After much fruitless searching I haven't been able to improve the red/blue/green/white stages there were a couple of close ideas that either cut off too much other stuff or used too many cards, or something similar.
The real problem is white.
White's activated abilities/kickers/flashbacks/etc are pretty awful at forming a stage. Especially compared to green and blue, which have multiple good options, though again, I couldn't improve that part of our deck either.
MKM is mostly a miss but there are some interesting cards: evidence examiner has potential for several ending layers, seems nearly better than mind's eye? (with swapping back to infectious inquiry for the initial draws)
break out is a creature tutor and haste enabler, it almost allows us to go back to phabine as it hits everything we need it to except master transmuter.
still hoping for a good white activated ability to fix the awkwardness of the stages.
edit: sludge titan would have been useful i think for an old version? It's slightly too good now, but not that far off.
Evidence examiner is better than mind's eye since ashnod can copy clue token abilities
3 opponent draw -> many minds eye trigger per opponent draw -> 1 card draw per mind's eye trigger
vs
1 combat -> many collect evidence trigger -> many clues per collect evidence trigger -> many card draw activation per clue
I don't think the "many clues per collect evidence trigger -> many card draw activation per clue" part works out. After each draw we need to go through the whole combo. That includes resolving some Worldfires to clear the board, which gets rid of the stockpiled clues. Activating all clues before the next worldfire gets rid of the full increase to the "many" draws, so it's not worth a layer.
(also I'm not entirely sure where the "many clues per collect evidence trigger" come from in the latest version of the deck.)
there aren't stockpiled clues on board at once (which wouldn't work); rather many copies of evidence examiner in play cause many investegate triggers per collect evidence trigger
OTJ has a few 'whenever you commit a crime' cards that might be useful, with Kaervek, the Punisher being the clear standout, potentially allowing for another higher order stage? (something like black instants between the foundry stage and the volute stage?) I'm not sure if we could get it to work perfectly but it definitely popped out at me from the spoilers. If this can be made to work it's a huge deal.
Ah, nice. It doesn’t look like Alaina is enough to put Mind’s Eye back in front; I think it adds another layer to the Mind’s Eye version, getting us to w^3 + w3 + 4, with an input of close to 300. But I think the evidence examiner version gets us also to w^3 + w3 + 4, with an input comparable to how make examiners we can put on the field at the beginning of combat.
Edit: unless Alania can replace multiple cards at the start? That would be sweet.
How can we get Alania, Divergent Storm triggers for creature types that we don't play as card in the deck? Artificial Evolution can change the type of spells after we cast them, but how do we cast a Pony?
How can we get Alania, Divergent Storm triggers for creature types that we don't play as card in the deck? Artificial Evolution can change the type of spells after we cast them, but how do we cast a Pony?
Isochron Scepter and Eye of the Storm cast the copies and would therefore trigger Alania, Divergent Storm. Unfortunately, they can't target a card that's been on the stack for a while which is when we have to modify it with Artificial Evolution. From what I could find the cards that copy on the stack, such as Fork, don't have language like "cast the copy". If we had a card that targets the stack and casts the copy, then we could:
Cast a creature spell
Cast Artificial Evolution changing the creature type
Cast the hypothetical stack-targeting caster copier to trigger Alania
Unfortunately, I don't think a card like that exists.
How can we get Alania, Divergent Storm triggers for creature types that we don't play as card in the deck? Artificial Evolution can change the type of spells after we cast them, but how do we cast a Pony?
Isochron Scepter and Eye of the Storm cast the copies and would therefore trigger Alania, Divergent Storm. Unfortunately, they can't target a card that's been on the stack for a while which is when we have to modify it with Artificial Evolution. From what I could find the cards that copy on the stack, such as Fork, don't have language like "cast the copy". If we had a card that targets the stack and casts the copy, then we could:
Cast a creature spell
Cast Artificial Evolution changing the creature type
Cast the hypothetical stack-targeting caster copier to trigger Alania
Unfortunately, I don't think a card like that exists.
No, the text-changing effect is not copiable.
707.2. When copying an object, the copy acquires the copiable values of the original object’s characteristics and, for an object on the stack, choices made when casting or activating it (mode, targets, the value of X, whether it was kicked, how it will affect multiple targets, and so on). The copiable values are the values derived from the text printed on the object (that text being name, mana cost, color indicator, card type, subtype, supertype, rules text, power, toughness, and/or loyalty), as modified by other copy effects, by its face-down status, and by “as . . . enters” and “as . . . is turned face up” abilities that set power and toughness (and may also set additional characteristics). Other effects (including type-changing and text-changing effects), status, counters, and stickers are not copied.
- Does it work? Having flash itself is a potential issue. I don't remember if there is a way to end up with it in our hand during a computation though and glancing over the list i don't see one, so could be fine.
- Does it help? I don't see it helping actually gain layers or card slots, but it may make the computation/stage recovery processes simpler. Which may make it easier to understand and easier to communicate in a writeup, as well as maybe consuming slightly fewer resources which is an infinitesimal damage improvement.
edit: actually the fact that orrery can be drawn by drake makes the startup easier which is the more important thing
I think I have a way to get F_{w^w} -- under a stronger no-infinite condition: there are no possible infinite loops, even if they don't produce damage, and on any turn (which rules out the current approach of iterated Busy Beaver functions).
