I'm not sure if this is a disproof of the indestructible setup, but it might be troubling: If we have three arcbonded creatures of the same creature type, one flying and indestructible (call it creature A), and the other two not (B and C), we can start by alternating arcbond triggers between B and C. This will result in B and C taking 1 damage every 2 triggers, which is perhaps enough just to hold steady. Then, after any amount of time we can switch to something like ABAC cycling, which will deal 3 damage every 4 turns to B and C, and can perhaps kill them at the same time, ending the computation. Yeah, that looks like an infinite.
Setup:
- non-flying A, flying A
- non-flying B, C, and D, each with arcbond; as well as some flyers
- When non-flying A dies, make non-flying A and 2 each of flying B, C, and D
Each newly created non-flying A has 3 toughness. Every 3 arcbonds triggers A, and raises the toughnesses of each of the non-flying B,C, and D by 2. If arcbonds are evenly distributed among them, they survive; but at an arbitrary point we can choose to stop resolving B's arcbonds, making it die.
This is because this setup lacks the property that is true of the flooding model (as well as previously thought to be true of the dralnu model) that an arcbond creature still damages other members of its type, outpacing the damage on it.
Massacre Girl is interesting. It also seems to be somewhat of a flooding model, since if X creatures die we get X Massacre Girl triggers; also, the ending condition doesn't seem to be when a particular clock goes to 0, but rather when we run out of decrements and nothing hits 0. This seems to be another model that we would have to prove Turing-complete.
I think Massacre girl model (with access to dralnu's crusade) is equivalent to the standard waterfall model.
Resolving a massacre girl trigger is one decrement step. For all relevant creature types A, we can have "when A dies make Z" (or 2 Zs) for some type Z. Thus the number of Zs / their toughness will be proportional to the number of massacre girl triggers still on the stack. The terminating condition can be that a certain type that's keeping alive all the Necromancers (i.e. Wizard) dies; so from that point no new creatures are created. At that point, all the massacre girl triggers resolve, leaving only Zs, which act as the output. We can also use a few control clocks ("when A dies make A") to ensure that each step can create enough triggers to not terminate early.
Looking at that version the answer to your question is: quite a few things
March of the Machines kills 0 mana artifacts (and makes 1 mana ones into 1/1s)
Dual Nature can give us death triggers via legend rule Soul separator makes 1/1s
panharmonicon and harmonic prodigy copy allow massacre girl to get more etb triggers.
we can bounce and replay Massacre Girl with Distorting Wake and aegis automaton.
Hmmm... so, I guess we would have to only be able to put Massacre Girl on the battlefield once, and give it to the opponent, where it stays forever. Not being able to play a creature more than once is really limiting... looks like no hyperstage sweeper is possible?
Edit: I suppose a possible hyperstage sweeper could be "destroy all creatures and lands you control", but there doesn't seem to any spell like that.
Perhaps switch to a bunch of stages then.
Edit: Oh, the trigger condition for Massacre Girl means that destroying all of our creatures won't have to kill our Vedalken Orrery. Is there something that destroys all of our creatures and artifacts? (or creatures and enchantments, or all nonland permanents)
If not, maybe we could make do with some of the creatures that partially give flash...
Yeah, we don't need the goblin boom keg damage anymore.
it seems difficult to ensure we lose flash...
On flooding waterfall model ais523 said: "...I still don't have much of an idea of whether it's TC or not – it seems like a hard language to do a TCness proof or disproof for"
It's not a no, but seems very complicated to prove.
We could also have up to 2 normal clocks if we can't copy enchantments, which makes it more likely to be TC, but more complicated.
Didn't see much to wipe out Vedalken Orrery as well... there's Kaervek's Spite, but as an instant we won't have the ability to recycle it normally.
Instead, perhaps we go with Hypersonic Dragon, Prophet of Kuphrix, and Raff Capashen to give sorceries, creatures, and historic spells flash, leaving just most enchantments (and planeswalkers, but we don't usually use those).
