I suppose another possibility is to be able to get by with one Mirrorweave cast. That means we need to get rid of Opalescence and put Bishop of Wings onto the battlefield, with the stack not empty. So, Disenchant to get rid of the Opalescence, and Quicksilver Amulet to put Bishop of Wings onto the battlefield. Would that work?
Edit: I don't think we can use a second Mirrorweave to turn everything into a random token, since the Arcbonded creatures will still be doing damage, and it will eventually kill the whole group. But, I guess we can turn them all into a Bishop of Wings that has been hacked so that when it dies it makes a token; then when everything dies, we'll get a fresh group of tokens with no Arcbond triggers anymore.
I believe I have an improvement - we can double dip on the Mountains, by putting them into play tapped with Rampant Growth, then untapping them many times with Amulet of Vigor. Then we can bounce them and exile them to Azor's Gateway.
So, in addition to the four flashback spells, I think that we can exile Channel, Forest, Primal Surge, Coat of Arms, Dowsing Dagger, Selesnya Signet, Galvanic Key, and the four Great Furnaces and 20 Mountains, allowing us to transform Azor's Gateway 31 times. The 24 red mana lands take us to more than 10^^^^2, so 31 more batches of red mana takes us to more than 10^^^^33.
Hmm, nice. I would still like to resolve the Turing-completeness of the variant model, since it's looking like we can get considerably more with that. But, it's good to have an alternative for now.
Edit: So, I think I can still get 4 layers, but I'd like to get 5. To do that, I need to figure out a way to use life to put a basic land onto the battlefield tapped. I guess the basic land can come from anywhere.
Unfortunately, I didn't figure out a way to get turn life payment into recycling the basic land, so this is four layers. With Pull from Eternity, we can bring exiled cards back to our graveyard, which we can fetch to our hand if they are sorceries or legendary permanents, but not otherwise. Nicely enough, we can bring back 4 different CMC's of cards. We can also exile and bring back Sage's Knowledge once, but not more than once, since after the second exile we will have no more copies of Sage's Knowledge in our hand, and we can no longer retrieve sorceries from the graveyard anymore. So we can exile one Sage's Knowledge, and each of the 27 extra cards, to transform Azor's Gateway into Sanctum of the Sun 28 times, which will get us to more than 10^^^^29.
Hmmm, so if we wanted a pure Waterfall Model, we would be limited to 12 waterclocks. (plus I suppose a halting clock and an output clock) That just seems too few - maybe it is universal, but imagine trying to emulate a 32-instruction universal Minsky machine with that.
I think we would have better luck with 240+ waterclocks using the variant model. We could make up to 12 of them have special creatures, but wouldn't that just complicate matters?
Yeah, I think it's Turing complete too. If it is needed, we'll just have to dig into the proof for the Waterfall Model and make modifications, I guess.
In the scenario where we have Bishops, the computation on each side is independent. (Well, we have the Arcbond triggers... but, we can still consider what happens to our side without worrying about the other side, we just have to allow for the Arcbond triggers to be from an opponent's creature.) The computation continues until both Arcbonded creatures are killed. If both Arcbonded creatures belong to the opponent, we don't have to care about what happens on our side. If one or both are on our side, then we do care. But, if we only have two Bishops, when we sacrifice one we get at most two tokens. If they are the same type as the Bishop, soon all three will die, and the computation halts. Or if the dying Bishop created a different type, then the first Bishop could create another type when that type dies - but that leads to a dead end as well. So two Bishops doesn't go anywhere.
We could probably check the possibilities for a haltable infinite with more Bishops. Looking at our current enchantments:
Dralnu's Crusade: this is the one that we're worrying about. Swarm Intelligence: This is nice for our sorcery recursion, but we could do without it if we need to. Runed Halo for us: This might be necessary, if we transform this, we have make sure that we can stay alive for arbtrarily long. Runed Halo for the opponent: Not actually necessary. Opalescence: This won't be a creature. Fastbond: This is nice, but we can also do without this if necessary.
