We can't recur Memory Jar, because we have to use it from the starting hand, and it gets stuck in the graveyard.
[EDIT] Hmmm, I think I disagree with Iijil's assessment that storm sorceries cannot save progress. It is true that you resolve the sorcery before you make use of it and progress further. But, that further progress will be reflected the _next_ time you resolve that storm sorcery, which is when we need the progress to show up. So, as far as I can tell, storm sorceries are okay; of course, I have been wrong before. So I will await Iijil's reply.
It seems to me that using a sorcery to save the progress runs into exactly the same problem as using a sorcery to get back the spent ressource did. I explained that in an old post. We need the progress restored before we put the transition on the stack, otherwise the layers in the intermediate stages will all have a fixed size, won't they?
I don't see a way to get around that problem, so if you think you can do it, can you go into more detail?
The layers in the intermediate stages will all have a fixed size - initially. That's perfectly fine. It would be okay to start with all stages having 2 layers, and all hyperstages having 2 stages, so long as the number of hyperstages increased without bound, and the number of stages in each hyperstages increase as the combo resolves.
Let's say we have a 3 hyperstage hyperhyperstage, e.g. 10 life to start.
We first put the triggers on the stack:
SHT (Spellweaver Helix trigger) for Empty the Warrens
SHT for Worldpurge
We resolve the SHT for ETW, and then resolve ETW:
SHT for Worldpurge
(X red mana)
We resolve the SHT for Worldpurge, leaving it on the stack:
Worldpurge
(X red mana)
We use our X red mana to create X Metallurgeon stages:
X stages
Worldpurge
We spend 3 more life two more times to repeat the process:
X stages
Worldpurge
X stages
Worldpurge
X stages
Worldpurge
We resolve the X stages, then Worldpurge, gaining many more spells cast
X stages
Worldpurge
X stages
Worldpurge
(X2 spells cast)
we resolve a Psychic Battle trigger in the second hyperstage to get needed resources, then spend 3 life to recreate the top stage
X2 stages
Worldpurge
X stages (minus 1 PB trigger)
Worldpurge
X stages
Worldpurge
We repeat this process, until the second hyperstage is completely expended. Then we have
X stages
Worldpurge
(X3 spells cast, 7 life)
We spend 6 life to recreate the top two stages
X3 stages
Worldpurge
X3 stages
Worldpurge
X stages (minus one PB trigger)
Worldpurge
and so on and so on. So, it does seem like the hyperstages are properly updating. Am I missing anything?
[EDIT] Hmm, I think I see what you are saying... we need the Psychic Battle / Bloodbond March triggers to update in the intermediate hyperstages. Yeah, that does look like a problem. So, we do need a "saving progress" package.
A group effort to make the most insane non-infinite combo is very interesting, and the current deck list looks like an absurd mess. I want to understand. I want to break my brain by trying to understand this! Could someone please try to help me understand this madness?
I think I have an idea that could compete with the one being talked about, but I don't know absurd yours is, so feel free to tell me if my combo isn't even close. My idea revolves around Selvala, Heart of the Wilds, Heroes' Bane, Paradox Engine, and Dragonlord Kolaghan. The idea is to use Selvala to make mana equal to Heroes' Bane Power, dump it all into its doubling ability, then untapping her with Paradox Engine. Kolaghan is the haste enabler because she has the highest potential for extra damage out of all the haste enablers. I'm thinking that you could probably use Mirror Gallery and spells that clone Selvala, which would make Doubling Season even crazier with this idea.
[Edit] Should have known it couldn't have been that easy... Ow, my brain.
Hi CTNC! It would be great to have another mind thinking about this.
The combo you talk about is indeed nowhere close, at least not yet. Each time Paradox Engine untaps Selvala, Selvala gets X mana where X is the power of Heroes' Bane, which allows us to double Heroes' Bane's power X/4 times, to go from X to X * (2^(X/4)). This is basically exponentiation, or the first layer of recursion. Let's say you had something (call it resource A) that allowed you could expend to cast X spells, where X is the power of Heroes' Bane or the amount of mana that you have. Then you could untap Selvala X times, allowing you to pump Heroes' Bane to about X^X^X^...^X with X X's. This is called tetration, and is the second layer of recursion. It is denoted by X^^X in Knuth arrow notation. Then, suppose that you had something (call it resource B) that you could expend to gain X of resource A, where again X is one of the things that is growing bigger and bigger. Then you could apply X -> X^^X X times, which would get you to about X^^^X; this is called pentation, and is the third layer of recursion. So this is the pattern; each additional layer of recursion means that you can repeat the previous layer X times, where X represents the current output so far.
To get a better idea of how the layer system works, you can take a look at at the following writeup of one of the earliest contestants in this challenge:
This achieved 36 layers, which seemed incredible at the time (and is more than enough to overpower everyone else's attempts at this challenge). Later, we came up with the following deck that achieved 408 layers:
But, shortly after that deck was written up, we came up with a combo that changed everything: a combo that could take X of something (mana or creature tokens) and produce X layers! You could then iterate the process, so if you started with say 10 mana, iterating the "stage" (that's what we're calling the combo that creates X layers) takes you to 10^^^^^^^^^^10, then a 10^^^...^^^10 where the number of ^'s is 10^^^^^^^^^^10, and so on.
Iijil wrote an excellent post summarizing the details:
In particular, follow the first link in the above link to my post on page 2, where I describe the stage combo in some detail. Basically, you build up the stack with groups of alternating triggers, each group representing a layer; once one group runs out, one of the next group creates X of that group at the expense of one trigger in the next group, so it functions just like the layer system. The amount of mana / creature tokens keeps track of which group you are at in the stack, so as you get lower in the stack your mana/tokens builds up, allowing you to build the stack back up to the top.
That was the cutting edge for a while, but then we figured out how to build a hyperstage; this is where you take X of a resource (mana or tokens) and build up X stages. That was the best we could do for a while; then later we figured out how to do two hyperstages. Finally, very recently we came up with ideas for how to implement a hyperhyperstage - as you may guess, this is where you take X of a resource and build up X hyperstages. So this is the current frontier. We are still in the midst of tinkering with the deck and checking for problems/infinities.
Okay, I still haven't gone through the current deck step by step. That will take some doing. I will go over it generally in a later post; in the meantime, feel free to ask any questions you may have about what I have linked so far.
[EDIT] Here's a possible save progress combo: Hunting Pack, to power Earthcraft into Plains. Unfortunately, this can't work with Allay, as for three mana we can cast Allay to generate many more enchantments to power Earthcraft. Generally speaking, we can't be able to create enchantments with white mana, or else we would be able to create many of them because of Dual Nature, and therefore be able to create lots more white mana. So it seems we need to get rid of Dual Nature for this to work. So I tried going back to Vitu-Ghazi Guildmage; this seems okay, as it costs four mana to populate, and that will give us only one mana back. So, the deck becomes:
The two Glimmerpost should generate 4 life, going infinite (put the second on the battlefield in response to the firsts trigger). There are a few lands like Kabira Crossroads that we can use to gain 2 life from one card.