Recall the way the pre-Ackermann decks worked, by having a sequence of resources, each of which can be spent to get lots of the previous resource. Then the Toralf, God of Fury method extends that to infinitely many resources, indexed by the natural numbers.
My new idea is to impose a more complicated well-order on a subset of the natural numbers, isomorphic to the desired ordinal w^w.
The mapping is: Take the binary expansion of a natural number. If it ends in a 1 bit, then count how many 0 bits there are in between each two consecutive 1 bits; those numbers are used as the coefficients of powers of w. For example, 10001101001(binary) maps to w^3*3+w+2.
Resource n is a number of creatures with n +1/+1 counters on them.
Now resolve the Rosnakht trigger, creating a 0/1 creature token, which triggers Remnant of the Rising Star and Halana, Kessig Ranger. Stack those triggers to resolve in that order, adding +1/+1 counters and then dealing the same amount of damage to the selected creature (which is a copy of Innocent Bystander).
We will need to draw two cards to get them back from the library. One will come from the Clue token produced by Innocent Bystander's ability, and the second will come from Taii Wakeen, Perfect Shot also triggering from the damage. This normally means dealing exactly enough damage to destroy the creature on its own, but this changes when we add wither using Massacre Girl, Known Killer, because Taii Wakeen's ability, like most triggered abilities, is checked after the event (rule 603.10), at which point the creature's toughness will be reduced by the -1/-1 counters. In this case, to trigger Taii Wakeen's ability, the amount of damage must be exactly half of the creature's original toughness (which thus must be even). Because Innocent Bystander has 1 base toughness, this means the number of +1/+1 counters goes from n to n-(n+1)/2 = (n-1)/2, removing a 1 bit from the end of the binary expansion.
We also have Rite of Passage triggering from the damage, which adds 1 to the number, which can change the last bit from 0 to 1; it is worse to resolve it when the last bit is 1, which results in a net loss in the number of 1 bits. If the last bit is still 1, repeat the procedure to also consume that 1 bit.
After all the final 1 bits have been consumed, start resolving the stuff on the stack. The Rite of Passage trigger changes a final 0 bit to 1, and then we get a bunch of copies of Solidarity of Heroes, which add a bunch of 0 bits to the end, and this repeats. However, after the last Rite of Passage trigger resolves, we do not want to add 0 bits at the ultimate end, so point the copies of Solidarity of Heroes elsewhere.
The net result on the ordinal is to reduce by 1 the first nonzero coefficient of a power of w, and to add to the lower coefficients.
Through the procedure, we have also been creating new creatures and adding +1/+1 counters to them. The first one has (n+1)/2 +1/+1 counters, corresponding to an ordinal that is either 1 less (if the binary expansion ends in 01, equivalently if the ordinal is a successor ordinal) or a lot less (otherwise, the number of 1 bits is lower, thus the highest power of w is lower), and the later ones all correspond to ordinals that have a lower highest power of w.
Note that Innocent Bystander has the condition "3 or more damage"; this cuts the repetition off after n=5, and prevents endlessly hitting an n=1 creature which restores back to n=1.
To rule out having Taii Wakeen's damage-increasing effect, Talisman of Curiosity is the provider of blue and green mana, with Paradox Engine to untap it, and Healer of the Pride gives 2 life with each iteration of the procedure, exactly offsetting the damage provided that the damage is not increased. Colorless mana is provided by Mana Echoes. This limits some of the iteration, but it doesn't make much of a difference; however, this rules out the possibility of omitting Massacre Girl, Known Killer, because in that case the number of creatures stays constant while their toughnesses have to keep increasing.
Note: It is important that Assimilation Aegis can be on only one creature at a time, and can only move at sorcery speed. Otherwise, it would be possible to send the last damage to a different creature, and then the key creature could get added 0 bits from Solidarity of Heroes copies before resolving its first Rite of Passage trigger, and end up with a higher ordinal.
Using the 0 bits in the binary expansion, seperated by the 1s to store information is some clever out of the box thinking. I would never have expected that to work, but this method with Solidarity of Heroes copies seperated by Rite of Passage triggers seems to accomplish it.
Could you give a more detailed explanation for how this gets to f_{w^w}? It looks plausible, but it seems easy to go wrong there.
I'd also like a closer look at Mana Echoes as the mana provider for Remnant of the Rising Star. At first glance those seem to put a limit on the size of the resources we can consume, which is scary.
When we crack a limit ordinal creature with storm count X we get up to X Solidarity of Heroes copies per 1 in the representation. So if we go all out the next representation has O(2^X) toughness. It would take that much mana to make full use of it without missing the Taii Wakeen, Perfect Shot card draw and fizzling out.
Energy Tap generates a constant amount of mana per copy, so at that point the total mana it has generated in the past is O(X^2) for at most X casts with at most X copies each. That's pretty much the same as Mana Echoes since we get 1 creature generating at most X mana every time we increase the storm count.
