benenfit by copying Doubling Season, since each one has an exponential effect, and that effect applies to pretty much everything. So our life increases by roughly 2^(cX^2) where c is a constant, or exponential growth. This puts us at layer 1.
Next we have the other effect of adding more creatures. Most of the time when we add a token creature, we will want all the tokens to copy Doubling Season. If we have X Doubling Seasons in play, the effect of a token creation is to actually create 2^X more tokens, which will usually be or turn into Doubling Seasons, so each such ability takes X to 2^X. So this is layer 1 again.
Going through the rest of the combo:
Layer 1: With X Doubling Seasons in play, an ability to create a token can create 2^X Doubling Seasons.
Layer 2: With X Riku of Two Reflections in play, putting a nontoken creature into play will trigger token creation X+1 times, each time taking X to 2^X. The last creation can create more Rikus, so we wind up taking X Rikus to 2^^(X+1) Rikus. Casting any noncreature spell will trigger the flickering of Ojutai Exemplars, and therefore this layer. Casting White Sun's Zenith will achieve this layer in another way, since if we copy it the spell X times we will take X to 2^^(X+1) as well.
Layer 3: Resolving a copy of Cackling Counterpart will create us to create 2^X Teferi, Temporal Archmages, where X is the number of Doubling Seasons on the battlefield. If we have Y Rowan Kenriths on the battlefield, then we can create 2^X(Y+1) Teferi abilities, each of which can draw and recast White Sun's Zenith, so we can take X to 2^^^(2^X(Y+1)). We could also copy Rowan Kenrith and use his ability to increase Y, but unfortunately, the number of emblems doesn't increase anywhere near as fast as everything else, since unfortunately emblem creation isn't affected by Doubling Season. So the best we can do is to use the first 2^X-1 copies to copy Rowan Kenrith, and the last copy to copy Teferi Temporal Archmage. We can use the activate the 2^X-1 Rowan Kenriths' ultimate abilities, and each one doubles the number of emblems. So we wind up with Y*2^(2^X - 1) emblems, so we get 1+Y*2^(2^X-1) copies of Teferi's ability and we take X to 2^^^[Y*2^(2^X-1)].
Layer 4: Each time we resolve a Precursor Golem trigger for Cackling Counterpart with X creatures in play, we get X-1 copies of Cackling Counterpart, and we can use Goblin Flectomancer to retarget the target to Mirage Mirror. So this takes X to 2^^^^(X-1).
Layer 5: Each time we cast Cackling Counterpart, we get X Precursor Golem triggers, where X is the number of Precursor Golems in play. So this takes X to 2^^^^^X.
Layer 6: Each time we cast Runic Repetition, we can copy it X times to retrieve Cackling Counterpart X+1 times, taking X to 2^^^^^^(X+1).
Layer 7: Each time we cast Timetwister we can copy it X times to redraw Runic Repetition X+1 times, taking X to 2^^^^^^^(X+1).
Layer 8: Each time we cast one the exile-for-retrieval spells, we can copy it X times to retrieve Timetwister X+1 times, taking X to 2^^^^^^^^(X+1).
There are various other effects and increases being done, but they are insignficant compared to this main sequence. Since we have 20 exile-for-retrieval spells, the final damage is between 2^^^^^^^^^22 and 2^^^^^^^^^23.
It would definitely help to add Thousand-Year Storm, since we get the Precursor Golem of effect of creating a bunch of spells, and each trigger copying the original spell a bunch of times. So that would change things to:
Layer 6: Each TYS trigger for Runic Repetition creates X copies of Runic Repetition.
Layer 7: Each time we cast Runic Repetition, we get X TYS triggers.
Layer 8: Each TYS trigger for Timetwister creates X copies of Timetweister.
Layer 9: Each time we cast Timetwister, we create X TYS triggers.
Layer 10: Each TYS trigger for Praetor's Counsel creates X copies of Praetor's Counsel.
Layer 11: Each time we cast Praetor's Counsel, we create X TYS Triggers.
