Highest Possible Crit Range?

[flabort]

Just found this, real intresting. I think I'll break this down page by page...

Psychic weapon master and disciple of dispater can't ever stack by the books- one requires a crystal weapon, and the other an iron/steel one.

I'm screaming inside: this is from page 1, and haven't gotten to page 2 yet.
he needs two weapons to use lightning mace, and two different types of weapons for these two feats, so using TWO aptitude rapiers made from each of the materials, or if they are a special material by being aptitude, counting as each of the materials due to APTITUDE rapier, would qualify for all three!

So the build is defeated by a smokestick?

Page 2, most interesting quote i found. it would appear to be true, by that point. interesting to see were the build is heading by this point. and also the conversation. concerning the conversation about ∞,

Spoiler

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I can hardly wrap my head around tesseracts and other 4 dimensional objects, or other n-dimensional objects, and related speres, let alone infinite and related spheres! although, i do know n∞=∞^n=∞/n=∞+n=∞-n=(∞+n)^n=∞... basicly, anything you do to ∞ results in ∞, even ∞/∞=∞. i also do know that 1+1/2+1/4+1/8+1/16... ect., halving the value each time, never quite reaches 2, but by the time it gets to the point were it might as well be, there's ∞ terms...

$%@#, could you imagine a ∞-dimensional object?

Interesting question....if you crit against a mirror image (as the spell), would you get an extra attack?

And since nobody answered, I'll ask again, do you roll once or twice to attack with a weapon with the splitting property?

ooh, cool. still page 2, I love mirror image abuse cheese. fill a room with yourself, ect.
Personally, i'd let the crit to the illusion count for another attack, so as not to tip the player off that it's an illusion. And, i've never heard of the splitting property. guess i'll find out later in the thread, maybe.

No, actually. Aptitude is feats, and those are classes.

I know it was a page back, but many folk don't seem to know that. Just wanted to get it out there.

Ooh, sucker punch to my comment on page 1. Ok, but if the weapons CAN be made of different materials still, that's still cool.

There is a feat from one of the completes that lets rogues treat nat 1s as nat 20s

Ok, page 2 is a GOLD MINE of good material. let's just pack that on....

reaching page three, i'm seeing some people are talking as if one miss will end the whole chain, and others talking about the attacks like a tree rather than a chain, and using different math. are we using permutations, combanations, or niether here? arg. i hate math 30 pre-probability.

Blood in the Water has already been mentioned; it grants you a cumulative +1 attack and damage each time you crit. So your weapon's base damage quickly becomes moot.

Finally a page 3 quote. so, that's what that does. and how are we packing so many feats into this build?! well, by the 27th hit, we're hitting +27 dammage, or more, due to this one feat, and are hitting on everything but a 1. hunng. and we're getting, like, 2.98337243294982749832734398273489x10^∞ hits.... or more. Mozza-swiss with a little cheddar, much?

Do not argue RAW with Curmudgeon, he is always right.

LOL. that is all from that quote.

-Arguing with Curmudgeon-

LOL. again.

Ok, so to recap, We're arguing about the chances and odds that this is indeed dealing infinite hits, i'm questioning trying to pass high school, we've got two Aptitude Rapiers with splitting property, one made of crystal and one of iron, quallifying for the Lightning mace feat, and the Psycic Weapon master and Disiple of dissasture classes, we've got some rouge with a feat so 1's act as 20's, we've got Blood in Water so no damage reduction can stand in our way, we've got some barbarian variant (street fighter?!), and a couple others, so that we hit with crits from 1-20, and as a result we kill if we're allowed one hit, and a simple smokestick foils all of our planning. and this is all pre-epic, and we can STILL kill a lvl. 100. harsh.

Limburger and monteray Jack indeed, even if I missed something!

[term1nally s1ck]

Oh, I'm not using the Psychic weaponmaster in my build (Which I will call the Chuck Norris if I want...the brackets were to explain why..), jeez, I can barely fit in the levels I've got...