There's one other "retrieve from exile" card left, Pull From Eternity. We can get a copy of it via Izzet Guildmage, and then just make blue mana obtainable from artifacts but not from Retract. An artifact creature like Silver Myr looks like the ticket. Izzet Guildmage doesn' mess with the other instants/sorceries except for Retract, which fortunately can be replaced by Rebuild. The fix adds two cards, but better than nothing.
But Silver Myr is not good enough to power the card retrieval. To produce a blue mana we need to create a Mimic Vat copy of Silver Myr. But that means we immediately have to use that blue mana to pull the original Myr back from exile. We never get anything done.
Shoot, this is getting difficult. I have an idea, which unfortunately is going to take more cards. Use Seat of the Synod, coupled with Nature's Revolt so that it can be bounced by Metallurgeon. Then Rebuild will bring Seat of the Synod back to our hand - but we don't have a cheap way of getting it back on to the battlefield; we need Walking Atlas. This seems promising.
[EDIT] could we save cards by going back to Saltcrusted Steppe? Maybe using Riftsweeper. I don't see that going infinite with Rebuild, but I also don't know how to get an intial token copy of it.
Yeah, the idea was to have create copies of Endbringer to get the rest of the cards. We have to check to make sure we can actually do it with just Five-Alarm Fire and World at War.
(On Lantern Control)"A guy who literally just sits there and mills cards he doesn't like from your library while he slowly, slowly kills you this way."
"If a person's profile includes anime or My Little Pony, feel free to ignore everything they say."
It's a contest to come up with the most damage you can on Turn 1 with a 60-card Vintage deck, without being able to deal arbitrary amounts of damage. It's just for fun; not everyone will be interested.
The deck currently starts out with Mox Sapphire, Mox Ruby, Radiant Fountain, Show and Tell, Omniscience, Consecrated Sphinx, and Memory Jar. Play the first four cards to put Omniscience on the battlefield, then play Consecrated Sphinx and Memory Jar. Tap Memory Jar, allowing us to draw 21 cards thanks to Consecrated Sphinx. We will later attack to put counters on Five-Alarm Fire, and use those counters to target Endbringer. Each targeting of Endbringer will destroy it thanks to Horobi, Death's Wail, and then we can imprint it on Mimic Vat and create a hasted token of Endbringer, which we can use to draw another card. We can retrieve Endbringer from the graveyard thanks to Bloodbond March, and repeat the process to draw out the rest of the deck.
The general method of the deck is rather complicated; see my reply to CTNC above, in particular this helpful post by Iijil. I do plan to write an a full article detailing the deck, but I don't want to do it until we have finished improving the deck (up to a point). But I guess it will be harder to bring new people in before I do the write-up; it's a bit of a Catch-22.
The deck currently starts out with Mox Sapphire, Mox Ruby, Radiant Fountain, Show and Tell, Omniscience, Consecrated Sphinx, and Memory Jar. Play the first four cards to put Omniscience on the battlefield, then play Consecrated Sphinx and Memory Jar. Tap Memory Jar, allowing us to draw 21 cards thanks to Consecrated Sphinx. We will later attack to put counters on Five-Alarm Fire, and use those counters to target Endbringer. Each targeting of Endbringer will destroy it thanks to Horobi, Death's Wail, and then we can imprint it on Mimic Vat and create a hasted token of Endbringer, which we can use to draw another card. We can retrieve Endbringer from the graveyard thanks to Bloodbond March, and repeat the process to draw out the rest of the deck.
The general method of the deck is rather complicated; see my reply to CTNC above, in particular this helpful post by Iijil. I do plan to write an a full article detailing the deck, but I don't want to do it until we have finished improving the deck (up to a point). But I guess it will be harder to bring new people in before I do the write-up; it's a bit of a Catch-22.
The way the hyperhyperstage works is quite similar to how the hyperstage works, which is modeled itself modeled after the stage combo, but in enacted using multiple spells to get the needed effect.
So our main stage combo (you can skip this paragraph if you understand how this works already) uses Metallurgeon tokens. If we have N metallurgeon tokens, we can cast and counter Metallurgeon (getting a bunch of Bloodbond March triggers), use one Bloodbond March trigger to bring it back to the battlefield, then tap a Metallurgeon token for a bunch of Psychic Battle triggers. Repeat N times for 2N alternating groups of Bloodbond March and Psychic Battle triggers. When we resolve the top stack of Psychic Battle triggers, we can destroy the original Metallurgeon with one of them, then use the rest to create more Psychic Battles. (In the latest deck, we can destroy Core Prowler to put a counter on Saltcrusted Steppe; once we have a bunch of counters on Saltcrusted Steppe, we can activate it and remove all but one counter to get a bunch of white and green mana, and then activate Vitu-Ghazi Guildmage a bunch of times to populate a Psychic Battle token a bunch of times.). The destruction of Metallurgeon allows us to imprint it on Mimic Vat, and create a new Metallurgeon token; we then use Mirror of Fate to bring it back from exile into the library, and Hair-Strung Koto to put it into the graveyard. Then, after the top group of Psychic Battle triggers are all gone, we use the topmost Bloodbond March trigger to bring Metallurgeon back to the battlefield, and we can use our newly created Metallurgeon token to target Metallurgeon and create a bunch more Psychic Battle triggers. We do this for each Bloodbond March trigger, until we get to the last one; after the last Bloodbond March trigger is used, we use the Psychic Battle triggers to create more Bloodbond March tokens, and we use the next Psychic Battle trigger on the stack (rather than tapping the latest Metallurgeon token) to bounce Metallurgeon back to our hand. This allows us to cast and counter Metallurgeon again, creating a bunch more Bloodbond March triggers, and we use each one to create a bunch of Psychic Battle triggers, and the process continues like this. Each group of X triggers in the stack iterates the group above it X times, so the top group adds some constant amount to the number of enchantments, the second group from the top multiplies by some constant amount, the third group exponentiates, the fourth applies tetration, and so on for 2N operations. This is what we call a "stage combo". A "second stage" will be similar, except the top group of Psychic Battle triggers will create X Metallurgeons, which allows us to increase the number of Psychic Battle tokens from X to F_w(X) in the fast-growing hierarchy. Then the second group increases from X to F_{w+1}(X), the third from X to F_{w+2}(X), and so on, so X tokens for the second group will take X to F_{w2}(X); N stages will take X to F_{wN}(X) in the fast-growing hierarchy.