So that ups the damage to between 2^^^^^^^^^^^^22 and 2^^^^^^^^^^^^23. (12 arrows)
As for improving the deck, it would be nice to extend the chain of retrieval cards. Here's a way to improve the chain by one: Replace Timetwister with Sage's Knowledge and Reiterate. Sage's Knowledge can fetch Runic Repetition. Reiterate can copy an exile-for-retrieval spell and fetch Sage's Knowledge. Then the exile-for-retrieval spells can fetch Reiterate. This adds two arrows, getting us to 2^^^^^^^^^^^^^^22 (14 arrows).To answer the intended question, in the 1998-1999 DCI tournament rules it states that decks must consist of all Alpha or all non-Alpha cards. In the 2000-2001 DCI tournament rules, it states that you could mix Alpha and non-Alpha cards provided you used opaque sleeves for your deck. I couldn't find the 1999-2000 tournament rules on the internet, but at least this narrows it down.
After casting the enchantments you cast Rite of Replication (unkicked, so that we can cast Reiterate the most times), which doesn't get copied. We can choose to target either Doubling Season or Thousand-Year Storm; I think it makes more sense to target Thousand-Year Storm, so that Radiate will make copies of Doubling Season, which we need to do in order to get exponential growth. Then you cast Radiate, which gets one copy from Thousand-Year Storm. That copy creates one copy if Rite of Replication, which targets Doubling Season. So we create two token copies of Doubling Season.
We don't want to resolve either Radiate or Rite, since we need to copy them. So we cast Reiterate next, with buyback. The Thousand-Year Storm triggers, creating two copies of Reiterate. So we get three Reiterates total, and each time we create a copy of Radiate. The next Reiterate will create three copies of Reiterate from Thousand-Year Storm, then 4,5,6,7. So we get a total of 3 + 4 + 5 + 6 + 7 + 8 = 33 Reiterates in total.
The first Radiate will target the three Doubling Seasons, getting us to 3 + 2^3 = 11, then 11 + 2^11 = 2059, then 2059 + 2^2059 ~ 6.6 * 10^619 Doubling Seasons.
The second Radiate will target all 6.6 * 10^619 Doubling Seasons, so we get more than 10 ^^ (10^619) > 10 ^^^ 2 Doubling Seasons in the end. Then the third will get more than 10 ^^ 10 ^^ (10^619) > 10 ^^^ 3 Doubling Seasons, and so on, until the 33rd generates more then 10 ^^^ 33 Doubling Seasons.
That's more than the old 7, but less than the improvements that I mentioned. The major problem is that we aren't getting more Thousand-Year Storms. So I guess it makes sense to use one of the Reiterates to copy Rite of Replication rather than Radiate, so that we can get both enchantments going.
So, let's go back to the first casting of Reiterate. We don't need more Thousand-Year Storms until the next Reiterate is cast, so let's use the first two Reiterate copies to copy Radiate; that takes us to more than 10 ^^ (10^619) Doubling Seasons. Then we use the original Reiterate to copy Thousand-Year Storm, which creates 10 ^^ (10^619) Thousand-Year Storms.
We then cast Reiterate the second time, and now we get more than 10 ^^ (10^619) copies. Each one takes X Doubling Seasons to 10 ^^ X Doubling Seasons, so by the end we will have more than 10 ^^^ (10 ^^ (10^619)) Doubling Seasons and Thousand-Year Storms. The same thing happens for each of the next Reiterate castings, so by then end we will have between 10 ^^^^ 6 and 10 ^^^^ 7 creatures. More than before, but still less than the last deck I mentioned. (Of course, you do have a seventh card to make things bigger!)
Edit: So, the following seems like a good entry into the seven card challenge:
Opalescence
Doubling Season
Thousand-Year Storm
Cackling Counterpart
Radiate
Increasing Vengeance
Mystic Retrieval
So, we start off by casting the enchantments. We cast Cackling Counterpart, targeting Doubling Season. We cast Radiate, getting one copy via Thousand-Year Storm. The copy copies Rite of Replication, targeting Thousand-Year Storm, creating two Thousand-Year Storm copies.