Either way we clearly need to hold back on the number of Solidarity of Heroes copies we actually target onto the key creatures, to prevent fizzling out. But the amount of mana we have still grows every time we get to such a point. So to me it looks like it is still fine to get to f_{w^w}. It's just that every time I try to be more formal about it I get lost.
That would go infinite because Energy Tap also targets a creature, giving an extra Rosnakht trigger, and the extra creature token can be set to any value without having to match half of an existing value minus 1.
For an ordinal a, let |a| denote the integer that represents it; i.e. the number of +1/+1 counters a creature needs to be representing it.
Let F(a,X) denote the creature count (of which storm count is proportional) that can be created starting from X and a creature with |a| counters.
Note that |a|/2+2 mana is required to start processing that creature.
For successor ordinals: Start with a creature of power |b+1| - its binary representation ends in 01.
It ends up with power |b+1|//2+1 = |b| [where // is integer division] and creates a creaure of power |b|//2 which is even and thus unusable. It does however create 2X mana from mana echoes which is more than sufficient to keep the chain going.
Thus, F(b+1,X) = F(b,X+1)
For limit ordinals:
Let a's lowest power of w be k; so |a| ends in a 0 followed by k+1 1s.
The process transforms the starting creature into one with |b| counters in which b is obtained by replacing the w^k in a by sum[w^(k-1-i) * a_i] for some sequence a_0..a_(k-1) that we can choose, for which each a_i < X.
Let A = sum[a_i]; then |b| = O(2^A * |a|) - so we must choose a_i for which we will have enough mana to process |b| when we get there.
(minor complications when a = w^k exactly and the minimum damage basecase (|a| <= 5) comes into play)
The process also creates k+1 creatures, k of which correspond to ordinals c_0 ... c_k (from largest to smallest) much smaller than a (smaller highest power of w), and the k+1th is even and thus unusable. (|c_i| = |a|//2^(i+1)) This creates ~2(k+1)X mana, plenty start processing them.
Let Y = F(c_0, F(c_1, ...F(c_k,X+k+1)...)) - i.e. the creature count we would get after processing all the residual creatures.
Then, we choose the maximal coefficients a_i for which |b| <= Y - for example we could chose a_0 = log_2(Y) if that's <= X.
Once we reach the point where processing the c_0 ordinals becomes stronger than exponentiation of X, we'd have the freedom to choose their maximal possible values which are the storm count.
So, F(a, X) = F(b, Y)
This feels like a variation of the Middle-Growing heirarchy) which does asymptotically behave like the FGH at w^w.
|0| = 11 (binary) is the base case - not damagable.
|n| = 100..01 with n 0s, for finite ordinal n<w
F(n, X) = X+n for n<w
F(w,X)
|w| = 1011
create 101, soidarity triggers, rite trigger
create 10, solidarity triggers, rite trigger
original is 10 -> resolve rite, becomes 11; resolve solidarity triggers and rite, becomes 1100..001; with a 0s for our choice of a<X.
usable byproduct is 101 = |1| F(1,X) = X+1. thus choose a<log(X+1).
our original became 110001.
sequences starting with multiple leading 1s according the the description of the ordinal representation would represent the same ordinal as if all but one leading 1 were removed.
let's check how they behave:
as before, 11 is undamagable
111:
create 11, solidarity triggers, rite trigger
remainder undamagable. resulve rite becomes 100 is unusable.
so we created 1 creature, 111 behaves like |1| = 101.
110001: goes through succ case as usual 3 times to 111; then above case can do 1 more. thus increments X by 4.
so 11(0^n)1 as a binary string behaves the same as 1(0^n+1)1 = |n+1| and can be considered another representation of it
back to above, from |w| = 1011 we ended up with 11(0^a)1 ~= |a+1|
so F(w,X) ~= X+log X
check how 1111 behaves:
create 111, solidarity triggers, rites trigger
create 11, solidarity triggers, rites trigger - remainder undamagable.
resolve rites -> 100, resolve solidarity+rites -> 100(0^a)1 for some a = |a+2|. remainders are |1| and |0|; creating 3 creatures total. so choose a=log(X+2)
1111 behaves like w.
Hypothesis: A binary sequence that starts with multiple leading 1s behaves the same as the one obtained by replacing the second 1 with a 0.
try F(w*2, X):
|w2| = 10011
create 1001, solidarity triggers, rites trigger
create 100, solidarity triggers, rites trigger
original becomes 101 -> 101(0^a)1 = |w+a|
usable residual |1001| = 2
using it up creates 4 creatures. choose a<log(X+4)
F(w*n,X) = O( the nth iteration of the function (X -> X+log(X))
call this gn
g1 > X + log(X)
g2 = g1 + log g1 > X + log X + log g1 > X + 2*logX
g3 = g2 + log g2 > X + 2logX + log g2 > X + 3logX
in general, gn > X + n*logX
F(w*n, X) >= X + n*log X
F(w^2,X):
|w^2| = 10111
create 1011 (=|w|), solidarity, rites
create 101 (=|1|), solidarity, rites
create 10, solidarity, rites
resolve: 11(0^a)1(0^b)1; for now choose b=0. we have a representation of w*(a+1) as per above analysis about sequences starting with multipe 1s.