So, if we get rid of Swarm Intelligence and Fastbond, we can convert two enchantments into Bishops. We would be without Runed Halo, so we would have to stay alive. I guess we can check this for infinite possibilities.
Edit: Oh whoops, I missed that Coat of Arms counts all creatures.
Oh good, a Bishop on our side won't trigger. That's good news then.
I do think we can scale back down? We do get a scaling factor, but that scaling factor also increases the number of decrements we need to get back down to 0. So in a loose sense, the scaling increases the time it takes for things to go back down exponentially, but eventually they do come back down.
Hmm... in that case, I guess we need to consider the computational model when we have no Dralnu's Crusades. I guess this is similar to what we have now, except that the incremental bumps will trigger for every creature that dies, and not just on the "special creature". So when N creatures die, each increment gets multiplied by N. What is the computational power of this?
If this is Turing-complete, then we can just do the combo with just one copy of Mirrorweave, and no Opalescence, I think. If it turns out to either die early or increase without bound, then perhaps we can stick with the current paradigm. If it's not Turing complete, but still allows haltable infinites, maybe we're screwed?
So, maybe we have to dig into the proof that the Waterfall Model is Turing-complete, and see whether or not it can be modified to fit the model above.
So, without any red mana, Dralnu's Crusade can be put into play by anything that does that, like Primal Surge. But, how do we cast Arcbond without any red mana?
Goblin Dark-Dwellers would give us an extra Goblin to sacrifice. Chandra Ablaze looks good, but it's first ability allows it to deal damage to a target. Maybe Sunforger? Then we could pair it with Fireblast, provided we had access to exactly one red mana.
Well, if we had just one red mana, we could cast Arcbond anyway. So maybe that's the plan.
Oh wait, but Skirk Volcanist allows us to have red mana - that sounds better.
Edit: So, it looks like we can switch to Skirk Volcanist with the rest of the combo largely unchanged. But, I am looking for an efficient way to have multiple Sanctum of the Suns. Pull from Eternity brings exiled cards to our graveyard, and at the moment we can only bring back legendary permanents and sorceries back from the graveyard. Unfortunately, that's enough to get five different CMC's already - 0,2,5,7,10. The 5,7,10 belong to Stunted Growth, Primevals' Glorious Rebirth, and Primal Surge. Stunted Growth and Primal Surge both seem replaceable, although confining ourselves to those CMC's might be difficult. What are alternate ways of putting a basic land from our graveyard or hand into our library, without being able to put Arcbond into our library?
Even though our computational power is extremely restricted, we can still set up infinite behavoir. For example, if we set up the death of Angels to create N more Angels, and at least N more Beasts, and have enough Beasts in the beginning to not die the first time around, then the Beasts will (ordinarily) live forever. However, the ordering of the triggers affects this somewhat, since whatever creature gets triggered does not take damage. If there are X Angels in total, then the resolution of X triggers can be divided among all the Angels, reducing the maximum damage taken by 1.
Hmmm... But if we create N Angels after death, it will take about N/5 triggers to kill them all. Then, we won't have the time to be able to distribute the triggers on all the Angels, and at least one will take damage every time, killing the Angels in the same amount of steps every time.
So, I guess this is okay?
What about the normal setup, do we have a problem there? There the computation is more complex, and each creature type can get incremented by many different other creature types. So I don't know there.
Edit: Actually, I guess it's not true that all creature of the given creature type have to die at the same time; the proof that we had doesn't work if we can save some damage from the Arcbond triggers. In that case, things become a lot more complex.
Hmmm, I guess the issue is we don't know exactly what we can do with only one copy of AE per creature. So their Bishop of Wings can either cause the death of Angels creating some copies of whatever creature type we choose, or the death of whatever creature type creating copies of Spirits. Is this enough to go infinite, but still haltable by the way we order Arcbond triggers?
Haha, that's actually a clever way of needing to kill two creatures! Let's be sure whether we need it or not, though.
