This guide has been a long time in coming. Some may remember my previous post, roughly outlining the mathematics behind sustaining Indigon's mana cost. I went back and formalized the mathematics behind it; if anyone is interested, I wrote it out in LaTeX and uploaded it here.

But for those who didn't read over that post or are unaware of what "sustained Indigon" refers to, I'll do a quick overview.

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What is a "sustained Indigon" build?

Indigon is a unique that scales Spell Damage from Mana spent Recently (the past 4 seconds).

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(50-60)% increased cost and (20-25)% increased spell damage are the most important ranges to pay attention to

Since the cost of skills increases as total Mana spent Recently increases, Indigon ramps up mana costs very quickly, as this graph demonstrates.

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Number of casts on x axis, Mana Cost on y axis

Naturally, this leads to a problem: we eventually ramp our mana cost above what we're able to spend, either because we don't have enough regen or because the mana cost is greater than our maximum mana! So this creates an uneven buff from Indigon—your damage becomes inconsistent if it heavily scales via Indigon. And since Indigon can scale up to 2000% increased Spell Damage, it has very high potential—if we can make it consistent.

Enter the concept of sustained Indigon builds via convergence of the scaling mana costs. The details are contained in the LaTeX proof I linked earlier, but it's possible, albeit with a lot of mathematical work required, to ensure that the Indigon ramping only ramps up to a specific cost, and not beyond that. You can see the difference in what a divergent Indigon mana sequence looks like vs. a convergent Indigon mana sequence.

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Blue is convergent mana sequence, Yellow is divergent mana sequence

If we're able to attain this in a build, then we can maintain the Indigon buff indefinitely, providing a consistently massive Spell Damage buff to scale our damage. Let's jump into an example build where this works.

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Scorching Ray Sustained Indigon - ranges from budget 2mil DPS to higher-end 7mil DPS

Budget (~5div) PoB: https://pobb.in/TY-5Hk6dK7vo

Higher-end (40div+) PoB: https://pobb.in/soXwYo109w-e

This build attains our goal of sustaining Indigon indefinitely; in the clip below, you can see the buff being maintained, keeping the mana cost stable and the Indigon buff applied continuously.

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Stable, Sustained Indigon Scorching Ray!

We begin by ramping the Indigon costs quickly through Flameblast + Archmage, which quickly eats up our available mana; then we switch to Scorching Ray and continue ramping until we hit our convergence mana cost value of 494 Mana. With 17 casts over 4 seconds, this gives us our 8398 Mana spent Recently, which requires over 2000 Mana regenerated per second. If you check the PoB, you'll see that Indigon by far contributes the most damage to the build—the build would lose over 70% damage if it were dropped!

Of course, this build has numerous other problems, so I don't recommend anyone actually play it. In order to get over 2000 Mana regenerated per second, we need a massive amount of Mana regeneration, so a great deal of our gear and most of our passive tree is dedicated to this task. But we'll get into those details later when discussing problems for sustained Indigon convergence builds; for now, let's deep dive into how to make a sustained Indigon build and what makes it work.

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How to Create a Convergent Indigon Mana Sequence for a Build

If there is nothing else to take away from this post, it will be this section, as it's the most relevant for build creators. Again, the proofs of what I'm about to mention are located in the LaTeX file linked at the top of the post, so please refer to that if you want to know why any of the following is true.

To restate the theorem in question:

>The Indigon Mana Cost Sequence converges if and only if bck < 200

where:

>b -> the base mana cost of the skill, multiplied by any More/Less modifiers (this does not include increased/reduced modifiers!)
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>c -> the "increased Cost of Skills" mod value for your Indigon (in my pictured Indigon above, c = 0.5)
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>k -> number of casts Recently (in the past 4 seconds)

A few notes here: to calculate b, Path of Exile doesn't straight up multiply the values all together; rather, they multiply each More/Less/Mana Reservation modifier together, round it down to the nearest hundredth, as this reddit post concludes after testing (i.e. 11.12 * 1.4 = 15.568 -> rounded down to 15.56 before being multiplied by the next modifier). Then the final modifier is multiplied to the mana cost, which is then rounded down to the nearest integer.

To demonstrate this: Scorching Ray at level 26 has a base mana cost of 6. It has the support multipliers in the following order: 1.3, 1.3, 1.3, 1.3, 1.4 (the order of the supports is the order of the multiplication as well). So we have 1.3 * 1.3 = 1.69 -> 1.69 * 1.3= 2.197 -> 2.19 * 1.3 = 2.847 -> 2.84 * 1.4 = 3.976.