our residuals are |1| and |w| which processing will take X to O(X+logX); choose a< log(X+log X)
F(w^2, X) > F(w*(log (X+logX)), X+logX) > (X+logX + log(X+logX)*log(X+logX)) > X + (log(X+logX))^2
F(w^2+w, X):
|w^2+w| = 101011
create 10101 (=|w+1|), solidarity, rites
create 1010, solidarity, rites
resolve: 1011(0^a)1 = |w^2 + a|
residuals |w+1| takes X to O(X+logX); choose a<log(X+logX)
F(w^2+w, X) ~= F(w^2 + log(X+logX), X+logX) = F(w^2, X+logX+log(X+logX)) = F(w^2, g2(X)) with the g function above
F(w^2*2, X):
|w^2*2| = 100111
create 10011 = |w2|, solidarity, rites
create 1001 = |2|, solidarity, rites
create 100, solidarity, rites
resolve: 101(0^a)1(0^b)1 = |w^2+w*a+b|. assume b=0
residuals |2| and |w2| take X to > X+2logX; choose a < log(X+2logX)
F(w^2*2, X) ~= F(w^2 + w*log(X+2logX))
F(w^2*k + w*n, X) (k,n=>1)
|w^2*k + w*n|
residuals are |w*k+n| and |k|
-> F(w^2*k + w*(n-1) + log F(w*k+n, X+k), F(w*k+n, X+k)) = F(w^2*k + w*(n-1), g1(F(w*k, X+k+n))
=> F(w^2*k, Y)
where Y is the nth iteration of the function X -> g1(X+k*log(X))
analysing this seems tricky
feels like it ought escape logs at some point (and thereby catch up to the FGH) but not quite sure.
Meanwhile the Innocent Bystander went to 9-5=4 +1/+1 counters with 5 toughness to draw the Taii Wakeen, Perfect Shot card. Then it gets one counter back from Rite of Passage to end up with 5 +1/+1 counters at 101 as well.
processing a 1000111 (=w^2*3), we target a Innocent Bystander copy with 64+3+2+1=71 +1/+1 counters. That's 72 toughness, so on the first hit we need to deal 36 damage. With 36 +1/+1 counters the byproduct from that hit is not usable. The main target is at 36 toughness with the Hardened Scales trigger on the stack. Second hit for 18, another unusable byproduct. Main target at 18 toughness. 3rd hit at 9. This last byproduct is a usable 1001 (=|2|).
The main target got down to 8 +1/+1 counters (1000 in binary) and resolves the Hardened Scales and Solidarity of Heroes to go to 10010...010...01 (=|w^2*2 + w*X + X)
So there's only 1 usable byproduct from a limit ordinal. Hopefully that is still enough to generate the mana to make storm the relevant limit eventually.
Meanwhile the byproduct from a successor ordinal is usable, and actually is as powerful as the main target. And that might actually be important. Without that extra boost this whole thing looks a lot like a weak Goodstein sequence to me (from looking at it without mana considerations), which apparently only grows like f_{w}.
Right; so we actually have
F(a+1,X) = F(a,F(a,X+1)),
and F(a,X) for limit a = F(b,(F(c,X)), where c is obtained by "dividing" the highest possible power of w and subtracting 1; and b is limited either by X or by log(F(c,X))
also leading 1s essentialy don't matter - processing 111 yields no usable byproducts. at most you could get a single extra step from an extra leading 1.
F(w,X)
proc 1011
create 101+1 = 110 (unusuable), solidarity, rite
create 10+1 = 11 = |0|, solidarity, rite
resolve: 11(0^a)1 = |a|. minimal extra value from residual so a=logX
F(w,X) = F(log X, X+1) = X+1+2^logX-1 = 2X
F(w+1,X) = 4X
F(w+2,X) = 16X
F(w+n,X) = 2^2^n * X
F(w2)
proc 10011
usable residual 101 = |1|; main becomes 101(0^a)1 = |w+a|
minimal residual value so take a=logX
F(w2) = F(w+logX,X) = 2^2^log X * X = 2^X * X > 2^X
F(w2+1) >= 2^2^X
F(w3) >= 2^^X
F(w4) >= 2^^^X
yeah, the extra power from the successor byproduct enables the scaling to counteract the logs immediately and by the time the limit byproducts are contributing enough to make back the full mana to be able to use all X of the storm rather than logX (which happens when the byproduct is w2; which is when the original ordinal is w^3+w) the difference is irrelevant.
The real problem is white.
White's activated abilities/kickers/flashbacks/etc are pretty awful at forming a stage. Especially compared to green and blue, which have multiple good options, though again, I couldn't improve that part of our deck either.