So next, a hyperstage. Here, the resource controlling the combo is red mana, supplied by Mox Ruby. We make use of three spells, Bump in the Night, Battle Cry, and Rebuild. We cast Battle Cry and Rebuild so that they are in the graveyard, and then cast Spellweaver Volutes on both of them. (One of the Spellweaver Volutes will be a copy so that we don't have to waste a card on a second Spellweaver Volute.) We also cast Bump in the Night so that it is in the graveyard, and from then on we use a red mana from Mox Ruby to flashback it. So, the combo proceeds as follows: We start M red mana. Also, we have N Metallurgeon tokens and the original Metallurgeon, which we cast and tap accordingly to set up a stack of 2N alternating groups of triggers, just like in the last paragraph. Then, we spend a red mana to flashback Bump in the Night, which triggers both Spellweaver Volutes. We resolve the Spellweaver trigger on Battle Cry first and resolve the Battle Cry; this untaps all N of the Metallurgeons. Resolve the Spellweaver trigger on Rebuild, but do not resolve the Rebuild; leave it on the stack, above the stack of 2N alternating Psychic Battle and Bloodbond March triggers. Next, we start casting and tapping Metallurgeons again to build up another stack of 2N alternating groups of triggers, then spend another red mana to flashback Bump in the Night again (we use Mirror of Fate and Hair-Strung Koto to put all our spells back in the graveyard, and use Copy Enchantment to replace missing Spellweaver Volutes as needed.) Repeat the process M times, creating M+1 copies of 2N alternating groups of triggers, separated by M copies of Rebuild. Now we start resolving our stack. After we resolve the topmost stage, we resolve the topmost Rebuild, which will destroy all the Metallurgeon tokens, and bring Metallurgeon and Mox Ruby back to our hand. The Mox Ruby gives us the mana to flashback Bump in the Night, but (fortunately) we cannot do so immediately; in order to reset our spell and Spellweaver Volute situation, we have to use Mirror of Fate a few times, and copy a Spellweaver Volute. To do this, we need to activate Vitu-Ghazi Guildmage a few times, and for this we need mana. So we need to use a few Psychic Battle triggers from the next stage down to get the needed mana, and then we can properly set up our spells, and use a red mana to flashback Bump in the Night, recreating the top stage. So, we get to apply the Ackermann function for a few Psychic Battle triggers in the second stage; all told the second stage gets us F_{w2}(X), the third stage F_w3(X), and the entire hyperstage gets us F_{w^2}(X).
Next is the hyperhyperstage. This time, the resource that controls the hyperhyperstage is life. The flashback spell will be Flash of Defiance (which costs 3 life to flashback), and casting it triggers Spellweaver Helixes imprinting Fungal Sprouting and Worldpurge. The setup is precisely analogous to the hyperstage. Let's say we start with between 3P+1 and 3P+3 life, at least 2 red mana, and at least 1 Metallurgeon token. First, we setup the first hyperstage using whatever red mana we currently have more than 2(this might just be one stage if we have only 2 red mana); do not use the last Metallurgeon token on the topmost stage. Next, we use a red mana to cast Bump in the Night with a Spellweaver Volute on Time and Tide. (You will see why we do this later) Then we spend 3 life and a red mana to flashback Flash of Defiance, triggering the Spellweaver Helixes for both Fungal Sprouting and Worldpurge. Resolve the Fungal Sprouting first, after we generate a large Auratog by sacrificing enchantments. This creates a bunch of Saprolings, which we can sacrifice to Utopia Mycon for a bunch of red mana. Now we can implement a second hyperstage, with a lot more red mana to use. Repeat this process P times, creating P+1 groups of hyperstages. Resolve as above; when the topmost hyperstage is completely expended, make sure that we have put a bunch of storage counters on Saltcrusted Steppe by targeting Core Prowler a bunch of thimes. Next, play Urborg, Tomb of Yawgmoth and Shimmer, naming Swamps, and finally Time and Tide (triggered by Bump in the Night) so that Urborg and Saltcrusted Steppe phase out. Then we play Glimmerpost and Radiant Fountain, gaining 3 life. Then we play Portcullis, and then bounce a bunch of our nontoken permanents, until we are down to at most seven nontoken permanents. Finally, we let Worldpurge resolve, emptying our mana pool (but we have spent it all already), and moving the permanents still on the battlefield back to our hand, including Glimmerpost, Radiant Fountain, Mox Ruby, and the Portcullis currently exiling most of our permanents, which now come back to the battlefield. Then, Time and Tide resolves, so that Saltcrusted Steppe and Urborg, Tomb of Yawgmoth phase back in; we can now remove storage counters from Saltcrusted Steppe to generate a bunch of mana, and recreate a bunch of Psychic Battles and Bloodbond Marches. We now have the resources to get 3 more life and one more red mana, so we can flashback Flash of Defiance again - but again, to reset Flash of Defiance we need to expend some of our Psychic Battles from the next hyperstage down. So each few Psychic Battles from the next hyperstage down allows us to restart the hyperstage above it. Each hyperstage is with w^2 in the fast growing hierarchy, so P hyperstages is worth F_{w^2*P}(X), and the whole hyperhyperstage is worth F_{w^3}(X).
Whew - well, if you have any questions on the above, or the deck in general, please feel free to ask!
I don't quite get how the fast growing hierarchy works (F_{w^3}(X) etc), can you link me to something explaining it, or explain it here? if it helps with explaining, I understand Knuth's arrow notation, and Conway's arrow notation, but not chained arrow notation.
Also, I haven't found a setup to do it yet, but I'm worried about Time and Tide being the source of life. Couldn't it be cast, using a Spellweaver Volute, during the hyperstage, getting 3 life, getting R red mana, getting R hyperstages from the hyperstage, getting 3R red mana?
The fast-growing hierarchy has the following rules:
F_0(n) = n+1
F_{a+1}(n) = (F_a)^n(n) (that is, it is equal to F_a(F_a(...(F_a(n)...)) with n F_a's)
If a is a limit ordinal (that is, it doesn't end with +n) F_a(n) = F_a[n](n), where a[n] is the fundamental sequence for a
The subscript for F is what we call an ordinal. In broad terms, ordinals are ways to measure the lengths of sequences that can be longer than the standard infinite sequence. So:
the empty sequence has order type of the ordinal 0
(a) has order type the ordinal 1
(a,b) has order type the ordinal 2
(a,b,c) has order type the ordinal 3
...
(a,b,c,...) has order type the ordinal w (this is the Greek letter omega, but apparently this forum doesn't allow the use of non-Latin unicode characters. WTF.)
(a,b,c,...,a2) has order type w+1 (this is an infinite sequence followed by one element that is bigger than the whole sequence)
(a,b,c,...,a2,b2) has order type w+2
...
(a,b,c,...,a2,b2,c2,...) has order type w*2 (this is written w2 rather than 2w for technical reasons; 2w is actually 2+2+2+... = w, whereas w2 = w+w, which is what we want)
...
(a,b,c,...,a2,b2,c2,...,a3,b3,c3,...) has order type w*3
...
an infinite sequence of infinite sequences has order type w^2
...
an infinite sequence of infinite sequence of infinite sequences has order type w^3
...
This goes on forever (and by forever, I really mean forever... anything that you can conceptualize is really just a really small initial segment of the ordinals. For example, w^w^w^... is peanuts. But for our purposes we need only worry about ordinals below w^w, which are basically polynomials over w with natural number coefficients. (Well, ordinal addition and multiplication is different from polynomial addition and multiplication, but that need not concern us.) For these ordinals, fundamental sequences are easy: if an ordinal a ends in w^m, a[n] is a except the final w^m is replaced by w^(m-1)*n. So for example:
F_3 applies F_2 n times to n, and each time the output is greater than 2 to the power of the input, so the result will be more than 2^2^2...^n with n 2's, which is more than 2^^(n+1).
In general, F_(m+1)(n) will be more than 2^...^(n+1) with m ^'s.
So, for finte subscripts the fast-growing hierarchy is similar to Knuth arrows, or Conway chains of length 3.