To be honest, this build has exactly the same issue as any ubercharger....if your DM throws a monster at you that you can't attack, or that has miss chances/huge AC, you're gonna struggle to get the hit off, even if that one hit would kill it. Discounting the AC due to ToB cheese, miss chances and anything that can keep out of melee range can't be killed...well, you COULD one-shot the earth, taking it down with it, but that's a last resort, surely :P

(although, the idea of having a quest to traverse a mountain range, as has happened to me before, and responding with 'I hit the mountain. The mountain dies.' would be most definitely follow the rule of cool.)

The whole miss chances thing was why I wanted Splitting...because then I could probably get seeking on a pile of arrows, too, and annihilate any miss chances. Splitting is an enhancement for a bow/crossbow, where it duplicates each arrow you fire. I just can't recall if you would roll the attacks seperately or just roll once,or roll once and count it twice...I think you'd get two chances to proc extra attacks with them, which would nearly guarantee that you kill whatever you attack. Plus, y'know....range...

I take no responsibility for anyone blowing all their WBL on arrows within the first 6 seconds of the first battle in a campaign. You could literally blot the sun out, provided you could carry enough arrows. It's like having a heavy machinegun to use.

[Eldariel]

Oh, I'm not using the Psychic weaponmaster in my build (Which I will call the Chuck Norris if I want...the brackets were to explain why..), jeez, I can barely fit in the levels I've got...

Since the name Chuck is already taken, you may wish to reconsider that lest you want a roundhouse in the face Tornado Throw to the other side of the continent.

[Fortuna]

Sorry to those who have been trying to explain advanced mathematics to everyone, I am just curious about the little things (I once had an interesting discussion about philosophers, thought for an hour, and then pointed out very succinctly that Bishop Berkely was a moron and why). At any rate, I would like a clarification of this please.

We have a system defined below. This is meant to be a simplified version which should be infinite if the system which we are working with is infinite (note that I have probably used more axioms than neccessary).

Axiom 1: A system is entirely defined by these axioms and two numbers p and m.

Axiom 2: A system begins containing one instance.

Axiom 3: A system's population is equal to the cardinality of the set of instances in the system.

Axiom 4: A system develops in discrete iterations.

Axiom 5: At any iteration, there is a probability that the system will fail equal to 1/p^population.

Axiom 6: If a system does not fail at a given iteration, its population is multiplied by m.

Axiom 7: If a system fails, no further iterations occur.

Hypothesis: That a system with finite m and 0<p<1 will never satisfy population=aleph-0.

Conceptual Proof: At an iteration i, the chance that the system will fail is equal to 1/p^population, and population=m^(i-1). The probability that the system fails is therefore equal to 1/p^(m^(i-1)).
The probability that a system exists at iteration i+1 is equal to the product of (1-1/p^(m^(x-1))) for all integer values of x from 1 to i inclusive. Let i equal an arbitrarily high finite number. Then the probability that the system exists is the product of an arbitrary number of values less than 1 but greater than zero. Therefore as i increases, the product of (1-1/p^(m^(x-1))) for x from 0 to i is strictly decreasing.

Now, this is as far as I can get. From here, i believe that the clincher is the limit of the product of (1-1/p^(m^(x-1))) for x from 0 to i as i increases without bound. Please show me how you calculate this limit, or at least outline your method, unless of course I have stuffed up my reasoning in which case please point out how.

[term1nally s1ck]

You won't be able to make this proof, I'm afraid.

if M = (say) 2, and p = 10

The product is equal to:

1 - SUM(over x)(1/p^(m^(x-1))) + SUM (over X and Y)((1/p^(m^(x-1)))(1/p^(m^(y-1)))
- .......


But that second sum is equal to (SUM(over x)(1/p^(m^(x-1))))^2, and the same is true as you go down the line, making the entire series equal to 1-S+(S^2)-(S^3).....which, since S is trivially provable to be of absolute value less than 1, converges to a finite value. In this case, i can't tell you what that value is, because I can't write down a nice number that's equal to that sum, but the final value is between 1 and 1-S (a bit under 0.9)
we know for certain that S<0.2 though, so [1, 1-S] does not include 0

we can get what the sum is by a nice trick though.

Let P be the Probability that the sequence starting with population = 1 ends.