So it seems like this is where we are:
1 Coat of Arms
2 Xathrid Necromancer
3 Artificial Evolution
4 Arcbond
5 Comeuppance
Starting a Computation
6 Opalescence
7 Wrong Turn
8 Kaervek's Spite
9 Goblin Boom Keg
10 March of the Machines
Utility
11 Vedalken Orrery
12 Metalworker
13 Twinning Glass
14 Thousand-Year Storm
15 Bloodbond March
16 Chancellor of the Annex
Foundry Stage
17 Martyr of Spores
18 Soul Foundry
19 Emblem of the Warmind
20 Cowardice
21 Riftsweeper
22 Chromescale Drake
23 Ashnod the Uncaring
24 Ratadrabik of Urborg
25 Spellweaver Volute
26 Veyran, Voice of Duality
27 Fated Infatuation
28 Lingering Souls
29 K'rrik, Son of Yawgmoth
30 Archon of Falling Stars
31 Pious Interdiction
Helix Megastage
32 Spellweaver Helix
33 Worldfire
34 Starlight
35 Abandon the Post
36 Abandon the Post
37 Mox Ruby
38 Yarok, the Desecrated
Blue Mana Stage
39 Master Transmuter
40 Floodchaser
41 Great Furnace
42 Brutal Suppression
43 Search for Glory
44 Snow-Covered Island
45 Arboreal Grazer
46 Firebrand Ranger
47 Groundskeeper
48 Forest
White Mana Stage
49 Ruin Ghost
50 Weathered Wayfarer
51 Archaeological Dig
52 Ghitu Encampment
Start
53 Mana Crypt
54 Epic Experiment
55 Resurgent Belief
56 Omniscience
57 Caress of Phyrexia
End Layers
58 Mind's Eye
59 Smoke Spirits' Aid
60 Precursor Golem
evidence examiner has potential for several ending layers, seems nearly better than mind's eye? (with swapping back to infectious inquiry for the initial draws)
break out is a creature tutor and haste enabler, it almost allows us to go back to phabine as it hits everything we need it to except master transmuter.
still hoping for a good white activated ability to fix the awkwardness of the stages.
edit: sludge titan would have been useful i think for an old version? It's slightly too good now, but not that far off.
3 opponent draw -> many minds eye trigger per opponent draw -> 1 card draw per mind's eye trigger
vs
1 combat -> many collect evidence trigger -> many clues per collect evidence trigger -> many card draw activation per clue
(also I'm not entirely sure where the "many clues per collect evidence trigger" come from in the latest version of the deck.)
Edit: unless Alania can replace multiple cards at the start? That would be sweet.
Isochron Scepter and Eye of the Storm cast the copies and would therefore trigger Alania, Divergent Storm. Unfortunately, they can't target a card that's been on the stack for a while which is when we have to modify it with Artificial Evolution. From what I could find the cards that copy on the stack, such as Fork, don't have language like "cast the copy". If we had a card that targets the stack and casts the copy, then we could:
Unfortunately, I don't think a card like that exists.
No, the text-changing effect is not copiable.
707.2. When copying an object, the copy acquires the copiable values of the original object’s characteristics and, for an object on the stack, choices made when casting or activating it (mode, targets, the value of X, whether it was kicked, how it will affect multiple targets, and so on). The copiable values are the values derived from the text printed on the object (that text being name, mana cost, color indicator, card type, subtype, supertype, rules text, power, toughness, and/or loyalty), as modified by other copy effects, by its face-down status, and by “as . . . enters” and “as . . . is turned face up” abilities that set power and toughness (and may also set additional characteristics). Other effects (including type-changing and text-changing effects), status, counters, and stickers are not copied.
- Does it work? Having flash itself is a potential issue. I don't remember if there is a way to end up with it in our hand during a computation though and glancing over the list i don't see one, so could be fine.
- Does it help? I don't see it helping actually gain layers or card slots, but it may make the computation/stage recovery processes simpler. Which may make it easier to understand and easier to communicate in a writeup, as well as maybe consuming slightly fewer resources which is an infinitesimal damage improvement.
edit: actually the fact that orrery can be drawn by drake makes the startup easier which is the more important thing
Recall the way the pre-Ackermann decks worked, by having a sequence of resources, each of which can be spent to get lots of the previous resource. Then the Toralf, God of Fury method extends that to infinitely many resources, indexed by the natural numbers.
My new idea is to impose a more complicated well-order on a subset of the natural numbers, isomorphic to the desired ordinal w^w.
The mapping is: Take the binary expansion of a natural number. If it ends in a 1 bit, then count how many 0 bits there are in between each two consecutive 1 bits; those numbers are used as the coefficients of powers of w. For example, 10001101001(binary) maps to w^3*3+w+2.
Resource n is a number of creatures with n +1/+1 counters on them.
There is an Assimilation Aegis exiling Innocent Bystander. Equip it to a creature with n +1/+1 counters, and resolve the resulting trigger. Now cast Solidarity of Heroes targeting Rosnakht, Heir of Rohgahh, which triggers Rosnakht and Thousand-Year Storm; put the Rosnakht trigger on top. Cast Memory Lapse, and use the copies from Thousand-Year Storm to put both cards back into the library.