Next, we cast Increasing Vengeance, which triggers all three Thousand-Year Storms, each one creating two copies of Increasing Vengeance. So we will get seven copies of Increasing Vengeance in total. We can use the first to copy Cackling Counterpart, getting two more Doubling Seasons. Then the next six copies can copy Radiate. The first targets two Doubling Seasons and three Thousand-Year Storms, getting 11 and then 2059 Doubling Seasons, and then 3 + 3*2^2059 Thousand-Year Storms. The second Radiate will create more than 10 ^^ 2059 Doubling Seasons and Thousand-Year Storms, the third 10 ^^ 10 ^^ 2059, and by the sixth we have more than 10 ^^^ 6 Doubling Seasons, and Thousand-Year Storms.
Finally we cast Mystic Retrieval, which will trigger more than 10 ^^^ 6 Thousand-Year Storms. We can use the first trigger to fetch the three spells from the graveyard, and then cast the three again. Keep the Cackling Counterpart and Radiate on the stack, and only resolve the Increasing Vengeance card. Then each copy of Mystic Retrieval can fetch Increasing Vengeance. Each Increasing Vengeance spell can copy Radiate, which takes X to 10 ^^ X; casting Increasing Vengeance triggers many copies, so we take X to 10 ^^^ X; casting Mystic Retrieval triggers many copies, each of which can fetch Increasing Vengeance, so we take X to 10 ^^^^ X. So we wind up with more than 10 ^^^^ (10 ^^^ 6) creatures after Mystic Retrieval fully resolves. Then we can flashback it from the graveyard, so we get a final number of more than 10 ^^^^ (10 ^^^^ (10 ^^^ 6)).
Eh, that's okay I guess. I'm pretty sure we can do more.
Edit: Wait, I'm being stupid. Each Increasing Vegeance Spell can copy Mystic Retrieval, which can fetch Radiate from the graveyard. A single Radiate spell takes X to 10 ^^ X; a casting of Radiate creates many Thousand-Year Storm copies, which takes X to 10 ^^^ X. Each Increasing Vengeance can fetch a Radiate to recast it. Each Increasing Vengeance cast creates many copies, so it takes X to 10 ^^^^ X. Each Thousand-Year Storm trigger for Mystic Retrieval creates many copies of Mystic Retrieval, each of which can fetch Increasing Vengeance, so it takes X to 10 ^^^^^ X. And finally, casting Mystic Retrieval creates many Thousand-Year Storm triggers, so it takes X to 10 ^^^^^^ X.
So we wind up with more than 10 ^^^^^^ (10 ^^^^^^ (10 ^^^ 6) creatures. Much better.
With those changes in hand, we bring in Godtoucher to bounce Boggart Mob. Godtoucher can also target Child of Alara, but now we can only bounce it back to our hand, which does not trigger the destruction of permanents. So I think this is safe.
So I was thinking of something like:
2 Psychic Battle
3 Cowardice
4 Brudiclad, Telchor Engineer
5 Bloodbond March
6 Cephalid Shrine
7 Soul Foundry
8 Omniscience
9 Vedalken Orrery
10 Pull from Eternity
11 Izzet Guildmage
12 Karn, Silver Golem
13 Perpetual Timepiece
14 Mimic Vat
15 Ray of Revelation
16 Skull of Orm
17 Mana Vault
18 Obelisk of Bant
19 Metallurgeon
20 Battle Cry
21 Rebuild
22 Goblin Dark-Dwellers
23 Boggart Mob
24 Godtoucher
25 Wirewood Herald
26 Engineered Explosives
27 Salvaging Station
29 Acorn Harvest
30 Smite the Monstrous
31 Thousand-Year Storm
32 Child of Alara
33 Tireless Mercenaries
34 Celestial Gatekeeper
35 Celestial Gatekeeper
36 Spellweaver Helix
37 Worldfire
38 Spider Spawning
39 Spider Spawning
40 Molderhulk
41 Divine Congregation
42 Bayou
43 Eureka
44 Mox Emerald
45 Cloud of Faeries
46 Nectar Faerie
47 Centaur Archer
48 Maze Glider
50 Frightshroud Courier
51 Simian Spirit Guide
52 Goblin Kites
53 Wilderness Hypnotist
54 Streambed Aquitects
55 Old Man of the Sea
56 Xathrid Gorgon
57 Tribal Unity
58 Military Intelligence
59 World at War
60 Reforge the Soul
One minor correction: You can't copy Opalescence with Riku, but I imagine that is just a typo.