So then we multiply 6 * 3.97 = 23.82, which is rounded down to 23 for our final mana cost. (Note that PoB seems to have a bug where it rounds up after increases/reductions are calculated but correctly rounds down after more/less values.) So our base mana cost is 23 for this Scorching Ray setup.

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Uncertainty about Cast Time Mechanics

Now, here is where I had a misconception:

>If we have 4.32 cast speed, then 4.32 * 4 = 17.28 casts per 4 seconds. We round this down to: k = 17 casts per 4 seconds. Since we now know all three variables: 23 * 0.5 * 17 = 195.5 < 200, so this setup of gems and Indigon and cast speed should ensure that our Indigon mana cost sequence will converge.

I thought cast rate was just 1/casts per second, so you'd have 1/4.32 = 0.23148148148 repeating cast time. But my testing showed this wasn't the case: it casts every 0.23 seconds, so it presumably truncates all past the hundredth digit.

This is where it gets hard for us to be precise, because we aren't sure what the tick rate of the PoE server is; for now, I've proceeded on the assumption that it rounds to the hundredth digit because the tick rate can accommodate that precisely enough.

So with that in mind, if we cast once every 0.23 seconds (presumably we spend the cost at the start of the 0.23 second cast), then we cast at 0, 0.23, 0.46, ..., 3.91, for a total of 18 times in a 4 second window. But since it should be the past 4 seconds, by the time it gets to the next cast instant (4.14), the first cast at 0 is excluded, and so on for future casts, so it should always be the past 17 casts.

However, when I tested this, we diverged! (Or, another possibility: it still converged, but it converged to a much higher number than 494, which I could not sustain with mana regen tailored for 494. This may be possible for ramping high values initially. However, testing without ramping still fails to converge to 494 when it should, so I suspect it does, in fact, diverge, or at least spends mana cost/calculates it differently than I expect.)

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We ramp up with Flameblast, and then it should ramp until it stabilizes at 494; instead, it diverged and we ran out of mana—which is the problem we wish to avoid

But when I reduced the cast rate by just 2%, it went to a 0.24s cast rate, and this time it converged, seemingly in a manner as described above, though it converged to a different value than we calculated. That is what is shown in the first clip of this post, converging at 345 mana per second—meaning we get around 5520 Mana spent Recently for a nice 675% increased Spell Damage buff from Indigon—a solid buff, but we were hoping for 8398 Mana spent Recently for over 1000% increased Spell Damage buff!

This honestly puzzles me, as it must involve the specifics of how cast rate interacts with Mana spent Recently and the server tick rate, which are things I have no idea how to test or figure out; but I am glad of one thing: the Indigon spell cost did, indeed, converge! The only problem is in our calculations as to which value it converges to and the specifics of calculating cast rate. I suspect if we learn more about how cast rate and Mana spent Recently are calculated, then we may be able to solve this. But at least we've shown that it does, in fact, converge, as we theorized!

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Determining Convergence Value

But what will it converge to? There is a formula to calculate this, though it's a bit complex, since it's in Mathematica formulation; I'll post it here in case anyone wants to calculate this themselves (you will need to adjust the variables to your values for b, c, and k (b = 23, c = 0.5, k = the 17 in "x - 17" and "l[19], l[10], ... l[17]"), as these are for the Scorching Ray build demonstrated here; running this gives us 494, the value to which we saw our example build converge):

>RecurrenceTable[{l[x] == Piecewise[{{Floor[11.5 Floor[Sum[l[k], {k, x - 17, x - 1}]/200] + 23], Floor[11.5 Floor[Sum[l[k], {k, x - 17, x - 1}]/200] + 23] < 20000}, {0, Floor[11.5 Floor[Sum[l[k], {k, x - 17, x - 1}]/200] + 23] >= 20000}}], l[1] == 0, l[2] == 0, l[3] == 0, l[4] == 0, l[5] == 0, l[6] == 0, l[7] == 0, l[8] == 0, l[9] == 0, l[10] == 0, l[11] == 0, l[12] == 0, l[13] == 0, l[14] == 0, l[15] == 0, l[16] == 0, l[17] == 0}, l, {x, 1, 2600}]

Where 2600 at the end is the number of instances it will show. This allows you to look for a converged value at the end of it, since it'll reach the same number over and over for a long while. As a general rule of thumb, if the convergence value is x, it's going to follow the inequality 200(b - bc - 1)/(200-bck) < x < 200b/(200-bck). As you can tell, getting bck as close to (but under) 200 means the range of possibly convergence values so higher; vice versa for further away/lower.

