Let’s start with the case of N undamaged tokens. Adding a token gives us N+1 undamaged tokens, which is the same case just choosing a larger N to start with. On the other hand, doing a tick of damage reduces us to N-1 toughness. Clearly, N ticks will kill all of them, zeroing the clock.
So let’s do less than N damage, call it M, and then create K tokens and see what the effect is. Well the old tokens have N+K-M toughness, and the new ones have N+K toughness.
Suppose we do the remaining N+K-M damage ticks to kill the original tokens, do the new tokens die as well? Well they would have N+K-(N+K-M) = M toughness left. However, all of the N original tokens are dying now, so they actually have M-N toughness after losing the coat of arms buffs. Since M<N, M-N<0 and the new tokens die as well.
This logic applies to any sequence of damage ticks and token creates.
Alternate proof:
We proceed by induction on N, the number of tokens and damage ticks.
base case: N=1 one tick kills all one of them
Inductive case: If N ticks kill all N of them, would N+1 ticks kill all N+1 of them?
Let’s make an extra one before the Nth tick, so we have N+1 of them after N ticks.
The N+1st tick kills the bottom token, reducing us to N of them. By the inductive hypothesis, this also kills the original N tokens, as the extra tick cancels out the coat of arms buff. Leaving us the new token with 1 damage on it and no friends, killing it as well.
If we make the N+1st token earlier, it just has more damage on it and still dies.
Now, there are some minor complications such as the buff from dralnu’s crusade, but that ends up being less and less of a factor as the numbers get bigger.
I also think we need a slightly better explanation of flooding + half-flooding clocks.
One more complication possibly worth mentioning is the weird stuff we can do with layers, like if we have crusades making all goblins zombies, all zombies crabs, and all crabs goblins, if we make a goblin a zombie and a crab, do we have three goblin crab zombies? No! It depends on timestamps. It can get even weirder than that, but no matter what it’s all handled deterministically by the rules without player intervention, so there’s nothing that can break computation.
Yeah I think that doesn't generate red that way, I was also having trouble getting it to explicitly make red while maintaining the stage structure. I might be wrong though.
As an alternate idea, what about swapping the order of the stages?
Yeah drake being more efficient than owl at drawing cards makes the start a lot easier.
It looks like we can get to full computations for the first profane because we should have access to the full volute stage well before then.
In terms of additional creatures for that, we only need necromancer, archon, krrik, pathmage, riftsweeper, chancellor. so 6. which along with the extra copy from the start, leaves us with just needing 5x path.
TYS and helix interact a bit strangely, but its to our benefit: the order we stack the TYS triggers in doesn't need to be the same order as we cast them off of helix, so we can get something like:
Edit:
Also looking at the 3 bonus stage deck, blood moon + gauntlet of might seems inefficient for making a bunch of red. does Katsumasa, the Animator work? It unfortunately blocks treasures for later so making initial colored mana becomes difficult...
edit2: hmm no looks like that goes infinite by just dropping both spite and worldfire from our combos.
The issue is getting the cephalid shrine in play after a worldfire in time to counter a spite that's saccing our archons.
start:
Yeah we can make a good chunk of sentry oaks, just doing the stupid suboptimal thing of recasting lingering souls for life to get volute trigger for fated makes something like a billion of them? Though doing no computation stage combos is actually a much better use so its more like a->b->c->7 or so. We unfortunately can't really loop in computation output without being able to do the worldfire transition to grab their inflated life total. and I think we need a bit more card draw to be able to resolve and rebuild worldfire without dead ending.
Yeah, I don't think we can get fully started with just three cards due to the difficulty of getting set up for the first worldfire transition. We can do a good amount of work but it gets dead ended by access to colored mana.
We can definitely use just 2 copies of II with this start. and possibly just one.
having the last three be something like:
panharmnonicon
owl -> tys+shrine
II x3 shrine counters original
draw 4:
fated
Path
HFM
WF
Is enough to get access to everything without using another II or going to combat or w/e.
epic experiment looks interesting though, and dig mox crypt casts it nicely without needing chancellor of the tangle. (though that card slot is used by the resurgent belief)
Actually, are other start options like show and tell still off limits?
I think show and tell messed with the wormfang behemoth version and could possibly be safe now that resolving it after a worldfire will truly do nothing?
Though its probably still not good for the land stages...
as for other start options Tibalt's trickery is clearly not allowed. Eureka seems worse than show and tell Chaos warp is also clearly a no Flash we just saw has problems
I don't think there's much else to consider?
We can I guess hardcast path of the schemer to reanimate a creature, but I don't think any one creature lets us omniscience.
I would want to mention that the setup with arcbond, coat of arms, bishops and crusades is sufficient to implement the Waterfall Model, which has been proven to be Turing Complete. (we actually implement a more complex variant thanks to flooding and partially flooding clocks, though those are not needed for TC-ness, they do make small computation run longer)
The full flooding variant is still likely not TC, though there's been no proof one way or the other. If it was we could cut Dralnu's Crusade and get down to a lucky 13 cards.
Gotta be careful to still let us cast Replenish in the start.
Speaking of, I've done some work on some lines with Path of the Schemer, and we don't need to waste more than one extra II in the setup anymore. I have a line that almost works wasting zero, but it is a bit short of being solid. Still that's 6 poison draws to power up before combat.
Also, remind me again of how grip of chaos and psychic battle interact? Grip of Chaos only triggers when the ability is initially put on the stack but each grip retriggers all the PBs?
Another thought is Sisay, weatherlight captain getting a tapped legendary land, though unfortunately it appears the only option that makes colored mana is somehow teferi's isle...