The number you get after four Parallel Evolutions is a lot more than 35 million digits long:
After the first Parallel Evolution, you copy the two Parallel lives tokens, and each copy gets multiplied by 2^3 (since you have 3 Parallel Lives total) so you wind up with 2 + 2*2^3 = 18 Parallel Lives tokens.
After the second, you get 18 + 18 * 2^19 = 9437202 tokens total.
After the third, you get 9437202 + 9437202 * 2^9437203 ~ 1.42 * 10^2,840,888
After the fourth, you apply the formula one more time to get more than 10^10^2,840,887 Parallel Lives tokens.
Next comes the Rites and Radiates. each copy of Rite will add another "10^" to the exponential tower (the highest exponent will drop by an infinitesimal amount each time, but it will never dip below 2,840,887), and the first Radiate creates more than 10^10^2,840,887 Rites, so you will wind up with more than an exponential tower of 10^10^2,840,887 10's. In Knuth arrow notation, this is written 10^^(10^10^2,840,887). (More generally, a^^b is defined as a^(a^(...(a^a)...)) with b copies of a.) Similarly, the second Radiate will create more than 10^^(10^10^2,840,887) copies of Rite, so we will get more than 10^^10^^(10^10^2,840,887) token copies in the end. Since we wind up with six copies of Radiate in total, the final number will be more than 10^^10^^10^^10^^10^^10^^(10^10^2,840,887) token copies. We can simplify this with triple arrow notation: a^^^b = a^^(a^^...(a^^a)...) with b copies of a. So the final number is more than 10^^^7 but less than 10^^^8.
If I may make a suggestion, we could improve the number by replacing Riku of Two Reflections with a spell copier that we can have multiples of, like Swarm Intelligence. This would mean we don't get copies of Parallel Lives to start with, so casting Parallel Lives early on doesn't help. So perhaps we could replace that with [c]Clone Legion. We'll get only two copies of Clone Legion; the first will take us to 27 Parallel Lives and Swarm Intelligences, the second to 3623878683 each. That is a lot less than before, but now look what happens with the Radiates; The first cast gets copied 3623878683 times so we wind up with more than 10^^^3623878684 Parallel Lives and Swarm Intelligences. We then cast Radiate two more times, so we wind up with more than 10^^^10^^^10^^^3623878684 creatures, or between 10^^^^4 and 10^^^^5 creatures in the end.
Another improvement: Replace Mystic Retrieval with Bound // Determined. We can only cast Bound once, but since it is an instant, we can cast it while Rite of Replication is still on the stack. So we cast it after all the Radiates resolve, but while the original Rite is still on the stack. I just noticed that we can resolve the copies of the original Rite before we start resolving Radiates; so after the Rites resolve we will have more than 10^^3623878684 Swarm Intelligences, then the Radiates resolve and we go to 10^^^10^^3623878684 copies. Then we cast Bound, getting more than 10^^^(10^^3623878684) copies, and each one can fetch Radiate and recast it, increasing the final number to 10^^^^(10^^^(10^^3623878684)) creatures.
Looking forward to your full deck!
EDIT: I've been wondering whether the "target artifact" combos like Arcum Dagsson or Goblin Trashmaster could be made to work after all. They would certainly go infinite in the old version of the combo, but in this new version we have a lot of new moving parts, and maybe those can't be refreshed as easily as the artifacts. But I'm not sure.
If it still goes infinite, then I think we can still make it work by getting rid of Mirrorworks. Just replace Mirror of Fate and Mirrorworks with Izzet Guildmage and Pull from Eternity, I think. That resolves the problem of requiring more Psychic Battle triggers of our stage artifact, since we will need lots of mana for the stage transition and one copy of Mana Vault and a Keyrune won't be enough. However, we may have the Muzzio problem of being able to bring a copy of whatever we are using to target/sacrifice an artifact below the hyperstage transition, and getting an extra use out of them. Stakfish, do you know an artifact combo that would be safe from this happening?