Hmm yeah, too bad about Karn. Squee, the Immortal is also out since he's a Goblin. Are there any other ways to get cards from the exile zone, but limited?
Hmm, from what I know of Busy Beaver candidates, the size of the output is generally around sqrt(N), where N is the number of executions. There can be some that are around log(N). For Waterfall machines, the state of the variables must not loop in a nonhalting machine, so if there are N steps in the execution, then the variables must at some point reach more than N^(1/k), where k is the number of waterclocks. But, I guess there is no theoretical reason that the waterclocks couldn't take a nosedive at the end. Certainly, it seems impossible that ALL sufficiently large inputs have the computation kill off almost all of the waterclocks at the end, so much so that we don't get Busy Beaver growth, but I guess we don't necessarily no for sure.
Hmm, but I guess we can insure it: have a particular creature type that only creates one of itself, and at least six of a second creature type. Then each Arcbond trigger creates 6 more tokens of the second creature type, so that it never dies. So the number of the second type that we have at the end is six times the number of triggers that we resolve. The total amount of life gain is a small constant times the number of triggers, based on how many Bishop of Wings that we have (which can be around 10^^^^^^57, but that's still very small compared to BB(10^^^^^^57). So yeah, some minimal recursion is enough to overcome that small constant factor.
Bishop of Wings should be fine with my setup, since the creatures are all coming in on the opponent side.
While Precognition Field is much cheaper than Primal Surge, we still have to pay the mana costs for our permanents. It does save one card in the opening, so if we can get started without using any more card slots, it would probably be better.
So, instead of Tolarian Academy hooking directly into life using Fastbond, we get it back with Primevals' Glorious Rebirth. This requires whate and black, so we use a Plains for the white mana, which we can get back using Rampant Growth; and then we get the green mana using Bountiful Promenade, which we finally recycle using Fastbond. So that adds two layers.
We get 16 untapping cards, and we need to use 3 to transform the Gateways, so 56 uses of Sanctum of the Sun for 10^^^^^^57.
Yup, pretty much every deck we've ever made, even decks that needed Grip of Chaos to keep hitting the right targets throughout the combo, had the average damage have the same lower bound as the max damage! Even when the chance of achieving the max damage was 1/X where X was larger than the lower bound given for the max damage, the max damage was still more than say X^X, so the average damage would be more than X^X/X >> X.
Hmm, it looks like I assumed that the Smothering Tithe tokens would get copies, but of course Mirroworks only triggers for nontoken artifacts. We can multiply the number of artifacts with things like Doubling Season, but since we don't have a way to get enchantment tokens, such improvements are limited, so they don't provide extra layers. So that loses a layer, although of course we can add all the token and mana doublers to bring the iteration number into the billions, I think.
But, at this point it is better to go to an old strategy from our last version of standard: Sanctum of the Sun. This gives us an amount of mana of any color equal to our life, which we can make very big, so this gives us another layer.
Another problem I noticed was that Channel the Suns is a sorcery, so we can keep recycling it to get all different kinds of mana, which is no good. I think we can cast Primal Surge instead.
We nicely have 4 different CMCs of artifacts currently, so we can use Karn, the Great Creator to bring 4 exiled artifacts back, and we only need to exile one other card of a different CMC. Looking through the deck, we can exile Primal Surge and... I guess that's it? So we need three other exile cards to get four Sanctum of the Suns. (We can try to pick useful ones, like Mana Reflection.)
We get 4 Sanctum of the Suns, and we have 23 cards to untap them. But, we will need 3 untapping spells to transform the Azor's Gateways, leaving 20 untapping spells, so we can use each Sanctum of the Sun 21 times each. So we get about 10^^^^85, I guess - a bit of a drop, but not bad.
I suppose another possibility is to be able to get by with one Mirrorweave cast. That means we need to get rid of Opalescence and put Bishop of Wings onto the battlefield, with the stack not empty. So, Disenchant to get rid of the Opalescence, and Quicksilver Amulet to put Bishop of Wings onto the battlefield. Would that work?