In terms of what resources to build, we don't have to option of normal creature copying... perhaps build up artifacts, via Mirrorworks and artifact bouncing?
Then there is the question of what stages to build... Mimic Vat involves sending creatures to the graveyard, so maybe that doesn't work. The alternative seems to be Soul Foundry, which ideally goes with Brudiclad, Telchor Engineer, but Brudiclad has a troublesome ability at the start of combat. Perhaps then it's back to the Brudiclad-less Soul Foundry decks, from early Iijil times?
Those involved bouncing specific creature types though, which Artificial Evolution can mess with... we only get one Artificial Evolution, presumably copied, but then we can use the copies to both set up an infinite computation, and also turn Massacre Girl into something that we can bounce, so that we can replay her and start the computation with our haste creatures still on the battlefield. (which can presumably be kept alive throughout the computation)
So... we can use Soul Foundry, or mana stages, but we can't use specific creature types to narrow our stage, is what I'm gathering at the moment.
Also, can't Soul foundry be used to create multiple massacre girls?
Edit: An alternative plan in decks that can only cast two arcbonds while still allowing multiple computations is to have the terminating condition be that a permanent with the text "damage can't be prevented" to die, after having cast something that prevents all damage, allowing the two arcbond creatures to survive for another computation. The only suitable permanents with that text however are enchantments (everlasting torment and leyline of punishment), so need to be given a creature type to survive. And being able to give creature types to things like goblin boom keg might be an issue.
Edit: Another idea involving mixing Bishop of Wings and Xathrid Necromancer is with Sanctifier en-Vec. Per waterclock group, there can be X black creatures and one nonblack creature, and we will only get death triggers from the nonblack one. This allows for the same implementation as the Sephara, Sky's Blade plan, but the fact that the black creatures still go away resolves the two main issues:
- Refreshing a waterclock always sets its value to a consistent number; meaning it's definitely the same computational model and thus is TC
- Damage is still spread across the black creatures in a way that outpaces any missed damage from arcbond; so the original proof of multiple arcbonds being safe still works
I don't think Soul Foundry can create multiple Massacre Girls, unless we can either untap the Soul Foundry, or be able to bounce the Massacre Girl.
I originally thought that Spellweaver Volute couldn't be used for stage level purposes, because we needed it to restart the computation - but perhaps we can have a normal primary stage, and Spellweaver Volute could help form a hyperstage, and we would simply perform the computation at the hyperstage transitions... a tantalizing possibility.
Edit: Ooh, nice! So does anything go wrong with just subbing that new combo into the most recent deck? i.e.:
The alternative I'm thinking of is the creature stage we had before, which is something that is worth revisiting. I think we seemed to have it working, but it wasn't as good as the life stage we most recently have used. With life gone now, it seems we are stuck with creatures/mana for stage/hyperstage. (it would be nice to replace creatures with artifact, but we'll see how things go)
Skirge Familiar seems like it could be good for mana, but if we moot mana at the stage level and life is already gone, there doesn't seem to be anywhere to go for the hyperstage.
Sleight of mind being an instant can mess with a flooding computation, but Trait doctoring looks like it works just as well. (we don't have any basic land types that it messes with)
Looking through the avialible cards, there does not seem to be anything that works like Sanctifier en-Vec, as it deals with tokens of a specific color. Most leyline of the void effects only care about cards.
I pretty sure the Sanctifier en-Vec plan goes infinite with 3 Arcbonds as well. While we can arrange things so that every creature type always has exactly 1 white representative, we aren't forced to to that. So the plan to go infinite is something like this:
Set up a normal waterclock A that empties every 3 Arcbonds. In addition to refilling itself it also produces 2 white B and 2 white C.