Next we have F_w(n). This is just F_w[n](n) = F_n(n), simple. So for example F_w(200) = F_200(200).
So what's the big deal? Well, let's look at F_{w+1}(n).
As you can see, we get subscripts of subscripts of subscripts. F_{w+1}(n) gets you an expression with n nested subscripts. In Knuth arrow notation, this would be something like "3^...^3, where the number of ^'s is 3^...^3, where the number of ^'s is 3^...^3,..." repeated n times. If you are familiar with Graham's number, you can see from the previous that F_{w+1}(64) is greater than Graham's number. So our decks were able to beat Graham's number once we created the stage combo, and added one or two layers to the first stage!
Since F_{w+1}(n) is similar to G(n) from Graham's number, F_{w+2}(n) is similar to G(G(G(...G(n)...)) with n G's. Then F_{w2}(n) = F_{w+n}(n), F_{w3}(n) = F_{w2+n}(n), F_{w^2}(n) = F_{wn}(n), and so on.
F{wm}(n) is similar to a Conway chain of length m+2, so once we get past F_{w^2} we are beyond what Conway chains can reasonably describe.
Time and Tide is not the source of life; Worldpurge bringing the life lands back to our hand so that we can replay it is the source of life. Keep in mind that phasing in does not count as entering the battlefield. Also, I don't see how Time and Tide could generate red mana.
Whoops, I forgot that phasing in doesn't count as entering the battlefield. It could have only generated red mana by giving you 3 life and letting you start the hyperhyperstage.
EDIT: What I don't get:
Your first two examples seem wrong, shouldn't they be w^5 + w^3 * 3 + w^2 * 5 + 35 and w^11 * 3 + w^8 * 5 + w^5 * 2 + w^4 * 33 respectively? And if not, why not?
I think I get the rest, so since F_w(n) = F_w[n](n) = F_n(n), would F_{w^2}(n) be F_{w*n}(n), which would be F_n^2(n)? And so, in general, F_{w^x}(n) would be equal to F_n^x(n)? This feels wrong because of what you wrote about F_{w+1}(2) and F_{w+1}(3), but seems to work using the same process from which F_w(n) = F_n(n) was derived. But I could see it going either way.
EDIT 2: I also don't get what you were doing with "Then F_{w2}(n) = F_{w+n}(n), F_{w3}(n) = F_{w2+n}(n), F_{w^2}(n) = F_{wn}(n), and so on.", that seems to just be infinitely growing, because the alternative is that F_{w*2}(n) is larger than F_{w*3}(n), and even larger than F_{w*n}(n).
If an ordinal ends in w^m*t, we don't convert the whole thing to w^(m-1) * tn. That would mean, for example, that F_{w5}[n] = F_{5n}[n], which would be smaller than F_{w+1}[n]. We want the fundamental sequence for a to be a sequence of ordinals that go all the way up to a (meaning that every ordinal less than a will be beaten by a[n] for n big enough; if (w*5)[n] = 5n, then w would be bigger than every ordinal in the fundamental sequence, which is no good.). So instead, if an ordinal ends with w^m*t, we convert one of the w^m's to w^(m-1) * n and leave the remaining w^m * (t-1) alone. Hence (w^m * t)[n] = (w^m * (t-1) + w^(m-1) * n); in particular, (*stuff* + w*5)[7] = *stuff* + w*4 + 7.
We definitely do not want F_{w^2}(n) = F_{n^2}(n); again, that would be less than F_{w+1}(n). Instead, F_{w^2}(n) = F_{w*n}(n) = F_{w * (n-1) + n}(n) = F_{w * (n-1) + n-1} (F_{w * (n-1) + n-1} (... (F_{w * (n-1) + n-1}(n))...)) with n F's. So for example F_{w^2}(9) = F_{w9}(9) = F_{w8+9}(9) = F_{w8+8}(F_{w8+8}(...(n)...)).
"Then F_{w2}(n) = F_{w+n}(n), F_{w3}(n) = F_{w2+n}(n), F_{w^2}(n) = F_{wn}(n), and so on." represented three different equations; I meant
[EDIT] Hmmm, I think I disagree with Iijil's assessment that storm sorceries cannot save progress. It is true that you resolve the sorcery before you make use of it and progress further. But, that further progress will be reflected the _next_ time you resolve that storm sorcery, which is when we need the progress to show up. So, as far as I can tell, storm sorceries are okay; of course, I have been wrong before. So I will await Iijil's reply.
In the meantime, let's go back to using Empty the Warrens:
2 Psychic Battle
3 Cowardice
4 Dismiss Into Dream
5 Bloodbond March
6 Cephalid Shrine
7 Mimic Vat
8 Omniscience
9 Leyline of Anticipation
10 Mirror of Fate
11 Dual Nature
12 Gold Myr
13 Grand Architect
14 Allay
15 Copy Enchantment
16 Karn, Silver Golem
17 Mox Ruby
18 Metallurgeon
19 Bump in the Night
20 Battle Cry
21 Retract
22 Spellweaver Volute
24 Empty the Warrens
25 Skirk Prospector
26 Worldpurge
27 Flash of Defiance
28 Flash of Defiance
29 Flash of Defiance
30 Spellweaver Helix
31 Hair-Strung Koto
32 Walking Atlas
33 Glimmerpost
34 Glimmerpost
35 Deathrite Shaman
36 Everglove Courier
37 Nectar Faerie
38 Ghosthelm Courier
39 Possessed Aven
40 Centaur Archer
41 Maze Glider
42 Paragon of Eternal Wilds
44 Frightshroud Courier
45 Paragon of Open Graves
46 Lightbringer
47 Goblin Tunneler
48 Possessed Barbarian
49 Endbringer
50 Korozda Gorgon
51 Chasm Skulker
52 Angelheart Vial
53 Justice
54 Repercussion
55 Five-Alarm Fire
56 World at War
56 Winding Constrictor
58 Show and Tell
59 Underground Sea
60 Memory Jar
A hyperhyperstage, 11 stages, and 8 layers; if Consecrated Sphinx is needed, remove Repercussion, reducing the number of layers to 7.
I don't see a way to get around that problem, so if you think you can do it, can you go into more detail?
Let's say we have a 3 hyperstage hyperhyperstage, e.g. 10 life to start.
We first put the triggers on the stack:
SHT (Spellweaver Helix trigger) for Empty the Warrens
SHT for Worldpurge
We resolve the SHT for ETW, and then resolve ETW:
SHT for Worldpurge
(X red mana)
We resolve the SHT for Worldpurge, leaving it on the stack:
Worldpurge
(X red mana)
We use our X red mana to create X Metallurgeon stages:
X stages
Worldpurge
We spend 3 more life two more times to repeat the process:
X stages
Worldpurge
X stages
Worldpurge
X stages
Worldpurge
We resolve the X stages, then Worldpurge, gaining many more spells cast
X stages
Worldpurge
X stages
Worldpurge
(X2 spells cast)
we resolve a Psychic Battle trigger in the second hyperstage to get needed resources, then spend 3 life to recreate the top stage
X2 stages
Worldpurge
X stages (minus 1 PB trigger)
Worldpurge
X stages
Worldpurge
We repeat this process, until the second hyperstage is completely expended. Then we have
X stages
Worldpurge
(X3 spells cast, 7 life)
We spend 6 life to recreate the top two stages
X3 stages
Worldpurge
X3 stages
Worldpurge
X stages (minus one PB trigger)
Worldpurge
and so on and so on. So, it does seem like the hyperstages are properly updating. Am I missing anything?