Also Yarok/Elesh Norn, MoM doubling up pious exactly once is fixed by switching to a double black flashback spell right? (moan of the unhallowed)
Let’s start with the case of N undamaged tokens. Adding a token gives us N+1 undamaged tokens, which is the same case just choosing a larger N to start with. On the other hand, doing a tick of damage reduces us to N-1 toughness. Clearly, N ticks will kill all of them, zeroing the clock.
So let’s do less than N damage, call it M, and then create K tokens and see what the effect is. Well the old tokens have N+K-M toughness, and the new ones have N+K toughness.
Suppose we do the remaining N+K-M damage ticks to kill the original tokens, do the new tokens die as well? Well they would have N+K-(N+K-M) = M toughness left. However, all of the N original tokens are dying now, so they actually have M-N toughness after losing the coat of arms buffs. Since M<N, M-N<0 and the new tokens die as well.
This logic applies to any sequence of damage ticks and token creates.
Alternate proof:
We proceed by induction on N, the number of tokens and damage ticks.
base case: N=1 one tick kills all one of them
Inductive case: If N ticks kill all N of them, would N+1 ticks kill all N+1 of them?
Let’s make an extra one before the Nth tick, so we have N+1 of them after N ticks.
The N+1st tick kills the bottom token, reducing us to N of them. By the inductive hypothesis, this also kills the original N tokens, as the extra tick cancels out the coat of arms buff. Leaving us the new token with 1 damage on it and no friends, killing it as well.
If we make the N+1st token earlier, it just has more damage on it and still dies.
Now, there are some minor complications such as the buff from dralnu’s crusade, but that ends up being less and less of a factor as the numbers get bigger.
I also think we need a slightly better explanation of flooding + half-flooding clocks.
One more complication possibly worth mentioning is the weird stuff we can do with layers, like if we have crusades making all goblins zombies, all zombies crabs, and all crabs goblins, if we make a goblin a zombie and a crab, do we have three goblin crab zombies? No! It depends on timestamps. It can get even weirder than that, but no matter what it’s all handled deterministically by the rules without player intervention, so there’s nothing that can break computation.
As an alternate idea, what about swapping the order of the stages?
-great furnace
-blood moon
-island
+tree of tales
+mountain
now green makes red via basics, and blue makes green via artifact land?
It looks like we can get to full computations for the first profane because we should have access to the full volute stage well before then.
In terms of additional creatures for that, we only need necromancer, archon, krrik, pathmage, riftsweeper, chancellor. so 6. which along with the extra copy from the start, leaves us with just needing 5x path.
TYS and helix interact a bit strangely, but its to our benefit: the order we stack the TYS triggers in doesn't need to be the same order as we cast them off of helix, so we can get something like:
tys path x5
tys resurgent x4
tys profane x6
helix profane
helix path
helix resurgent
for more profanes.
Edit:
Also looking at the 3 bonus stage deck, blood moon + gauntlet of might seems inefficient for making a bunch of red. does Katsumasa, the Animator work? It unfortunately blocks treasures for later so making initial colored mana becomes difficult...
edit2: hmm no looks like that goes infinite by just dropping both spite and worldfire from our combos.
The issue is getting the cephalid shrine in play after a worldfire in time to counter a spite that's saccing our archons.
start:
Yeah we can make a good chunk of sentry oaks, just doing the stupid suboptimal thing of recasting lingering souls for life to get volute trigger for fated makes something like a billion of them? Though doing no computation stage combos is actually a much better use so its more like a->b->c->7 or so. We unfortunately can't really loop in computation output without being able to do the worldfire transition to grab their inflated life total. and I think we need a bit more card draw to be able to resolve and rebuild worldfire without dead ending.
We can definitely use just 2 copies of II with this start. and possibly just one.
having the last three be something like:
panharmnonicon
owl -> tys+shrine
II x3 shrine counters original
draw 4:
fated
Path
HFM
WF
Is enough to get access to everything without using another II or going to combat or w/e.
epic experiment looks interesting though, and dig mox crypt casts it nicely without needing chancellor of the tangle. (though that card slot is used by the resurgent belief)
I'll look into lines from there.
And the problem with indestructability is we can bounce it to kill a damaged boom keg while we have necromancers.
I think karn also had some odd issue but I don't remember the specific interaction.
I think show and tell messed with the wormfang behemoth version and could possibly be safe now that resolving it after a worldfire will truly do nothing?
Though its probably still not good for the land stages...
as for other start options
Tibalt's trickery is clearly not allowed.
Eureka seems worse than show and tell
Chaos warp is also clearly a no
Flash we just saw has problems
I don't think there's much else to consider?
We can I guess hardcast path of the schemer to reanimate a creature, but I don't think any one creature lets us omniscience.
I would want to mention that the setup with arcbond, coat of arms, bishops and crusades is sufficient to implement the Waterfall Model, which has been proven to be Turing Complete. (we actually implement a more complex variant thanks to flooding and partially flooding clocks, though those are not needed for TC-ness, they do make small computation run longer)
The full flooding variant is still likely not TC, though there's been no proof one way or the other. If it was we could cut Dralnu's Crusade and get down to a lucky 13 cards.
Speaking of, I've done some work on some lines with Path of the Schemer, and we don't need to waste more than one extra II in the setup anymore. I have a line that almost works wasting zero, but it is a bit short of being solid. Still that's 6 poison draws to power up before combat.
Also, remind me again of how grip of chaos and psychic battle interact? Grip of Chaos only triggers when the ability is initially put on the stack but each grip retriggers all the PBs?
Edit: hmm that's a great point about shigeki
It looks like it lets us smuggle extra pathmage tokens down. causing the transition to go infinite.