B and C are self refilling: Whenever a B dies we create a white B and whenever a C dies we create a white C. So thanks to the waterclock A the number of tokens of type B and C is always 2/3 of the resolved arcbond triggers. Occasionally the damage marked on a B or C token catches up with their number, causing them all to die and be replaced by fresh tokens.
We can arrange starting conditions so that we stagger the refill of B and C so that all C's die and get replaced one arcbond after all B's die and get replaced. That way there is always one of the types on the battlefield.
The Arcbonded creatures can be type B and C (all the Xathrid Necromancer esque cards have to creature types, so we can use them for this). That way they stay alive as long as we choose to resolve arcbond triggers for all of them equally often, since at any time they get boosted by B or C.
At an arbitrary point we can choose to unbalance the arcbonds. The next time B or C gets refilled the remaining tokens are barely not enough to keep the more damaged arcbond creature alive, causing all of those to die in a chainreaction.
Setting up the starting conditions to ensure the staggered refills and toughness on arcbonded creatures line up properly seems tedious but very possible.
For Massacre Girl as an alternative to Arcbond: I don't understand how we ensure the correct order of triggers? Massacre Girl and Xathrid Necromancer both trigger at the same time, so we'd have to guarantee that one of those can only belong to us and the other can only belong to the opponent. I don't see how we could ever do that. But if they both belong to the same player, that means we can change the behavior of the program at arbitrary points via trigger ordering.
I think the best hope for salvaging repeated Busy Beaver calculations is the approach using 2 Arcbonds + alternate halting condition. We should be able to do that using Leonin Relic-Warder. Once it dies we return a Defang to the battlefield and put it on one of the Arcbond creatures to stop the triggers.
To get ready for the next calculation we then "just" have to exile Defang under a fresh relic-warder and donate that to the opponent.
We can choose not to halt if we put the Defang somewhere else, but we can't delay that choice to an arbitrary point in the calculation, so that shouldn't be a problem. We only need to make sure that we can't use the Relic Warder to sneak artifacts or enchantments into the calculation that would allow meaningful choices. I think that mostly means no Vedalken Orrery.
Just to make sure I am following
You are suggesting the following setup:
when a white A dies we have 2 necromancers making black As, 1 making a white A and, 2 making white Bs and white Cs
When a white B dies we remake it as a white B
when a white C dies we remake it as a white C
Starting with a white A, n white Bs and n+1 white Cs
With the arcbonds on three BCs (all of the Xathrid Necromancer variants have two creature types, so we have options)
So every three arcbond triggers we empty A, while both B and C are decremented by one and the three BCs increase by one.
eventually B zeroes and we flood B up to some bigger number with fresh tokens. when B zeroes, the BCs are back to their original value
the tick after, C zeroes and gets flooded with fresh tokens as well. the BCs are still fine here.
But what if A and B zero at the same time? Which is guaranteed to happen quickly in this setup regardless of n. Then C gets desynced from B and the analysis is not clear.
Due to parity B and C never zero at the same time, so it won't halt automatically.
(edit: we can always kill one of the BCs by alternating arcbonds between the other two of them, but then they have half the damage that other creatures would and not die with the third BC, and then the program never halts. So I guess the question is if B and C zero one apart an infinite number of times, which seems likely but not obvious.)
Damn; if that's correct then I think that even nukes flooding waterclocks with multiple arcbonds; so proving that they're TC wouldn't help. The main issue being that creatures with multiple types can be kept alive in an unintended way.
I guess now the only hope for multiple arcbond decks now is if at some point they print a xathid necromancer variant that only has one creature type? and then the entire deck would also be restricted to creatures with only one type?
The reason that Massacre girl hopefully works is that her death trigger is a delayed triggered ability created by the ETB. Provided that it's not possible to have massacre girl enter under the opponent's control, that trigger should only ever belong to us. And then the whole setup ensures that necromancer death triggers can only belong to them.