Edit: I don't think we can use a second Mirrorweave to turn everything into a random token, since the Arcbonded creatures will still be doing damage, and it will eventually kill the whole group. But, I guess we can turn them all into a Bishop of Wings that has been hacked so that when it dies it makes a token; then when everything dies, we'll get a fresh group of tokens with no Arcbond triggers anymore.
I believe I have an improvement - we can double dip on the Mountains, by putting them into play tapped with Rampant Growth, then untapping them many times with Amulet of Vigor. Then we can bounce them and exile them to Azor's Gateway.
Updated deck:
2 Channel
3 Forest
4 Primal Surge
5 Dralnu's Crusade
6 Coat of Arms
7 Bishop of Wings
8 Arcbond
9 Artificial Evolution
10 Izzet Guildmage
11 Swarm Intelligence
12 Angel's Grace
13 Mirrorweave
14 Mirrorweave
15 Opalescence
16 Dowsing Dagger
17 Westvale Abbey
18 Armed // Dangerous
19 Defy Gravity
21 Thran Dynamo
22 Sculpting Steel
23 Dematerialize
24 Sage's Knowledge
25 Sage's Knowledge
26 Tolarian Academy
27 Necrogen Spellbomb
28 Primevals' Glorious Rebirth
29 K'rrik, Son of Yawgmoth
30 Selesnya Signet
31 Energy Bolt
32 Azor's Gateway
33 Galvanic Key
34 Runic Repetition
35 Deep Reconnaissance
36 Parallel Evolution
38 Rampant Growth
22 Mountains
We can put 22 Mountains into play, then exile 31 cards, so we get to more than 10^^^^55.
Of course, better would be to find a way to pay life to recycle basic lands, so that we could get 5 layers.
Updated deck:
2 Channel
3 Forest
4 Primal Surge
5 Dralnu's Crusade
6 Coat of Arms
7 Bishop of Wings
8 Arcbond
9 Artificial Evolution
10 Izzet Guildmage
11 Swarm Intelligence
12 Angel's Grace
13 Mirrorweave
14 Mirrorweave
15 Opalescence
16 Dowsing Dagger
17 Westvale Abbey
18 Provoke
19 Defy Gravity
21 Thran Dynamo
22 Sculpting Steel
23 Dematerialize
24 Sage's Knowledge
25 Sage's Knowledge
26 Tolarian Academy
27 Necrogen Spellbomb
28 Primevals' Glorious Rebirth
29 K'rrik, Son of Yawgmoth
30 Selesnya Signet
31 Energy Bolt
33 Galvanic Key
34 Runic Repetition
35 Deep Reconnaissance
36 Parallel Evolution
4 Great Furnace
20 Mountain
So, in addition to the four flashback spells, I think that we can exile Channel, Forest, Primal Surge, Coat of Arms, Dowsing Dagger, Selesnya Signet, Galvanic Key, and the four Great Furnaces and 20 Mountains, allowing us to transform Azor's Gateway 31 times. The 24 red mana lands take us to more than 10^^^^2, so 31 more batches of red mana takes us to more than 10^^^^33.
Edit: So, I think I can still get 4 layers, but I'd like to get 5. To do that, I need to figure out a way to use life to put a basic land onto the battlefield tapped. I guess the basic land can come from anywhere.