EDIT: It just occurred to me that getting rid of Mirrorworks can also solve the problem we had with the Soul Foundry deck. That deck also used Muzzio, so we have to switch that out as well, but perhaps we can squeeze out an extra stage or two with that deck.
Skull of Orm looks like a good cut! As for fetching Academy Rector, we can just use Misery Charm, can we not? That will allow the fifth stage back in, getting us to w^4 + w5 + 4.
Concerning the problem FireRogue517 mentioned, there was indeed an error with the early setup, which came about because I was greedy and tried to squeeze in another couple of copies of Dual Nature by having Copy Enchantment copy Dual Nature at strategic points. The additional Dual Nature means one more Dual Nature token is created, but when we bounce the Copy Enchantment (which is copying Dual Nature) card, we trigger the Dual Nature destruction abilities. I believe there was one point where doing this is okay, but there was another point where it was not. That problem could be easily fixed by simply not having Copy Enchantment copy Dual Nature at the wrong time; this will cost us a Dual Nature token, but not a big deal. Of course, the gigastage deck doesn't currently have Dual Nature in it, although perhaps we should keep an eye for its possible inclusion whenever it may become legal, since it may be more efficient.
If not, then it looks like the best we can do is to remove the Merfolk stage and end with World at War, getting to w^4+w4+4. I'm not seeing any cuts, although the need for both Salvaging Station and Goblin Welder is driving me nuts.
Heh, the ultracombo article. When I started writing it, I had every intention of finishing it. But, as the gigastage deck started coalescing, I lost the motivation to complete a description of what would (perhaps) become an outdated deck. Maybe if this deck never quite comes together, I will write up the older version.
As for this version, we haven't started writing up anything for it, as it is still in the development stages. I can summarize it briefly I suppose.
The setup is a little different from the previous version. Like before, the basic mechanic is to generate more and more Psychic Battle and Bloodbond March enchantments, and the number of these enchantments are basically our "current number" throughout the combo. However, in this version we removed Dual Nature, since it causes certain problems (currently the problem is with Child of Alara I think). So to get lots of copies of enchantments, we destroy them (using Psychatog in the above deck), and then imprint them on a Mimic Vat to generate a token copy. Then we use Skull of Orm to bring the enchantment back so we can sacrifice it again. This is less efficient then Dual Nature, but that won't affect the final estimate.
The primary stage is similar to the previous version - we have an artifact creature (either Metallurgeon or Rust Tick) that functions as our stage resource, and the stage is built up with alternating Psychic Battle and Bloodbond March groups of triggers.
The hyperstage is the biggest difference in the new version. In the previous version, we had Spellweaver Volute in the hyperstage andSpellweaver Helix in the megastage - in this version, those two cards are bumped up to the megastage and gigastage respectively, and the hyperstage is what's new here. The new spell triggerer is Goblin Dark-Dwellers, which can cast an instant or sorcery of CMC 3 or less when it enters the battlefield. With Panharmonicon, it can cast multiple spells. So we use Battle Cry, or Goryo's Vengeance couple with Muzzio, Visionary Architect, to get lots of Metallurgeons or Rust Ticks; we use Rebuild to regain our hyperstage resource, but at the cost of losing all our stage creatures.
The hyperstage mechanism is actually pretty complicated, as we couldn't get anything simple to work out. So, Rebuild has to replenish our hyperstage resource (which will turn out to be Moggcatcher tokens). It does this by bringing Engineered Explosives back to our hand, so that it can be cast again with a counter, and sacrificed. (Note that if we just bring Engineered Explosives back from the graveyard to the battlefield, it won't get a counter and we can't use it properly - this is quite handy.) Sacrificing Explosives will destroy Scattershot Archer, which can then be imprinted on a Mimic Vat to create a hasted copy. (Engineered Explosives can be brought back from the graveyard using Muzzio or Salvaging Station, while Scattershot Archer can be brought back using Verdant Succession.) The Scattershot Archer can then be activated to destroy Moggcatcher and Strongarm Thug, with the aid of Dearly Departed, Abzan Falconer, and Gratuitous Violence. The Moggcatcher can be imprinted to create a Moggcatcher token, and our hyperstage resource is renewed; the Strongarm Thug can be imprinted to create a token copy, which along with Panharmonicon can bring back both the Moggcatcher and Strongarm Thug to our hand. The Moggcatcher can then be tapped to bring Goblin Dark-Dwellers from the graveyard, which triggers the casting of our cheap instants, including Rebuild to bring Engineered Explosives back to our hand again.