Using something like Leonin relic warder is tricky; as even without vedalken orrery it could be used to sneak a necromancer onto our side. (edit: actually, leonin only hits artifacts and enchantments)
Edit: so to make it clear:
at every step the toughness-damage of the BCs is (number of Bs + number of Cs - number of pings + number of missed arcbonds) + constant. Number of Bs is 2/3*number of pings+starting number, as is number of Cs; except when they zero.
Number of missed arcbonds can be up to 1/3*number of pings, but can arbitrarily be frozen.
So normally total BC toughness is usually (4/3-1+1/3)*number of pings + constant = 2/3*number of pings + constant, and when the Bs or Cs zero it temporarily dips down by 2/3*number of pings, becomeing constant; so survives. Then when we freeze number of missed arcbonds this dips down to 0; killing one BC.
If it terminates after this it's infinite. (but it doesn't look like it does? as now number of missed arcbonds tends to 1/2*number of pings nd can no longer be varied)
Does B and A zereoing at the same time matter? regardless of ordering you're going to get the same number of creatures in the end, right?
hmm, yes A zeroing at the same time as B or C desyncs them by 2 arcbounds. But the parity argument should be correct and they never zero at the same time. It does not effect the number of creatures.
to actually halt we can either set the start up so that the constant part of (toughness - damage) is 1 for the arcbond creatures. then a single unbalanced arcbond kills them all in a chain reaction from Coat of Arms. Alternatively we can use arcbond triggers in a 1 2 1 3 pattern, so that both 2 and 3 take more than 2/3 of the damage. (Wait until the refills of B and C are far enough apart, so you can follow that pattern long enough to overcome the constant.) Only 1 will survive the next zeroing of B and arcbond on that one creature cannot trigger itself.
Ba nd A zeroing at the same time matters because in the next step C won't zero. When C is desynched, the BCs have higher toughness when the Bs die, and so would not all die.
They would only die if the Cs are also about to trigger. When B and C synch up is the only time where we have the possibility of killing one BC to kill the rest of them as they get 1 smaller from their friend dying.
edit: ah the 1213 pattern kills my argument... this looks like RIP to flooding clocks too.
So the Leonin Relic-Warder idea would eliminate Vedalken Orrery, and also Soul Separator, which wound up being very useful to the deck. Would going with the Leyline of Punishment idea work better? We wouldn't be able to prevent damage to the players, so perhaps lifelink for the opponent, and an Angel's Grace for ourselves, just mooting life as resource I guess.
We are still limited to not recasting any instants, so no Spellweaver Volute.
Does Archon of falling stars work when our opponent controls it? I don't think we are able to control it ourselves without also being able to control a Xathrid Necromancer.
Leyline of Punishment also prevents lifegain so to keep the game going we need some "can't lose" effect.
Massacre Girl setting up a delayed trigger is nifty. I completely missed that aspect of the effect. That takes care of the trigger order.
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Setup:
- non-flying A, flying A
- non-flying B, C, and D, each with arcbond; as well as some flyers
- When non-flying A dies, make non-flying A and 2 each of flying B, C, and D
Each newly created non-flying A has 3 toughness. Every 3 arcbonds triggers A, and raises the toughnesses of each of the non-flying B,C, and D by 2. If arcbonds are evenly distributed among them, they survive; but at an arbitrary point we can choose to stop resolving B's arcbonds, making it die.
This is because this setup lacks the property that is true of the flooding model (as well as previously thought to be true of the dralnu model) that an arcbond creature still damages other members of its type, outpacing the damage on it.
Edit: yeah you found a similar issue.
Resolving a massacre girl trigger is one decrement step. For all relevant creature types A, we can have "when A dies make Z" (or 2 Zs) for some type Z. Thus the number of Zs / their toughness will be proportional to the number of massacre girl triggers still on the stack. The terminating condition can be that a certain type that's keeping alive all the Necromancers (i.e. Wizard) dies; so from that point no new creatures are created. At that point, all the massacre girl triggers resolve, leaving only Zs, which act as the output. We can also use a few control clocks ("when A dies make A") to ensure that each step can create enough triggers to not terminate early.