Edit: Okay, a prospective deck:
2 Channel
3 Mox Emerald
4 Primal Surge
5 Dralnu's Crusade
6 Coat of Arms
7 Bishop of Wings
8 Arcbond
9 Artificial Evolution
10 Izzet Guildmage
11 Swarm Intelligence
12 Angel's Grace
14 Mirrorweave
15 Mirrorweave
16 Opalescence
17 Dowsing Dagger
18 Westvale Abbey
19 Blaze of Glory
20 Jump
22 Thran Dynamo
23 Eye of Nowhere
24 Sage's Knowledge
25 Sage's Knowledge
26 Tolarian Academy
27 Necrogen Spellbomb
28 Primevals' Glorious Rebirth
29 K'rrik, Son of Yawgmoth
30 Mox Pearl
31 Energy Bolt
33 Pull from Eternity
27 cards to exile
Unfortunately, I didn't figure out a way to get turn life payment into recycling the basic land, so this is four layers. With Pull from Eternity, we can bring exiled cards back to our graveyard, which we can fetch to our hand if they are sorceries or legendary permanents, but not otherwise. Nicely enough, we can bring back 4 different CMC's of cards. We can also exile and bring back Sage's Knowledge once, but not more than once, since after the second exile we will have no more copies of Sage's Knowledge in our hand, and we can no longer retrieve sorceries from the graveyard anymore. So we can exile one Sage's Knowledge, and each of the 27 extra cards, to transform Azor's Gateway into Sanctum of the Sun 28 times, which will get us to more than 10^^^^29.
I think we would have better luck with 240+ waterclocks using the variant model. We could make up to 12 of them have special creatures, but wouldn't that just complicate matters?
In the scenario where we have Bishops, the computation on each side is independent. (Well, we have the Arcbond triggers... but, we can still consider what happens to our side without worrying about the other side, we just have to allow for the Arcbond triggers to be from an opponent's creature.) The computation continues until both Arcbonded creatures are killed. If both Arcbonded creatures belong to the opponent, we don't have to care about what happens on our side. If one or both are on our side, then we do care. But, if we only have two Bishops, when we sacrifice one we get at most two tokens. If they are the same type as the Bishop, soon all three will die, and the computation halts. Or if the dying Bishop created a different type, then the first Bishop could create another type when that type dies - but that leads to a dead end as well. So two Bishops doesn't go anywhere.
We could probably check the possibilities for a haltable infinite with more Bishops. Looking at our current enchantments:
Dralnu's Crusade: this is the one that we're worrying about.
Swarm Intelligence: This is nice for our sorcery recursion, but we could do without it if we need to.
Runed Halo for us: This might be necessary, if we transform this, we have make sure that we can stay alive for arbtrarily long.
Runed Halo for the opponent: Not actually necessary.
Opalescence: This won't be a creature.
Fastbond: This is nice, but we can also do without this if necessary.
So, if we get rid of Swarm Intelligence and Fastbond, we can convert two enchantments into Bishops. We would be without Runed Halo, so we would have to stay alive. I guess we can check this for infinite possibilities.
Edit: Oh whoops, I missed that Coat of Arms counts all creatures.
I do think we can scale back down? We do get a scaling factor, but that scaling factor also increases the number of decrements we need to get back down to 0. So in a loose sense, the scaling increases the time it takes for things to go back down exponentially, but eventually they do come back down.
If this is Turing-complete, then we can just do the combo with just one copy of Mirrorweave, and no Opalescence, I think. If it turns out to either die early or increase without bound, then perhaps we can stick with the current paradigm. If it's not Turing complete, but still allows haltable infinites, maybe we're screwed?
So, maybe we have to dig into the proof that the Waterfall Model is Turing-complete, and see whether or not it can be modified to fit the model above.
Goblin Dark-Dwellers would give us an extra Goblin to sacrifice. Chandra Ablaze looks good, but it's first ability allows it to deal damage to a target. Maybe Sunforger? Then we could pair it with Fireblast, provided we had access to exactly one red mana.
Well, if we had just one red mana, we could cast Arcbond anyway. So maybe that's the plan.
Oh wait, but Skirk Volcanist allows us to have red mana - that sounds better.