The megastage, like the hyperstage of our previous version, uses Spellweaver Volute as our spell triggerer, and it triggers on the casting of an sorcery. So we use Acorn Harvest as our sorcery (which has CMC greater than 3, so no conflict with Goblin Dark-Dweller), which we can flashback for mana (which we can get cheaply) and 3 life. So life becomes are megastage resource. The instant that renews our hyperstage resource, and the instant that renews our megstage resource but eliminates all of our hyperstage resource, are combined into one: Smite the Monstrous. When we cast Acorn Harvest, we get many copies of Smite the Monstrous via Thousand-Year Storm. Most of the copies can be used to bounce Goblin Dark-Dwellers, which can then be replayed to cast our cheap instants and build up the hyperstage. (Note: we can't rebuild the hyperstage all at once, rather it gradually gets built as it resolves, with the lower Smite the Monstrous triggers standing as placeholders for hyperstage layers until the eventually get resolved and replaced by Battle Cry/Rebuild.) A few more Smite the Monstrous can be used to bounce large creatures that we need to return to our hand. The last Smite the Monstrous is used to destroy Child of Alara, which destroys all nonland permanents, and therefore all of our hyperstage resources (Moggcatcher tokens), but also destroys Centaur Safeguard, which allows us to gain 3 life, renewing our megastage resource. Note that we can't use multiple copies of Smite the Monstrous to destroy Child of Alara and therefore Centaur Safeguard multiple times, because - hmm, I'm forgetting the exact way that works. Stakfish, could you go over that again?
The gigastage uses Spellweaver Helix to cast sorceries, which can only be triggered by a sorcery we have more than one of, so no flashbacking Acorn Harvest to trigger it. So only Spider Spawning can trigger it, which requires a black mana to cast it via flashback. The we get a Frightshroud Courier stage that produces black mana, and a few stages after that; the last deck gets us to around F_{w^4 + w5 + 4}, assuming it works properly.
As for the challenges the deck presents, there have been many. Having four different stage types in the main combo leads to a lot of possible conflict with one level potentially affecting another in undesirable ways. And the newest hyperstage is partcularly complicated; the more complicated, the greater the chance of a flaw or a hidden infinite. Probably the greatest lingering problem is the potential infinite that can come from the megastage and gigastage transitions potentially renewing lower level resources improperly. For example, the way the megastage transition is supposed to work is that we spend three life, go to a higher hyperstage, and gain three life back; but to do so we need to use up a minimal amount of resources from below the megastage transition so as not to go infinite. (Basically, we need to use a few Metallurgeon/Rust Tick triggers to set up the megastage transition). Unfortunately, passing to the higher hyperstage allows us to bring Engineered Explosives and Goblin Dark-Dwellers back to our hand, and keep them in our hand when we pass back down to the lower hyperstage. This allows us to add a couple Moggcatcher tokens, which more then compensates for a couple of lost Metallurgeons, so that goes infinite. So we need to find a way for the megastage transition to use up those extra uses of Engineered Explosives that we gain, and that has been plaguing us for a while. We may have solved it with the necessity to resolve Verdant Succession to gain certain green creatures back, but I'm not totally sure there isn't a hidden infinite somewhere.
Another problem is that Iijil has left, and he was the best one at finding problems and infinities in our prospective decks.
Stakfish, feel free to point out any further details that are worth mentioning.
Yeah, I also would prefer to use Psychic Battle over Grip of Chaos, so that the probabilities remain "reasonable". I had a discussion with Iijil over this, and he was of the opinion that we should just present the deck that had the maximum possible damage. I guess that makes sense. I think, if our deck winds up favoring Grip of Chaos, we would write up the article about that deck, but include an addendum on how things change if we want to replace Grip of Chaos with Psychic Battle.