So, just to get the feel for things, what goes wrong if we just substitute Massacre Girl for Arcbond?
So: (making a couple changes to remove 1 toughness creatures):
2 Xathrid Necromancer
3 Massacre Girl
4 Artificial Evolution
5 Wrong Turn
6 Dralnu's Crusade
7 Opalescence
8 Desolation Giant
9 Goblin Boom Keg
10 March of the Machines
11 Vedalken Orrery
12 Spellweaver Volute
13 Pull from Eternity
14 Copy Enchantment
15 Dual Nature
16 Bloodbond March
17 Cephalid Shrine
18 Thousand-Year Storm
19 Mana Echoes
21 Urborg Syphon-Mage
22 Thousand-Year Elixir
23 K'rrik, Son of Yawgmoth
24 Traitor's Clutch
25 Stern proctor
26 Safe Passage
27 Spellweaver Helix
28 Worldfire
29 Gravitic Punch
30 Gravitic Punch
31 Recoup
32 Restore
33 Cragcrown Pathway
34 Harmonic Prodigy
35 Panharmonicon
36 Wormfang Behemoth
37 Distorting Wake
38 Chrome Mox
40 Guardian of the Guildpact
41 Kurkesh, Onakke Ancient
42 Alhammarret's Archive
43 Black Lotus
44 Show and Tell
45 Omniscience
46 Floating-Dream Zubera
47 Break Through the Line
48 Crucible of the Spirit Dragon
49 Amulet of Vigor
50 Animist's Awakening
51 Arbiter of the Ideal
52 Tolarian Kraken
53 Blue Sun's Zenith
54 Search the Premises
55 Moraug, Fury of Akoum
56 World at War
57 Doubling Season
58 Drafna's Restoration
59 Uproot
Oh, I guess we have ways to copy Massacre Girl. So first, no Dual Nature.
Looking at that version the answer to your question is: quite a few things
March of the Machines kills 0 mana artifacts (and makes 1 mana ones into 1/1s)
Dual Nature can give us death triggers via legend rule
Soul separator makes 1/1s
panharmonicon and harmonic prodigy copy allow massacre girl to get more etb triggers.
we can bounce and replay Massacre Girl with Distorting Wake and aegis automaton.
Edit: I suppose a possible hyperstage sweeper could be "destroy all creatures and lands you control", but there doesn't seem to any spell like that.
Perhaps switch to a bunch of stages then.
Edit: Oh, the trigger condition for Massacre Girl means that destroying all of our creatures won't have to kill our Vedalken Orrery. Is there something that destroys all of our creatures and artifacts? (or creatures and enchantments, or all nonland permanents)
If not, maybe we could make do with some of the creatures that partially give flash...
it seems difficult to ensure we lose flash...
On flooding waterfall model ais523 said: "...I still don't have much of an idea of whether it's TC or not – it seems like a hard language to do a TCness proof or disproof for"
It's not a no, but seems very complicated to prove.
We could also have up to 2 normal clocks if we can't copy enchantments, which makes it more likely to be TC, but more complicated.
Instead, perhaps we go with Hypersonic Dragon, Prophet of Kuphrix, and Raff Capashen to give sorceries, creatures, and historic spells flash, leaving just most enchantments (and planeswalkers, but we don't usually use those).
In terms of what resources to build, we don't have to option of normal creature copying... perhaps build up artifacts, via Mirrorworks and artifact bouncing?
Then there is the question of what stages to build... Mimic Vat involves sending creatures to the graveyard, so maybe that doesn't work. The alternative seems to be Soul Foundry, which ideally goes with Brudiclad, Telchor Engineer, but Brudiclad has a troublesome ability at the start of combat. Perhaps then it's back to the Brudiclad-less Soul Foundry decks, from early Iijil times?