[EDIT] Hmm, I think I see what you are saying... we need the Psychic Battle / Bloodbond March triggers to update in the intermediate hyperstages. Yeah, that does look like a problem. So, we do need a "saving progress" package.
I think I have an idea that could compete with the one being talked about, but I don't know absurd yours is, so feel free to tell me if my combo isn't even close. My idea revolves around Selvala, Heart of the Wilds, Heroes' Bane, Paradox Engine, and Dragonlord Kolaghan. The idea is to use Selvala to make mana equal to Heroes' Bane Power, dump it all into its doubling ability, then untapping her with Paradox Engine. Kolaghan is the haste enabler because she has the highest potential for extra damage out of all the haste enablers. I'm thinking that you could probably use Mirror Gallery and spells that clone Selvala, which would make Doubling Season even crazier with this idea.
[Edit] Should have known it couldn't have been that easy... Ow, my brain.
The
Necromancer (or Noob)
Cat.
Don't ask, I don't know why ether...
The combo you talk about is indeed nowhere close, at least not yet. Each time Paradox Engine untaps Selvala, Selvala gets X mana where X is the power of Heroes' Bane, which allows us to double Heroes' Bane's power X/4 times, to go from X to X * (2^(X/4)). This is basically exponentiation, or the first layer of recursion. Let's say you had something (call it resource A) that allowed you could expend to cast X spells, where X is the power of Heroes' Bane or the amount of mana that you have. Then you could untap Selvala X times, allowing you to pump Heroes' Bane to about X^X^X^...^X with X X's. This is called tetration, and is the second layer of recursion. It is denoted by X^^X in Knuth arrow notation. Then, suppose that you had something (call it resource B) that you could expend to gain X of resource A, where again X is one of the things that is growing bigger and bigger. Then you could apply X -> X^^X X times, which would get you to about X^^^X; this is called pentation, and is the third layer of recursion. So this is the pattern; each additional layer of recursion means that you can repeat the previous layer X times, where X represents the current output so far.
To get a better idea of how the layer system works, you can take a look at at the following writeup of one of the earliest contestants in this challenge:
Metroidcomposite's deck
This achieved 36 layers, which seemed incredible at the time (and is more than enough to overpower everyone else's attempts at this challenge). Later, we came up with the following deck that achieved 408 layers:
Our best layer deck
But, shortly after that deck was written up, we came up with a combo that changed everything: a combo that could take X of something (mana or creature tokens) and produce X layers! You could then iterate the process, so if you started with say 10 mana, iterating the "stage" (that's what we're calling the combo that creates X layers) takes you to 10^^^^^^^^^^10, then a 10^^^...^^^10 where the number of ^'s is 10^^^^^^^^^^10, and so on.
Iijil wrote an excellent post summarizing the details:
Iijil's summary
In particular, follow the first link in the above link to my post on page 2, where I describe the stage combo in some detail. Basically, you build up the stack with groups of alternating triggers, each group representing a layer; once one group runs out, one of the next group creates X of that group at the expense of one trigger in the next group, so it functions just like the layer system. The amount of mana / creature tokens keeps track of which group you are at in the stack, so as you get lower in the stack your mana/tokens builds up, allowing you to build the stack back up to the top.
That was the cutting edge for a while, but then we figured out how to build a hyperstage; this is where you take X of a resource (mana or tokens) and build up X stages. That was the best we could do for a while; then later we figured out how to do two hyperstages. Finally, very recently we came up with ideas for how to implement a hyperhyperstage - as you may guess, this is where you take X of a resource and build up X hyperstages. So this is the current frontier. We are still in the midst of tinkering with the deck and checking for problems/infinities.
Okay, I still haven't gone through the current deck step by step. That will take some doing. I will go over it generally in a later post; in the meantime, feel free to ask any questions you may have about what I have linked so far.
The latest deck:
2 Psychic Battle
3 Cowardice
4 Dismiss Into Dream
5 Bloodbond March
6 Cephalid Shrine
7 Mimic Vat
8 Omniscience
9 Leyline of Anticipation
10 Mirror of Fate
11 Dual Nature
12 Tooth of Ramos
13 Mana Vault
14 Allay
15 Copy Enchantment
16 Karn, Silver Golem
17 Mox Ruby
18 Metallurgeon
19 Bump in the Night
20 Battle Cry
21 Retract
22 Spellweaver Volute
24 Saltcrusted Steppe
25 Core Prowler
26 Shimmer
27 Urborg, Tomb of Yawgmoth
28 Time and Tide
29 Fungal Sprouting
30 Utopia Mycon
31 Endless One
32 Worldpurge
33 Flash of Defiance
34 Flash of Defiance
35 Flash of Defiance
36 Spellweaver Helix
37 Hair-Strung Koto
38 Walking Atlas
39 Glimmerpost
40 Glimmerpost
41 Deathrite Shaman
42 Everglove Courier
44 Ghosthelm Courier
45 Possessed Aven
46 Centaur Archer
47 Maze Glider
48 Paragon of Eternal Wilds
49 Eastern Paladin
50 Frightshroud Courier
51 Paragon of Open Graves
52 Lightbringer
53 Goblin Tunneler
54 Possessed Barbarian
55 Endbringer
56 Five-Alarm Fire
57 World at War
58 Show and Tell
59 Mox Sapphire
60 Memory Jar
[EDIT] Here's a possible save progress combo: Hunting Pack, to power Earthcraft into Plains. Unfortunately, this can't work with Allay, as for three mana we can cast Allay to generate many more enchantments to power Earthcraft. Generally speaking, we can't be able to create enchantments with white mana, or else we would be able to create many of them because of Dual Nature, and therefore be able to create lots more white mana. So it seems we need to get rid of Dual Nature for this to work. So I tried going back to Vitu-Ghazi Guildmage; this seems okay, as it costs four mana to populate, and that will give us only one mana back. So, the deck becomes:
2 Psychic Battle
3 Cowardice
4 Dismiss Into Dream
5 Bloodbond March
6 Cephalid Shrine
7 Mimic Vat
8 Omniscience
9 Leyline of Anticipation
10 Mirror of Fate
11 Vitu-Ghazi Guildmage
12 Copy Enchantment
13 Auratog
14 Skull of Orm
15 Karn, Silver Golem
16 Mox Ruby
17 Metallurgeon
18 Bump in the Night
19 Battle Cry
20 Retract
21 Spellweaver Volute
23 Hunting Pack
24 Earthcraft
25 Plains
26 Forest
27 Fungal Sprouting
28 Utopia Mycon
29 Worldpurge
30 Flash of Defiance
31 Flash of Defiance
32 Flash of Defiance
33 Spellweaver Helix
34 Hair-Strung Koto
35 Walking Atlas
36 Glimmerpost
37 Glimmerpost
38 Deathrite Shaman
39 Everglove Courier
40 Nectar Faerie
41 Ghosthelm Courier
43 Centaur Archer
44 Maze Glider
45 Paragon of Eternal Wilds
46 Eastern Paladin
47 Frightshroud Courier
48 Paragon of Open Graves
49 Lightbringer
50 Goblin Tunneler
51 Possessed Barbarian
52 Endbringer
53 Reality Spasm
54 Angelheart Vial
55 Five-Alarm Fire
56 World at War
57 Show and Tell
58 Mox Sapphire
59 Memory Jar
60 Consecrated Sphinx
If we make it work by using Riftsweeper or March of the Machines we can go infinite by looping Retract+Hunting Pack. All cards needed to refresh the Spellweaver Volute can have a token copy present to populate or are artifacts that are reusable enough from one retract.