Now resolve the Rosnakht trigger, creating a 0/1 creature token, which triggers Remnant of the Rising Star and Halana, Kessig Ranger. Stack those triggers to resolve in that order, adding +1/+1 counters and then dealing the same amount of damage to the selected creature (which is a copy of Innocent Bystander).
We will need to draw two cards to get them back from the library. One will come from the Clue token produced by Innocent Bystander's ability, and the second will come from Taii Wakeen, Perfect Shot also triggering from the damage. This normally means dealing exactly enough damage to destroy the creature on its own, but this changes when we add wither using Massacre Girl, Known Killer, because Taii Wakeen's ability, like most triggered abilities, is checked after the event (rule 603.10), at which point the creature's toughness will be reduced by the -1/-1 counters. In this case, to trigger Taii Wakeen's ability, the amount of damage must be exactly half of the creature's original toughness (which thus must be even). Because Innocent Bystander has 1 base toughness, this means the number of +1/+1 counters goes from n to n-(n+1)/2 = (n-1)/2, removing a 1 bit from the end of the binary expansion.
We also have Rite of Passage triggering from the damage, which adds 1 to the number, which can change the last bit from 0 to 1; it is worse to resolve it when the last bit is 1, which results in a net loss in the number of 1 bits. If the last bit is still 1, repeat the procedure to also consume that 1 bit.
After all the final 1 bits have been consumed, start resolving the stuff on the stack. The Rite of Passage trigger changes a final 0 bit to 1, and then we get a bunch of copies of Solidarity of Heroes, which add a bunch of 0 bits to the end, and this repeats. However, after the last Rite of Passage trigger resolves, we do not want to add 0 bits at the ultimate end, so point the copies of Solidarity of Heroes elsewhere.
The net result on the ordinal is to reduce by 1 the first nonzero coefficient of a power of w, and to add to the lower coefficients.
Through the procedure, we have also been creating new creatures and adding +1/+1 counters to them. The first one has (n+1)/2 +1/+1 counters, corresponding to an ordinal that is either 1 less (if the binary expansion ends in 01, equivalently if the ordinal is a successor ordinal) or a lot less (otherwise, the number of 1 bits is lower, thus the highest power of w is lower), and the later ones all correspond to ordinals that have a lower highest power of w.
Note that Innocent Bystander has the condition "3 or more damage"; this cuts the repetition off after n=5, and prevents endlessly hitting an n=1 creature which restores back to n=1.
To rule out having Taii Wakeen's damage-increasing effect, Talisman of Curiosity is the provider of blue and green mana, with Paradox Engine to untap it, and Healer of the Pride gives 2 life with each iteration of the procedure, exactly offsetting the damage provided that the damage is not increased. Colorless mana is provided by Mana Echoes. This limits some of the iteration, but it doesn't make much of a difference; however, this rules out the possibility of omitting Massacre Girl, Known Killer, because in that case the number of creatures stays constant while their toughnesses have to keep increasing.
Note: It is important that Assimilation Aegis can be on only one creature at a time, and can only move at sorcery speed. Otherwise, it would be possible to send the last damage to a different creature, and then the key creature could get added 0 bits from Solidarity of Heroes copies before resolving its first Rite of Passage trigger, and end up with a higher ordinal.
Using the 0 bits in the binary expansion, seperated by the 1s to store information is some clever out of the box thinking. I would never have expected that to work, but this method with Solidarity of Heroes copies seperated by Rite of Passage triggers seems to accomplish it.
Funny how that is enabled by making all even numbers unusable with the Taii Wakeen, Perfect Shot + Massacre Girl, Known Killer combo.
Could you give a more detailed explanation for how this gets to f_{w^w}? It looks plausible, but it seems easy to go wrong there.
I'd also like a closer look at Mana Echoes as the mana provider for Remnant of the Rising Star. At first glance those seem to put a limit on the size of the resources we can consume, which is scary.
(the existing paradox engine untaps the big creatures for energy tap)
Energy Tap generates a constant amount of mana per copy, so at that point the total mana it has generated in the past is O(X^2) for at most X casts with at most X copies each. That's pretty much the same as Mana Echoes since we get 1 creature generating at most X mana every time we increase the storm count.
Either way we clearly need to hold back on the number of Solidarity of Heroes copies we actually target onto the key creatures, to prevent fizzling out. But the amount of mana we have still grows every time we get to such a point. So to me it looks like it is still fine to get to f_{w^w}. It's just that every time I try to be more formal about it I get lost.
That would go infinite because Energy Tap also targets a creature, giving an extra Rosnakht trigger, and the extra creature token can be set to any value without having to match half of an existing value minus 1.
for mana echoes:
For an ordinal a, let |a| denote the integer that represents it; i.e. the number of +1/+1 counters a creature needs to be representing it.
Let F(a,X) denote the creature count (of which storm count is proportional) that can be created starting from X and a creature with |a| counters.
Note that |a|/2+2 mana is required to start processing that creature.
For successor ordinals: Start with a creature of power |b+1| - its binary representation ends in 01.