Edit: So, it looks like we can switch to Skirk Volcanist with the rest of the combo largely unchanged. But, I am looking for an efficient way to have multiple Sanctum of the Suns. Pull from Eternity brings exiled cards to our graveyard, and at the moment we can only bring back legendary permanents and sorceries back from the graveyard. Unfortunately, that's enough to get five different CMC's already - 0,2,5,7,10. The 5,7,10 belong to Stunted Growth, Primevals' Glorious Rebirth, and Primal Surge. Stunted Growth and Primal Surge both seem replaceable, although confining ourselves to those CMC's might be difficult. What are alternate ways of putting a basic land from our graveyard or hand into our library, without being able to put Arcbond into our library?
Hmmm... But if we create N Angels after death, it will take about N/5 triggers to kill them all. Then, we won't have the time to be able to distribute the triggers on all the Angels, and at least one will take damage every time, killing the Angels in the same amount of steps every time.
So, I guess this is okay?
What about the normal setup, do we have a problem there? There the computation is more complex, and each creature type can get incremented by many different other creature types. So I don't know there.
Edit: Actually, I guess it's not true that all creature of the given creature type have to die at the same time; the proof that we had doesn't work if we can save some damage from the Arcbond triggers. In that case, things become a lot more complex.
Haha, that's actually a clever way of needing to kill two creatures! Let's be sure whether we need it or not, though.
Hmm yeah, too bad about Karn. Squee, the Immortal is also out since he's a Goblin. Are there any other ways to get cards from the exile zone, but limited?
Hmm, from what I know of Busy Beaver candidates, the size of the output is generally around sqrt(N), where N is the number of executions. There can be some that are around log(N). For Waterfall machines, the state of the variables must not loop in a nonhalting machine, so if there are N steps in the execution, then the variables must at some point reach more than N^(1/k), where k is the number of waterclocks. But, I guess there is no theoretical reason that the waterclocks couldn't take a nosedive at the end. Certainly, it seems impossible that ALL sufficiently large inputs have the computation kill off almost all of the waterclocks at the end, so much so that we don't get Busy Beaver growth, but I guess we don't necessarily no for sure.
Hmm, but I guess we can insure it: have a particular creature type that only creates one of itself, and at least six of a second creature type. Then each Arcbond trigger creates 6 more tokens of the second creature type, so that it never dies. So the number of the second type that we have at the end is six times the number of triggers that we resolve. The total amount of life gain is a small constant times the number of triggers, based on how many Bishop of Wings that we have (which can be around 10^^^^^^57, but that's still very small compared to BB(10^^^^^^57). So yeah, some minimal recursion is enough to overcome that small constant factor.
Bishop of Wings should be fine with my setup, since the creatures are all coming in on the opponent side.
While Precognition Field is much cheaper than Primal Surge, we still have to pay the mana costs for our permanents. It does save one card in the opening, so if we can get started without using any more card slots, it would probably be better.
So, I've managed to make an improvement:
2 Channel
3 Elvish Spirit Guide
4 Primal Surge
5 Dralnu's Crusade
6 Coat of Arms
7 Bishop of Wings
8 Arcbond
9 Artificial Evolution
10 Radiance
11 Swarm Intelligence
12 Runed Halo
13 Runed Halo
14 Goblin Grenade
15 Mirrorweave
16 Mirrorweave
17 Opalescence
18 Dowsing Dagger
19 Donate
21 Thran Dynamo
22 Eye of Nowhere
23 Sage's Knowledge
24 Sage's Knowledge
25 Tolarian Academy
26 Primevals' Glorious Rebirth
27 Necrogen Spellbomb
28 Plains
29 Amulet of Vigor
30 Charcoal Diamond
31 Rampant Growth
32 Stunted Growth
33 Bountiful Promenade
34 Fastbond
36 Azor's Gateway
37 Azor's Gateway
38 Azor's Gateway
39 Azor's Gateway
40 Mirror Gallery
41 Karn, the Great Creator
42 random exile
43 random exile
44 random exile
16 untapping cards
So, instead of Tolarian Academy hooking directly into life using Fastbond, we get it back with Primevals' Glorious Rebirth. This requires whate and black, so we use a Plains for the white mana, which we can get back using Rampant Growth; and then we get the green mana using Bountiful Promenade, which we finally recycle using Fastbond. So that adds two layers.