The two Glimmerpost should generate 4 life, going infinite (put the second on the battlefield in response to the firsts trigger). There are a few lands like Kabira Crossroads that we can use to gain 2 life from one card.
2 Psychic Battle
3 Cowardice
4 Dismiss Into Dream
5 Bloodbond March
6 Cephalid Shrine
7 Mimic Vat
8 Omniscience
9 Leyline of Anticipation
10 Pull From Eternity
11 Izzet Guildmage
12 Silver Myr
13 Vitu-Ghazi Guildmage
14 Copy Enchantment
15 Auratog
16 Skull of Orm
17 Karn, Silver Golem
18 Mox Ruby
19 Metallurgeon
20 Bump in the Night
21 Battle Cry
22 Rebuild
23 Spellweaver Volute
25 Hunting Pack
26 Earthcraft
27 Plains
28 Forest
29 Fungal Sprouting
30 Utopia Mycon
31 Worldpurge
32 Flash of Defiance
33 Flash of Defiance
34 Flash of Defiance
35 Spellweaver Helix
36 Hair-Strung Koto
37 Walking Atlas
38 Glimmerpost
39 Radiant Fountain
40 Deathrite Shaman
41 Everglove Courier
42 Nectar Faerie
43 Ghosthelm Courier
45 Centaur Archer
46 Maze Glider
47 Paragon of Eternal Wilds
48 Eastern Paladin
49 Frightshroud Courier
50 Paragon of Open Graves
51 Lightbringer
52 Goblin Tunneler
53 Possessed Barbarian
54 Endbringer
55 Five-Alarm Fire
56 World at War
57 Show and Tell
58 Mox Sapphire
59 Memory Jar
60 Consecrated Sphinx
But Silver Myr is not good enough to power the card retrieval. To produce a blue mana we need to create a Mimic Vat copy of Silver Myr. But that means we immediately have to use that blue mana to pull the original Myr back from exile. We never get anything done.
[EDIT] could we save cards by going back to Saltcrusted Steppe? Maybe using Riftsweeper. I don't see that going infinite with Rebuild, but I also don't know how to get an intial token copy of it.
I think going back to Saltcrusted Steppe can work, and we can then use Mirror of Fate combined with March of the Machines. We will be able to get copies of Mirror of Fate after resolving Rebuild, but we need to destroy and retrieve March of the Machines, and also make a copy of Spellweaver Volute, and both of those cost mana. We have no other way of getting mana besides targeting Core Prowler, so I think it works.
2 Psychic Battle
3 Cowardice
4 Horobi, Death's Wail
5 Bloodbond March
6 Cephalid Shrine
7 Mimic Vat
8 Omniscience
9 Leyline of Anticipation
10 Mirror of Fate
11 Vitu-Ghazi Guildmage
12 Copy Enchantment
13 Auratog
14 Skull of Orm
15 March of the Machines
16 Mox Ruby
17 Metallurgeon
18 Bump in the Night
19 Battle Cry
20 Rebuild
21 Spellweaver Volute
23 Saltcrusted Steppe
24 Core Prowler
25 Shimmer
26 Urborg, Tomb of Yawgmoth
27 Time and Tide
28 Fungal Sprouting
29 Utopia Mycon
30 Worldpurge
31 Flash of Defiance
32 Flash of Defiance
33 Flash of Defiance
34 Spellweaver Helix
35 Hair-Strung Koto
36 Walking Atlas
37 Glimmerpost
38 Radiant Fountain
39 Deathrite Shaman
40 Everglove Courier
41 Nectar Faerie
43 Possessed Aven
44 Centaur Archer
45 Maze Glider
46 Paragon of Eternal Wilds
47 Eastern Paladin
48 Frightshroud Courier
49 Paragon of Open Graves
50 Lightbringer
51 Goblin Tunneler
52 Possessed Barbarian
53 Endbringer
54 Five-Alarm Fire
55 World at War
56 Winding Constrictor
57 Show and Tell
58 Mox Sapphire
59 Memory Jar
60 Consecrated Sphinx
With Winding Constrictor, we hopefully have enough to work with for Five-Alarm Fire to target Endbringer many times.
The general method of the deck is rather complicated; see my reply to CTNC above, in particular this helpful post by Iijil. I do plan to write an a full article detailing the deck, but I don't want to do it until we have finished improving the deck (up to a point). But I guess it will be harder to bring new people in before I do the write-up; it's a bit of a Catch-22.
Where was the post on hyperhyperstages?
There isn't one single post explaining it yet, but discussion on it started at post #970.
So our main stage combo (you can skip this paragraph if you understand how this works already) uses Metallurgeon tokens. If we have N metallurgeon tokens, we can cast and counter Metallurgeon (getting a bunch of Bloodbond March triggers), use one Bloodbond March trigger to bring it back to the battlefield, then tap a Metallurgeon token for a bunch of Psychic Battle triggers. Repeat N times for 2N alternating groups of Bloodbond March and Psychic Battle triggers. When we resolve the top stack of Psychic Battle triggers, we can destroy the original Metallurgeon with one of them, then use the rest to create more Psychic Battles. (In the latest deck, we can destroy Core Prowler to put a counter on Saltcrusted Steppe; once we have a bunch of counters on Saltcrusted Steppe, we can activate it and remove all but one counter to get a bunch of white and green mana, and then activate Vitu-Ghazi Guildmage a bunch of times to populate a Psychic Battle token a bunch of times.). The destruction of Metallurgeon allows us to imprint it on Mimic Vat, and create a new Metallurgeon token; we then use Mirror of Fate to bring it back from exile into the library, and Hair-Strung Koto to put it into the graveyard. Then, after the top group of Psychic Battle triggers are all gone, we use the topmost Bloodbond March trigger to bring Metallurgeon back to the battlefield, and we can use our newly created Metallurgeon token to target Metallurgeon and create a bunch more Psychic Battle triggers. We do this for each Bloodbond March trigger, until we get to the last one; after the last Bloodbond March trigger is used, we use the Psychic Battle triggers to create more Bloodbond March tokens, and we use the next Psychic Battle trigger on the stack (rather than tapping the latest Metallurgeon token) to bounce Metallurgeon back to our hand. This allows us to cast and counter Metallurgeon again, creating a bunch more Bloodbond March triggers, and we use each one to create a bunch of Psychic Battle triggers, and the process continues like this. Each group of X triggers in the stack iterates the group above it X times, so the top group adds some constant amount to the number of enchantments, the second group from the top multiplies by some constant amount, the third group exponentiates, the fourth applies tetration, and so on for 2N operations. This is what we call a "stage combo". A "second stage" will be similar, except the top group of Psychic Battle triggers will create X Metallurgeons, which allows us to increase the number of Psychic Battle tokens from X to F_w(X) in the fast-growing hierarchy. Then the second group increases from X to F_{w+1}(X), the third from X to F_{w+2}(X), and so on, so X tokens for the second group will take X to F_{w2}(X); N stages will take X to F_{wN}(X) in the fast-growing hierarchy.