It ends up with power |b+1|//2+1 = |b| [where // is integer division] and creates a creaure of power |b|//2 which is even and thus unusable. It does however create 2X mana from mana echoes which is more than sufficient to keep the chain going.
Thus, F(b+1,X) = F(b,X+1)
For limit ordinals:
Let a's lowest power of w be k; so |a| ends in a 0 followed by k+1 1s.
The process transforms the starting creature into one with |b| counters in which b is obtained by replacing the w^k in a by sum[w^(k-1-i) * a_i] for some sequence a_0..a_(k-1) that we can choose, for which each a_i < X.
Let A = sum[a_i]; then |b| = O(2^A * |a|) - so we must choose a_i for which we will have enough mana to process |b| when we get there.
(minor complications when a = w^k exactly and the minimum damage basecase (|a| <= 5) comes into play)
The process also creates k+1 creatures, k of which correspond to ordinals c_0 ... c_k (from largest to smallest) much smaller than a (smaller highest power of w), and the k+1th is even and thus unusable. (|c_i| = |a|//2^(i+1)) This creates ~2(k+1)X mana, plenty start processing them.
Let Y = F(c_0, F(c_1, ...F(c_k,X+k+1)...)) - i.e. the creature count we would get after processing all the residual creatures.
Then, we choose the maximal coefficients a_i for which |b| <= Y - for example we could chose a_0 = log_2(Y) if that's <= X.
Once we reach the point where processing the c_0 ordinals becomes stronger than exponentiation of X, we'd have the freedom to choose their maximal possible values which are the storm count.
So, F(a, X) = F(b, Y)
This feels like a variation of the Middle-Growing heirarchy) which does asymptotically behave like the FGH at w^w.
|0| = 11 (binary) is the base case - not damagable.
|n| = 100..01 with n 0s, for finite ordinal n<w
F(n, X) = X+n for n<w
F(w,X)
|w| = 1011
create 101, soidarity triggers, rite trigger
create 10, solidarity triggers, rite trigger
original is 10 -> resolve rite, becomes 11; resolve solidarity triggers and rite, becomes 1100..001; with a 0s for our choice of a<X.
usable byproduct is 101 = |1| F(1,X) = X+1. thus choose a<log(X+1).
our original became 110001.
sequences starting with multiple leading 1s according the the description of the ordinal representation would represent the same ordinal as if all but one leading 1 were removed.
let's check how they behave:
as before, 11 is undamagable
111:
create 11, solidarity triggers, rite trigger
remainder undamagable. resulve rite becomes 100 is unusable.
so we created 1 creature, 111 behaves like |1| = 101.
110001: goes through succ case as usual 3 times to 111; then above case can do 1 more. thus increments X by 4.
so 11(0^n)1 as a binary string behaves the same as 1(0^n+1)1 = |n+1| and can be considered another representation of it
back to above, from |w| = 1011 we ended up with 11(0^a)1 ~= |a+1|
so F(w,X) ~= X+log X
check how 1111 behaves:
create 111, solidarity triggers, rites trigger
create 11, solidarity triggers, rites trigger - remainder undamagable.
resolve rites -> 100, resolve solidarity+rites -> 100(0^a)1 for some a = |a+2|. remainders are |1| and |0|; creating 3 creatures total. so choose a=log(X+2)
1111 behaves like w.
Hypothesis: A binary sequence that starts with multiple leading 1s behaves the same as the one obtained by replacing the second 1 with a 0.
try F(w*2, X):
|w2| = 10011
create 1001, solidarity triggers, rites trigger
create 100, solidarity triggers, rites trigger
original becomes 101 -> 101(0^a)1 = |w+a|
usable residual |1001| = 2
using it up creates 4 creatures. choose a<log(X+4)
F(w2,X) = F(w+log X, X+4) = F(w, X+logX+4) = O(X+log(X)+log(X+logX))
F(w*n,X) = O( the nth iteration of the function (X -> X+log(X))
call this gn
g1 > X + log(X)
g2 = g1 + log g1 > X + log X + log g1 > X + 2*logX
g3 = g2 + log g2 > X + 2logX + log g2 > X + 3logX
in general, gn > X + n*logX
F(w*n, X) >= X + n*log X
F(w^2,X):
|w^2| = 10111
create 1011 (=|w|), solidarity, rites
create 101 (=|1|), solidarity, rites
create 10, solidarity, rites
resolve: 11(0^a)1(0^b)1; for now choose b=0. we have a representation of w*(a+1) as per above analysis about sequences starting with multipe 1s.