Those involved bouncing specific creature types though, which Artificial Evolution can mess with... we only get one Artificial Evolution, presumably copied, but then we can use the copies to both set up an infinite computation, and also turn Massacre Girl into something that we can bounce, so that we can replay her and start the computation with our haste creatures still on the battlefield. (which can presumably be kept alive throughout the computation)
So... we can use Soul Foundry, or mana stages, but we can't use specific creature types to narrow our stage, is what I'm gathering at the moment.
Also, can't Soul foundry be used to create multiple massacre girls?
Edit: An alternative plan in decks that can only cast two arcbonds while still allowing multiple computations is to have the terminating condition be that a permanent with the text "damage can't be prevented" to die, after having cast something that prevents all damage, allowing the two arcbond creatures to survive for another computation. The only suitable permanents with that text however are enchantments (everlasting torment and leyline of punishment), so need to be given a creature type to survive. And being able to give creature types to things like goblin boom keg might be an issue.
Edit: Another idea involving mixing Bishop of Wings and Xathrid Necromancer is with Sanctifier en-Vec. Per waterclock group, there can be X black creatures and one nonblack creature, and we will only get death triggers from the nonblack one. This allows for the same implementation as the Sephara, Sky's Blade plan, but the fact that the black creatures still go away resolves the two main issues:
- Refreshing a waterclock always sets its value to a consistent number; meaning it's definitely the same computational model and thus is TC
- Damage is still spread across the black creatures in a way that outpaces any missed damage from arcbond; so the original proof of multiple arcbonds being safe still works
I originally thought that Spellweaver Volute couldn't be used for stage level purposes, because we needed it to restart the computation - but perhaps we can have a normal primary stage, and Spellweaver Volute could help form a hyperstage, and we would simply perform the computation at the hyperstage transitions... a tantalizing possibility.
Edit: Ooh, nice! So does anything go wrong with just subbing that new combo into the most recent deck? i.e.:
2 Xathrid Necromancer
3 Bishop of Wings
4 Sanctifier en-Vec
5 Arcbond
6 Artificial Evolution
7 Wrong Turn
8 Opalescence
9 Desolation Giant
10 Goblin Boom Keg
11 March of the Machines
12 Vedalken Orrery
13 Spellweaver Volute
14 Pull from Eternity
15 Copy Enchantment
17 Bloodbond March
18 Cephalid Shrine
19 Thousand-Year Storm
20 Mana Echoes
21 Soul Separator
22 Urborg Syphon-Mage
23 Thousand-Year Elixir
24 K'rrik, Son of Yawgmoth
25 Lingering Souls
26 Stern Proctor
27 Safe Passage
28 Spellweaver Helix
29 Worldfire
30 Gravitic Punch
31 Gravitic Punch
32 Recoup
33 Restore
34 Cragcrown Pathway
35 Harmonic Prodigy
36 Panharmonicon
37 Wormfang Behemoth
39 Chrome Mox
Did we have any alternatives for that stage?
Edit: I wonder if there's any way to use Requiem angel instead?
Edit: can Skirge familiar replace Urborg syphon-mage? looks like it doesn't quite work...
Skirge Familiar seems like it could be good for mana, but if we moot mana at the stage level and life is already gone, there doesn't seem to be anywhere to go for the hyperstage.
So we have 1 necromancer making white creatures, and the rest still on black
I think this still implements proper waterclocks.