So next, a hyperstage. Here, the resource controlling the combo is red mana, supplied by Mox Ruby. We make use of three spells, Bump in the Night, Battle Cry, and Rebuild. We cast Battle Cry and Rebuild so that they are in the graveyard, and then cast Spellweaver Volutes on both of them. (One of the Spellweaver Volutes will be a copy so that we don't have to waste a card on a second Spellweaver Volute.) We also cast Bump in the Night so that it is in the graveyard, and from then on we use a red mana from Mox Ruby to flashback it. So, the combo proceeds as follows: We start M red mana. Also, we have N Metallurgeon tokens and the original Metallurgeon, which we cast and tap accordingly to set up a stack of 2N alternating groups of triggers, just like in the last paragraph. Then, we spend a red mana to flashback Bump in the Night, which triggers both Spellweaver Volutes. We resolve the Spellweaver trigger on Battle Cry first and resolve the Battle Cry; this untaps all N of the Metallurgeons. Resolve the Spellweaver trigger on Rebuild, but do not resolve the Rebuild; leave it on the stack, above the stack of 2N alternating Psychic Battle and Bloodbond March triggers. Next, we start casting and tapping Metallurgeons again to build up another stack of 2N alternating groups of triggers, then spend another red mana to flashback Bump in the Night again (we use Mirror of Fate and Hair-Strung Koto to put all our spells back in the graveyard, and use Copy Enchantment to replace missing Spellweaver Volutes as needed.) Repeat the process M times, creating M+1 copies of 2N alternating groups of triggers, separated by M copies of Rebuild. Now we start resolving our stack. After we resolve the topmost stage, we resolve the topmost Rebuild, which will destroy all the Metallurgeon tokens, and bring Metallurgeon and Mox Ruby back to our hand. The Mox Ruby gives us the mana to flashback Bump in the Night, but (fortunately) we cannot do so immediately; in order to reset our spell and Spellweaver Volute situation, we have to use Mirror of Fate a few times, and copy a Spellweaver Volute. To do this, we need to activate Vitu-Ghazi Guildmage a few times, and for this we need mana. So we need to use a few Psychic Battle triggers from the next stage down to get the needed mana, and then we can properly set up our spells, and use a red mana to flashback Bump in the Night, recreating the top stage. So, we get to apply the Ackermann function for a few Psychic Battle triggers in the second stage; all told the second stage gets us F_{w2}(X), the third stage F_w3(X), and the entire hyperstage gets us F_{w^2}(X).
Next is the hyperhyperstage. This time, the resource that controls the hyperhyperstage is life. The flashback spell will be Flash of Defiance (which costs 3 life to flashback), and casting it triggers Spellweaver Helixes imprinting Fungal Sprouting and Worldpurge. The setup is precisely analogous to the hyperstage. Let's say we start with between 3P+1 and 3P+3 life, at least 2 red mana, and at least 1 Metallurgeon token. First, we setup the first hyperstage using whatever red mana we currently have more than 2(this might just be one stage if we have only 2 red mana); do not use the last Metallurgeon token on the topmost stage. Next, we use a red mana to cast Bump in the Night with a Spellweaver Volute on Time and Tide. (You will see why we do this later) Then we spend 3 life and a red mana to flashback Flash of Defiance, triggering the Spellweaver Helixes for both Fungal Sprouting and Worldpurge. Resolve the Fungal Sprouting first, after we generate a large Auratog by sacrificing enchantments. This creates a bunch of Saprolings, which we can sacrifice to Utopia Mycon for a bunch of red mana. Now we can implement a second hyperstage, with a lot more red mana to use. Repeat this process P times, creating P+1 groups of hyperstages. Resolve as above; when the topmost hyperstage is completely expended, make sure that we have put a bunch of storage counters on Saltcrusted Steppe by targeting Core Prowler a bunch of thimes. Next, play Urborg, Tomb of Yawgmoth and Shimmer, naming Swamps, and finally Time and Tide (triggered by Bump in the Night) so that Urborg and Saltcrusted Steppe phase out. Then we play Glimmerpost and Radiant Fountain, gaining 3 life. Then we play Portcullis, and then bounce a bunch of our nontoken permanents, until we are down to at most seven nontoken permanents. Finally, we let Worldpurge resolve, emptying our mana pool (but we have spent it all already), and moving the permanents still on the battlefield back to our hand, including Glimmerpost, Radiant Fountain, Mox Ruby, and the Portcullis currently exiling most of our permanents, which now come back to the battlefield. Then, Time and Tide resolves, so that Saltcrusted Steppe and Urborg, Tomb of Yawgmoth phase back in; we can now remove storage counters from Saltcrusted Steppe to generate a bunch of mana, and recreate a bunch of Psychic Battles and Bloodbond Marches. We now have the resources to get 3 more life and one more red mana, so we can flashback Flash of Defiance again - but again, to reset Flash of Defiance we need to expend some of our Psychic Battles from the next hyperstage down. So each few Psychic Battles from the next hyperstage down allows us to restart the hyperstage above it. Each hyperstage is with w^2 in the fast growing hierarchy, so P hyperstages is worth F_{w^2*P}(X), and the whole hyperhyperstage is worth F_{w^3}(X).
Whew - well, if you have any questions on the above, or the deck in general, please feel free to ask!
Also, I haven't found a setup to do it yet, but I'm worried about Time and Tide being the source of life. Couldn't it be cast, using a Spellweaver Volute, during the hyperstage, getting 3 life, getting R red mana, getting R hyperstages from the hyperstage, getting 3R red mana?
F_0(n) = n+1
F_{a+1}(n) = (F_a)^n(n) (that is, it is equal to F_a(F_a(...(F_a(n)...)) with n F_a's)
If a is a limit ordinal (that is, it doesn't end with +n) F_a(n) = F_a[n](n), where a[n] is the fundamental sequence for a
The subscript for F is what we call an ordinal. In broad terms, ordinals are ways to measure the lengths of sequences that can be longer than the standard infinite sequence. So:
the empty sequence has order type of the ordinal 0
(a) has order type the ordinal 1
(a,b) has order type the ordinal 2
(a,b,c) has order type the ordinal 3
...
(a,b,c,...) has order type the ordinal w (this is the Greek letter omega, but apparently this forum doesn't allow the use of non-Latin unicode characters. WTF.)
(a,b,c,...,a2) has order type w+1 (this is an infinite sequence followed by one element that is bigger than the whole sequence)
(a,b,c,...,a2,b2) has order type w+2
...
(a,b,c,...,a2,b2,c2,...) has order type w*2 (this is written w2 rather than 2w for technical reasons; 2w is actually 2+2+2+... = w, whereas w2 = w+w, which is what we want)
...