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Difficulties with Sustained Indigon Builds and Future Considerations

1. The amount of investment necessary to scale and sustain mana precludes sufficient investment in defenses. This is doubly true because, unless you are able to scale your mana regen even higher than needed for sustained Indigon, proportional to your Life, you won't be able to use Mind over Matter as a form of damage mitigation without interrupting your damage. Unless GGG either lessens damage requirements, adds more damage mitigation with mana/mana regen/Indigon synergies, and/or greatly reduces the amount of investment necessary to attain the mana regen we need, this will continue to be an on-going problem.
2. Very precise mana cost management—you can't let extra mana costs slip in anywhere. Movement skills? Need to be cast with Lifetap or cast with no mana cost. Molten Shell? Lifetap. And these Lifetap costs are themselves scaled by Indigon, so you'll be spending hundreds, possibly even thousands, of Life to cast these spells. There may be an argument for using something like a Battery Staff if one really must cast something, but even then, if you run out of Energy Shield and it casts mana, it may cause the mana sequence to diverge, and then you'll have to start over and ramp back up, losing a bunch of damage uptime.
3. Time investment into calculating and testing the above—as shown earlier, the cast speed calculations aren't accurate. I'm not certain why, because I don't know enough about the server. In my dreams, Mark shows up to kindly inform us of how this would be calculated, but absent that, we can only continue testing. If we can deduce this mystery, then we can be more confident in our PoB calculations before moving to test in the game.
4. Ramp time—it can take some time to ramp up to our sustained Indigon mana values. We can speed this up with the likes of Arcane Cloak + a mana ramping skill (like Flameblast + Archmage), but it's still not quick enough to near instantly be ready for max damage.

There are some future build ideas which may be of use to explore further; I've gone as far as I want to with these ideas, so I'll post the concept with a working PoB, but none of them are really functional as of yet, mainly due to the above difficulties.

1. Using Witch's Nine Lives and some form of self-damage like Heartbound Loop, we generate massive amounts of Life, Mana, and Energy Shield Recoup; this gives us amazing defenses against damage over time, so when combined with Petrified Blood, we have strong defenses, while also giving us mana regen proportional to the amount of self-damage we're taking. This naturally synergizes very well with Wardloop; the only problem is that so much investment into the self-damage/recoup loop is necessary that there's not much investment remaining for Life, Armour, or damage scaling. PoB: https://pobb.in/g_Tta0xCFe2Z
2. Battlemage's Cry increased Spell Damage scaling: we can hyper-buff attack damage by using the Battlemage's Cry increased Spell Damage -> Attack Damage conversion. This was popularly used in an Occultist Replica Alberon's Warpath Cyclone build previously, where they stacked warcry effect and spell damage and Strength (converted to spell damage, then converted back to attack damage, which then multiplies Replica Alberon's Warpath's chaos damage which also scaled from Strength) to get massive damage. We can do something similar: scale Strength, 1000% increased Spell Damage via Indigon, and then scale that all back into attack damage. This can work well with skills like Doryani's Touch, which struggle with damage scaling. But this runs into the same defensive problems, as well as uptime/ramping concerns. PoB: https://pobb.in/8ngmCW9LFZ6X
3. Auto-casting perpetual Indigon engine—this one saves us the problem with ramping by having the spell always be casting, so it's always at max power! This is probably one of the most promising concepts, but I'm not sure how to implement it apart from something like an awakened cast while channeling setup as I have in my PoB for #1.

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tl;dr: if you want your Indigon to not ramp to infinity, your base mana cost * increased cost mod * number of casts per 4 seconds must be less than 200. Then it will converge to some number; you can determine this through the formula posted above.

These builds take a lot of effort to construct, because if your math is off, then it may: a) diverge instead of converge (and thus will never be consistent); b) converge but you will lack the mana regen needed for it; or c) will converge much earlier than you want, giving you a much smaller buff and potentially rending the investment into Indigon useless. You have to be specific and detailed with the above mathematical calculations for it to work.

But if it does, then you have an incredible and unique build, uniquely different from every other build out there running those skills! And, if we find the perfect storm of a build, we may be able to use this tech to scale damage far beyond what a skill is normally capable of.

If anyone learns more about the cast speed calculations or makes a sustained Indigon build, tag me/let me know! I'm hoping some wizened build masters will be able to find an interesting build idea that makes this work, as I'm exhausted from investigating all of the above.