So, how about:
2 Xathrid Necromancer
3 Sleight of Mind
4 Sanctifier en-Vec
5 Arcbond
6 Artificial Evolution
7 Wrong Turn
8 Opalescence
9 Desolation Giant
10 Goblin Boom Keg
11 March of the Machines
12 Vedalken Orrery
13 Spellweaver Volute
14 Pull from Eternity
15 Copy Enchantment
16 Dual Nature
17 Bloodbond March
18 Cephalid Shrine
19 Thousand-Year Storm
20 Mana Echoes
22 Urborg Syphon-Mage
23 Thousand-Year Elixir
24 K'rrik, Son of Yawgmoth
25 Lingering Souls
26 Stern Proctor
27 Safe Passage
28 Spellweaver Helix
29 Worldfire
30 Gravitic Punch
31 Gravitic Punch
32 Restore
33 Cragcrown Pathway
34 Harmonic Prodigy
35 Panharmonicon
36 Wormfang Behemoth
37 Aegis Automaton
38 Kurkesh, Onakke Ancient
39 Alhammarret's Archive
41 Show and Tell
42 Omniscience
43 Break Through the Line
44 Crucible of the Spirit Dragon
45 Amulet of Vigor
46 Animist's Awakening
47 Arbiter of the Ideal
48 Tolarian Kraken
49 Blue Sun's Zenith
50 Hama Pashar, Ruin Seeker
51 Thorough Investigation
52 Moraug, Fury of Akoum
53 World at War
54 Doubling Season
55 Drafna's Restoration
56 Temporal Spring
57 Floating-Dream Zubera
We could get the second additional stage back, if we found a nonwhite alternative for Sanctifier en-Vec
Set up a normal waterclock A that empties every 3 Arcbonds. In addition to refilling itself it also produces 2 white B and 2 white C.
B and C are self refilling: Whenever a B dies we create a white B and whenever a C dies we create a white C. So thanks to the waterclock A the number of tokens of type B and C is always 2/3 of the resolved arcbond triggers. Occasionally the damage marked on a B or C token catches up with their number, causing them all to die and be replaced by fresh tokens.
We can arrange starting conditions so that we stagger the refill of B and C so that all C's die and get replaced one arcbond after all B's die and get replaced. That way there is always one of the types on the battlefield.
The Arcbonded creatures can be type B and C (all the Xathrid Necromancer esque cards have to creature types, so we can use them for this). That way they stay alive as long as we choose to resolve arcbond triggers for all of them equally often, since at any time they get boosted by B or C.
At an arbitrary point we can choose to unbalance the arcbonds. The next time B or C gets refilled the remaining tokens are barely not enough to keep the more damaged arcbond creature alive, causing all of those to die in a chainreaction.
Setting up the starting conditions to ensure the staggered refills and toughness on arcbonded creatures line up properly seems tedious but very possible.
For Massacre Girl as an alternative to Arcbond: I don't understand how we ensure the correct order of triggers? Massacre Girl and Xathrid Necromancer both trigger at the same time, so we'd have to guarantee that one of those can only belong to us and the other can only belong to the opponent. I don't see how we could ever do that. But if they both belong to the same player, that means we can change the behavior of the program at arbitrary points via trigger ordering.
I think the best hope for salvaging repeated Busy Beaver calculations is the approach using 2 Arcbonds + alternate halting condition. We should be able to do that using Leonin Relic-Warder. Once it dies we return a Defang to the battlefield and put it on one of the Arcbond creatures to stop the triggers.
To get ready for the next calculation we then "just" have to exile Defang under a fresh relic-warder and donate that to the opponent.
We can choose not to halt if we put the Defang somewhere else, but we can't delay that choice to an arbitrary point in the calculation, so that shouldn't be a problem. We only need to make sure that we can't use the Relic Warder to sneak artifacts or enchantments into the calculation that would allow meaningful choices. I think that mostly means no Vedalken Orrery.
You are suggesting the following setup:
when a white A dies we have 2 necromancers making black As, 1 making a white A and, 2 making white Bs and white Cs
When a white B dies we remake it as a white B
when a white C dies we remake it as a white C
Starting with a white A, n white Bs and n+1 white Cs
With the arcbonds on three BCs (all of the Xathrid Necromancer variants have two creature types, so we have options)
So every three arcbond triggers we empty A, while both B and C are decremented by one and the three BCs increase by one.
eventually B zeroes and we flood B up to some bigger number with fresh tokens. when B zeroes, the BCs are back to their original value
the tick after, C zeroes and gets flooded with fresh tokens as well. the BCs are still fine here.