(a,b,c,...,a2,b2,c2,...,a3,b3,c3,...) has order type w*3
...
an infinite sequence of infinite sequences has order type w^2
...
an infinite sequence of infinite sequence of infinite sequences has order type w^3
...
This goes on forever (and by forever, I really mean forever... anything that you can conceptualize is really just a really small initial segment of the ordinals. For example, w^w^w^... is peanuts. But for our purposes we need only worry about ordinals below w^w, which are basically polynomials over w with natural number coefficients. (Well, ordinal addition and multiplication is different from polynomial addition and multiplication, but that need not concern us.) For these ordinals, fundamental sequences are easy: if an ordinal a ends in w^m, a[n] is a except the final w^m is replaced by w^(m-1)*n. So for example:
(w^5 + w^3 * 3 + w^2 * 5 + w * 5)[7] = w^5 + w^3 * 3 + w^2 * 5 + w * 4 + 7
(w^11 * 3 + w^8 * 5 + w^5 * 3)[11] = w^11 * 3 + w^8 * 5 + w^5 * 2 + w^4 * 11
w^4[6] = w^3 * 6
So let's work out some examples of the fast-growing hierarchy.
F_1(3) = F_0(F_0(F_0(3))) = F_0(F_0(4)) = F_0(5) = 6
In general, F_1 applies F_0 n times to n, and each time it adds one, so we get 2n in the end.
F_2(2) = F_1(F_1(2)) = F_1(4) = 8
F_2(3) = F_1(F_1(F_1(3))) = F_1(F_1(6)) = F_1(12) = 24
In general, F_2 applies F_1 n times to n, and each time it doubles the input, so we get n * 2 * 2... * 2 = n * 2^n.
F_3(2) = F_2(F_2(2)) = F_2(8) = 8 * 2^8 = 2048
F_3(3) = F_2(F_2(F_2(3))) = F_2(F_2(24)) = F_2(24 * 2^24) = (24 * 2^24) * 2^(24 * 2^24) > 10^121,000,000
F_3 applies F_2 n times to n, and each time the output is greater than 2 to the power of the input, so the result will be more than 2^2^2...^n with n 2's, which is more than 2^^(n+1).
In general, F_(m+1)(n) will be more than 2^...^(n+1) with m ^'s.
So, for finte subscripts the fast-growing hierarchy is similar to Knuth arrows, or Conway chains of length 3.
Next we have F_w(n). This is just F_w[n](n) = F_n(n), simple. So for example F_w(200) = F_200(200).
So what's the big deal? Well, let's look at F_{w+1}(n).
F_{w+1}(2) = F_w(F_w(2)) = F_w(F_2(2)) = F_w(8) = F_8(8)
F_{w+1}(3) = F_w(F_w(F_w(3))) = F_w(F_w(F_3(3)))) = F_w(F_{F_3(3)}(F_3(3))) = F_{F_{F_3(3)}(F_3(3))}(F_{F_3(3)}(F_3(3)))
As you can see, we get subscripts of subscripts of subscripts. F_{w+1}(n) gets you an expression with n nested subscripts. In Knuth arrow notation, this would be something like "3^...^3, where the number of ^'s is 3^...^3, where the number of ^'s is 3^...^3,..." repeated n times. If you are familiar with Graham's number, you can see from the previous that F_{w+1}(64) is greater than Graham's number. So our decks were able to beat Graham's number once we created the stage combo, and added one or two layers to the first stage!
Since F_{w+1}(n) is similar to G(n) from Graham's number, F_{w+2}(n) is similar to G(G(G(...G(n)...)) with n G's. Then F_{w2}(n) = F_{w+n}(n), F_{w3}(n) = F_{w2+n}(n), F_{w^2}(n) = F_{wn}(n), and so on.
F{wm}(n) is similar to a Conway chain of length m+2, so once we get past F_{w^2} we are beyond what Conway chains can reasonably describe.
Time and Tide is not the source of life; Worldpurge bringing the life lands back to our hand so that we can replay it is the source of life. Keep in mind that phasing in does not count as entering the battlefield. Also, I don't see how Time and Tide could generate red mana.
Whoops, I forgot that phasing in doesn't count as entering the battlefield. It could have only generated red mana by giving you 3 life and letting you start the hyperhyperstage.
EDIT: What I don't get:
Your first two examples seem wrong, shouldn't they be w^5 + w^3 * 3 + w^2 * 5 + 35 and w^11 * 3 + w^8 * 5 + w^5 * 2 + w^4 * 33 respectively? And if not, why not?
I think I get the rest, so since F_w(n) = F_w[n](n) = F_n(n), would F_{w^2}(n) be F_{w*n}(n), which would be F_n^2(n)? And so, in general, F_{w^x}(n) would be equal to F_n^x(n)? This feels wrong because of what you wrote about F_{w+1}(2) and F_{w+1}(3), but seems to work using the same process from which F_w(n) = F_n(n) was derived. But I could see it going either way.
EDIT 2: I also don't get what you were doing with "Then F_{w2}(n) = F_{w+n}(n), F_{w3}(n) = F_{w2+n}(n), F_{w^2}(n) = F_{wn}(n), and so on.", that seems to just be infinitely growing, because the alternative is that F_{w*2}(n) is larger than F_{w*3}(n), and even larger than F_{w*n}(n).
How are you making up for your opponent losing life to Bump in the Night? Or is Bump in the Night strictly not resolving via Cephalid Shrine?
If an ordinal ends in w^m*t, we don't convert the whole thing to w^(m-1) * tn. That would mean, for example, that F_{w5}[n] = F_{5n}[n], which would be smaller than F_{w+1}[n]. We want the fundamental sequence for a to be a sequence of ordinals that go all the way up to a (meaning that every ordinal less than a will be beaten by a[n] for n big enough; if (w*5)[n] = 5n, then w would be bigger than every ordinal in the fundamental sequence, which is no good.). So instead, if an ordinal ends with w^m*t, we convert one of the w^m's to w^(m-1) * n and leave the remaining w^m * (t-1) alone. Hence (w^m * t)[n] = (w^m * (t-1) + w^(m-1) * n); in particular, (*stuff* + w*5)[7] = *stuff* + w*4 + 7.
We definitely do not want F_{w^2}(n) = F_{n^2}(n); again, that would be less than F_{w+1}(n). Instead, F_{w^2}(n) = F_{w*n}(n) = F_{w * (n-1) + n}(n) = F_{w * (n-1) + n-1} (F_{w * (n-1) + n-1} (... (F_{w * (n-1) + n-1}(n))...)) with n F's. So for example F_{w^2}(9) = F_{w9}(9) = F_{w8+9}(9) = F_{w8+8}(F_{w8+8}(...(n)...)).
"Then F_{w2}(n) = F_{w+n}(n), F_{w3}(n) = F_{w2+n}(n), F_{w^2}(n) = F_{wn}(n), and so on." represented three different equations; I meant
F_{w2}(n) = F_{w+n}(n)
F_{w3}(n) = F_{w2+n}(n)
F_{w^2}(n) = F_{wn}(n)
@osieorb18: Yes, we always counter Bump in the Night. Sorry, I forgot to mention that.