So long as Grip of Chaos is not in the deck, there is really nothing that requires chance other than card drawing, and with Mirror of Fate we can arrange the library however we want to. So the probability of getting the combo off only includes the cards we draw up until we start using Mirror of Fate, so really it can be no worse (and in fact will be much better than) 1 in 60!, which I would consider firmly in "imaginable numbers".
About your question: Yes, any possible infinite damage combo is prohibited; I'm not sure what you mean by "optimal luck" never reaching that state, could you clarify?
I played around certain times with combos that would be finite if the opponent always acted in a way to keep the maximum damage down, rather than cooperate to go infinite; but Iijil and Stakfish seem to like the rules as they are.
Yes, the fast-growing hierarchy is consistent with the definition of ordinals. Now, F_w (64) is not greater than F_m (64) for all finite ordinals m, and cannot possibly be, since F_m (64) can be arbitrarily large. But, F_w (n), as a function of n, is faster growing than F_m (n) for any fixed m; we have F_w (n) = F_n (n) < F_m (n) for n < m, but once n passes m we have F_w (n) = F_n (n) > F_m (n). So past a certain point F_w (n) will always be greater. More generally, if we have two ordinals a < b < epsilon_0, then F_b will be a faster growing function than F_a. So the ordinals match up orderwise with their corresponding functions.
This might strike one as an academic point, since F_w (n) = F_n (n) doesn't seem like a major leap forward from the finite ordinals. But really it is. To see the big difference, look at what happens with F_{w+1} (n). For example, take F_{w+1} (4). This is equal to F_w (F_w (F_w (F_w (4)))). First we have F_w (4) = F_4 (4) ~ an exponential stack of 10's of height an exponential stack of 10's of height an exponential stack of 10's of height about 10^10^10^21. Big, but not that big given F_m on finite ordinals. But then we go to F_w (F_w (4)) = F_{F_4 (4)) (F_4 (4)) - now, that big number we just described goes into both the variable and the subscript of F, so now we are at an extremely high level in the finite hierarchy. Then we plug the extremely big result of that into the next F_w, so the extremely big result goes into both the variable and the subscript again. Then we plug that value into F_w one final time. Clearly, this is much bigger than an F_m (n) that we can easily describe without this iteration process. That's the power of diagonalization at limit ordinals.
If I recall correctly, the problem with Rust Tick was the same as with Metallurgeon; targeting the Human (in this case Muzzio, Visionary Architect) with Skyshroud Archer. So I think there isn't an obvious problem with Muzzio and Scattershot Archer.
If that is okay, we can maybe save three cards and get 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 Phantatog
14 Skull of Orm
15 Mirrorworks
16 Mana Vault
17 Simic Keyrune
18 Rust Tick
19 Muzzio, Visionary Architect
20 Goryo's Vengeance
21 Rebuild
22 Goblin Dark-Dwellers
23 Moggcatcher
24 Strongarm Thug
25 Panharmonicon
26 Engineered Explosives
27 Scattershot Archer
28 Gratuitous Violence
29 Thousand-Year Storm
30 Dearly Departed
31 Abzan Falconer
32 Wrap in Vigor
34 Acorn Harvest
35 Smite the Monstrous
36 Child of Alara
37 Centaur Safeguard
38 Centaur Safeguard
39 Verdant Succession
40 Spellweaver Helix
41 Worldfire
42 Spider Spawning
43 Spider Spawning
44 Molderhulk
45 Natural Order
46 Bayou
47 Mox Emerald
48 Eureka
49 Polluted Dead
50 Frightshroud Courier
51 Simian Spirit Guide
52 Goblin Kites
54 Streambed Aquitects
55 Old Man of the Sea
56 Xathrid Gorgon
57 Tribal Unity
58 Military Intelligence
59 World at War
60 Reforge the Soul
So it looks like we save two cards, using the extra card slot to get a partial layer. Do you see any problem with the above?
I wonder if it might be possible to replace Verdant Succession; that might allow an Everglove Courier layer prior to the Frightshroud Courier layer.