We get 16 untapping cards, and we need to use 3 to transform the Gateways, so 56 uses of Sanctum of the Sun for 10^^^^^^57.
Hmm, it looks like I assumed that the Smothering Tithe tokens would get copies, but of course Mirroworks only triggers for nontoken artifacts. We can multiply the number of artifacts with things like Doubling Season, but since we don't have a way to get enchantment tokens, such improvements are limited, so they don't provide extra layers. So that loses a layer, although of course we can add all the token and mana doublers to bring the iteration number into the billions, I think.
But, at this point it is better to go to an old strategy from our last version of standard: Sanctum of the Sun. This gives us an amount of mana of any color equal to our life, which we can make very big, so this gives us another layer.
Another problem I noticed was that Channel the Suns is a sorcery, so we can keep recycling it to get all different kinds of mana, which is no good. I think we can cast Primal Surge instead.
We nicely have 4 different CMCs of artifacts currently, so we can use Karn, the Great Creator to bring 4 exiled artifacts back, and we only need to exile one other card of a different CMC. Looking through the deck, we can exile Primal Surge and... I guess that's it? So we need three other exile cards to get four Sanctum of the Suns. (We can try to pick useful ones, like Mana Reflection.)
So, we have:
2 Channel
3 Mox Diamond
4 Primal Surge
5 Dralnu's Crusade
6 Coat of Arms
7 Rotlung Reanimator
8 Arcbond
9 Artificial Evolution
10 Radiance
11 Swarm Intelligence
12 Runed Halo
13 Runed Halo
14 Goblin Grenade
15 Mirrorweave
16 Mirrorweave
17 Opalescence
18 Dowsing Dagger
19 Donate
21 Thran Dynamo
22 Eye of Nowhere
23 Sage's Knowledge
24 Sage's Knowledge
25 Tolarian Academy
26 Fastbond
27 Divine Congregation
28 Azor's Gateway
29 Azor's Gateway
30 Azor's Gateway
31 Azor's Gateway
32 Mirror Gallery
33 Karn, the Great Creator
34 random exile
35 random exile
36 random exile
23 untapping cards
We get 4 Sanctum of the Suns, and we have 23 cards to untap them. But, we will need 3 untapping spells to transform the Azor's Gateways, leaving 20 untapping spells, so we can use each Sanctum of the Sun 21 times each. So we get about 10^^^^85, I guess - a bit of a drop, but not bad.
2 Channel
3 Mox Diamond
4 Channel the Suns
5 Squee's Revenge
6 Eureka
7 Dralnu's Crusade
8 Coat of Arms
9 Rotlung Reanimator
10 Arcbond
11 Artificial Evolution
12 Radiance
13 Mirari
14 Runed Halo
15 Runed Halo
16 Goblin Grenade
17 Mirrorweave
18 Mirrorweave
19 Opalescence
20 Dowsing Dagger
22 Thran Dynamo
23 Eyes of Nowhere
24 Sage's Knowledge
25 Tolarian Academy
26 Fastbond
27 Divine Congregation
28 Smothering Tithe
29 Smothering Tithe
30 Smothering Tithe
31 Smothering Tithe
32 Copy Enchantment
33 Copy Enchantment
34 Copy Enchantment
35 Copy Enchantment
37 Mirrormade
38 Mirrormade
39 Mirrormade
40 Unmoored Ego
41 Divine Congregation
42 Rakdos's Return
43 Donate
So the layer system is now:
Layer 0: Putting an artifact into play can create many copies thanks to Mirrorworks and Panharmonicon
Layer 1: Tapping a Tolarian Academy can generate a lot of blue mana.
Layer 2: Casting Divine Congegration gains us a lot of life.
Layer 3: Resolving a Smothering Tithe trigger can generate a lot of mana.
so this can create 10^^^^637 tokens, roughly.