But what if A and B zero at the same time? Which is guaranteed to happen quickly in this setup regardless of n. Then C gets desynced from B and the analysis is not clear.
Due to parity B and C never zero at the same time, so it won't halt automatically.
(edit: we can always kill one of the BCs by alternating arcbonds between the other two of them, but then they have half the damage that other creatures would and not die with the third BC, and then the program never halts. So I guess the question is if B and C zero one apart an infinite number of times, which seems likely but not obvious.)
I guess now the only hope for multiple arcbond decks now is if at some point they print a xathid necromancer variant that only has one creature type? and then the entire deck would also be restricted to creatures with only one type?
The reason that Massacre girl hopefully works is that her death trigger is a delayed triggered ability created by the ETB. Provided that it's not possible to have massacre girl enter under the opponent's control, that trigger should only ever belong to us. And then the whole setup ensures that necromancer death triggers can only belong to them.
Using something like Leonin relic warder is tricky; as even without vedalken orrery it could be used to sneak a necromancer onto our side. (edit: actually, leonin only hits artifacts and enchantments)
Edit: so to make it clear:
at every step the toughness-damage of the BCs is (number of Bs + number of Cs - number of pings + number of missed arcbonds) + constant. Number of Bs is 2/3*number of pings+starting number, as is number of Cs; except when they zero.
Number of missed arcbonds can be up to 1/3*number of pings, but can arbitrarily be frozen.
So normally total BC toughness is usually (4/3-1+1/3)*number of pings + constant = 2/3*number of pings + constant, and when the Bs or Cs zero it temporarily dips down by 2/3*number of pings, becomeing constant; so survives. Then when we freeze number of missed arcbonds this dips down to 0; killing one BC.
If it terminates after this it's infinite. (but it doesn't look like it does? as now number of missed arcbonds tends to 1/2*number of pings nd can no longer be varied)
Does B and A zereoing at the same time matter? regardless of ordering you're going to get the same number of creatures in the end, right?
to actually halt we can either set the start up so that the constant part of (toughness - damage) is 1 for the arcbond creatures. then a single unbalanced arcbond kills them all in a chain reaction from Coat of Arms. Alternatively we can use arcbond triggers in a 1 2 1 3 pattern, so that both 2 and 3 take more than 2/3 of the damage. (Wait until the refills of B and C are far enough apart, so you can follow that pattern long enough to overcome the constant.) Only 1 will survive the next zeroing of B and arcbond on that one creature cannot trigger itself.
They would only die if the Cs are also about to trigger. When B and C synch up is the only time where we have the possibility of killing one BC to kill the rest of them as they get 1 smaller from their friend dying.
edit: ah the 1213 pattern kills my argument... this looks like RIP to flooding clocks too.
So the Leonin Relic-Warder idea would eliminate Vedalken Orrery, and also Soul Separator, which wound up being very useful to the deck. Would going with the Leyline of Punishment idea work better? We wouldn't be able to prevent damage to the players, so perhaps lifelink for the opponent, and an Angel's Grace for ourselves, just mooting life as resource I guess.
We are still limited to not recasting any instants, so no Spellweaver Volute.
Edit: Hmm, with no access to Arcbond, how do we set up our computation again? I guess with Izzet Guildmage or Isochron Scepter for Artificial Evolution...
Edit: Oh Izzet Guildmage is bad. I guess Isochron Scepter then?
Aura thief also seems to have potential
Leyline of Punishment also prevents lifegain so to keep the game going we need some "can't lose" effect.
Massacre Girl setting up a delayed trigger is nifty. I completely missed that aspect of the effect. That takes care of the trigger order.