our residuals are |1| and |w| which processing will take X to O(X+logX); choose a< log(X+log X)
F(w^2, X) > F(w*(log (X+logX)), X+logX) > (X+logX + log(X+logX)*log(X+logX)) > X + (log(X+logX))^2
F(w^2+w, X):
|w^2+w| = 101011
create 10101 (=|w+1|), solidarity, rites
create 1010, solidarity, rites
resolve: 1011(0^a)1 = |w^2 + a|
residuals |w+1| takes X to O(X+logX); choose a<log(X+logX)
F(w^2+w, X) ~= F(w^2 + log(X+logX), X+logX) = F(w^2, X+logX+log(X+logX)) = F(w^2, g2(X)) with the g function above
F(w^2+w2, X):
|w^2+w2| = 1010011
create 101001 (=|w+2|, solidarity, rites
create 10100, solidarity, rites
resolve: 10101(0^a)1 = |w^2 + w + a|
choose a < log (X+logX)
F(w^2 + w2, X) ~= F(w^2 + w + log(X+logX), X+logX) = F(w^2, g3(X))
F(w^2 + w*n, X) ~= F(w^2, gn(X)) > F(w^2, X+n*log(X)) > X+nlogX + [log(g1(X+nlogX))]^2 > X + nlogX + [log(g{n+1}X)]^2
F(w^2*2, X):
|w^2*2| = 100111
create 10011 = |w2|, solidarity, rites
create 1001 = |2|, solidarity, rites
create 100, solidarity, rites
resolve: 101(0^a)1(0^b)1 = |w^2+w*a+b|. assume b=0
residuals |2| and |w2| take X to > X+2logX; choose a < log(X+2logX)
F(w^2*2, X) ~= F(w^2 + w*log(X+2logX))
F(w^2*k + w*n, X) (k,n=>1)
|w^2*k + w*n|
residuals are |w*k+n| and |k|
-> F(w^2*k + w*(n-1) + log F(w*k+n, X+k), F(w*k+n, X+k)) = F(w^2*k + w*(n-1), g1(F(w*k, X+k+n))
=> F(w^2*k, Y)
where Y is the nth iteration of the function X -> g1(X+k*log(X))
analysing this seems tricky
feels like it ought escape logs at some point (and thereby catch up to the FGH) but not quite sure.
Though I think you might have the usability of the byproducts backwards. Let's slowly go through some examples:
processing a 1001(=|2|), we target a Innocent Bystander copy with 9 +1/+1 counters. That's 10 toughness. We need to deal 5 damage with Halana, Kessig Ranger. So Remnant of the Rising Star needs to put 5 +1/+1 counters on the Rosnakht, Heir of Rohgahh token to get it to 5 power. That is a usable 101 (=|1|) as the by product.
Meanwhile the Innocent Bystander went to 9-5=4 +1/+1 counters with 5 toughness to draw the Taii Wakeen, Perfect Shot card. Then it gets one counter back from Rite of Passage to end up with 5 +1/+1 counters at 101 as well.
processing a 1000111 (=w^2*3), we target a Innocent Bystander copy with 64+3+2+1=71 +1/+1 counters. That's 72 toughness, so on the first hit we need to deal 36 damage. With 36 +1/+1 counters the byproduct from that hit is not usable. The main target is at 36 toughness with the Hardened Scales trigger on the stack. Second hit for 18, another unusable byproduct. Main target at 18 toughness. 3rd hit at 9. This last byproduct is a usable 1001 (=|2|).
The main target got down to 8 +1/+1 counters (1000 in binary) and resolves the Hardened Scales and Solidarity of Heroes to go to 10010...010...01 (=|w^2*2 + w*X + X)
So there's only 1 usable byproduct from a limit ordinal. Hopefully that is still enough to generate the mana to make storm the relevant limit eventually.
Meanwhile the byproduct from a successor ordinal is usable, and actually is as powerful as the main target. And that might actually be important. Without that extra boost this whole thing looks a lot like a weak Goodstein sequence to me (from looking at it without mana considerations), which apparently only grows like f_{w}.
F(a+1,X) = F(a,F(a,X+1)),
and F(a,X) for limit a = F(b,(F(c,X)), where c is obtained by "dividing" the highest possible power of w and subtracting 1; and b is limited either by X or by log(F(c,X))
also leading 1s essentialy don't matter - processing 111 yields no usable byproducts. at most you could get a single extra step from an extra leading 1.
F(1,X) = X+1
F(2,X) = F(1,F(1,X+1)) = X+3
F(3,X) = X+7
F(n,X) = X+2^n-1
F(w,X)
proc 1011
create 101+1 = 110 (unusuable), solidarity, rite
create 10+1 = 11 = |0|, solidarity, rite
resolve: 11(0^a)1 = |a|. minimal extra value from residual so a=logX
F(w,X) = F(log X, X+1) = X+1+2^logX-1 = 2X
F(w+1,X) = 4X
F(w+2,X) = 16X
F(w+n,X) = 2^2^n * X
F(w2)
proc 10011
usable residual 101 = |1|; main becomes 101(0^a)1 = |w+a|
minimal residual value so take a=logX
F(w2) = F(w+logX,X) = 2^2^log X * X = 2^X * X > 2^X
F(w2+1) >= 2^2^X
F(w3) >= 2^^X
F(w4) >= 2^^^X
yeah, the extra power from the successor byproduct enables the scaling to counteract the logs immediately and by the time the limit byproducts are contributing enough to make back the full mana to be able to use all X of the storm rather than logX (which happens when the byproduct is w2; which is when the original ordinal is w^3+w) the difference is irrelevant.