Looking at the first generation, we get that P = 1/10 (extinct after 1 try) + (9/10)*(P^2) (pop = 2 now). Solving quadratically, we get that P = 1 or 1/9

we have already established above that (1-P) =/= 0, so P =/= 1, therefore the probability in this case is 8/9 that it reaches aleph-0

Bad luck, you had a tougher job there. I had to only find an exception, you wanted to prove a general rule.

If I want to now find the set of values that can lead to infinities, I need to find the values of m and p that give me S between -1 and 1.

[Zincorium]

I'm screaming inside: this is from page 1, and haven't gotten to page 2 yet.
he needs two weapons to use lightning mace, and two different types of weapons for these two feats, so using TWO aptitude rapiers made from each of the materials, or if they are a special material by being aptitude, counting as each of the materials due to APTITUDE rapier, would qualify for all three!

Fail. Specifically, failure to read the Aptitude property. Actually, failure to even skim the description in the tome of battle.

Aptitude counts for feats, and only feats. It's not even a bit misleading- feats are all that are ever mentioned. Class features that are not bonus feats are not feats at all. So it still *must* be made primarily of iron for the class feature of the Disciple of Dispater to increase it's crit range. Aptitude does not change this. The psychic weapon master requires a *standard* crystal weapon- one without any special property- in order to use the class ability gained. Aptitude, again, does not change this.


Sorry for the rant, but calling someone out without even bothering to research it is MY schtick. Don't be a copycat.

(actually, I feel really guilty when I do it, but that rarely prevents me from flying off the handle the next time)

[ocdscale]

The probability that a system exists at iteration i+1 is equal to the product of (1-1/p^(m^(x-1))) for all integer values of x from 1 to i inclusive. Let i equal an arbitrarily high finite number. Then the probability that the system exists is the product of an arbitrary number of values less than 1 but greater than zero. Therefore as i increases, the product of (1-1/p^(m^(x-1))) for x from 0 to i is strictly decreasing.

Forgive me if I misunderstand your post, and forgive me if this post is somewhat superfluous given the terminally sick's calculations above.
I wanted to just explain why the section in bold doesn't quite support your hypothesis.

The fact that the probability of existence inexorably decreases does not necessarily mean the probability of existence converges to 0, only that it converges to some point.
As a simple proof of concept, imagine a system that has a probability of existence of:
0.75/(t+1) + 0.25
Where t is the time in rounds. Clearly the probability of existence decreases over time, but equally clear that it has a healthy 25% chance of existence even as t approaches infinity.

[Fortuna]

I am aware of this. As stated below that, that is where my reasoning breaks down. What I am asking for is a simple demonstration of the limit to infinity of that product. I notice that there have been many mentions of sums to infinity in this thread, but I am fairly sure that this is a case for products to infinity, and I have little to no experience with those.

[lvl 1 sharnian]

Fail. Specifically, failure to read the Aptitude property. Actually, failure to even skim the description in the tome of battle.

Aptitude counts for feats, and only feats. It's not even a bit misleading- feats are all that are ever mentioned. Class features that are not bonus feats are not feats at all. So it still *must* be made primarily of iron for the class feature of the Disciple of Dispater to increase it's crit range. Aptitude does not change this. The psychic weapon master requires a *standard* crystal weapon- one without any special property- in order to use the class ability gained. Aptitude, again, does not change this.


Sorry for the rant, but calling someone out without even bothering to research it is MY schtick. Don't be a copycat.

(actually, I feel really guilty when I do it, but that rarely prevents me from flying off the handle the next time)

Well, the main point of his post wasn't that Aptitude would bypass the restriction, just an addition. His main point was that one could wield a Crystal Aptitude Rapier in one hand and a Steel Aptitude Rapier in the other so as to gain the benefit of both.

However, that wouldn't be worth it, as it would cause differing threat ranges that don't apply.

Someone mentioned a crysteel material from Ebberon that functioned as both steel and crystal. Source?

Edit: I can't recall, does nonlethal damage do anything besides knock you out? Because if not, just Non-lethal damage to keep it infinite. Also, if you coup-de-grace, then you automatically hit, no need for a roll, would that also apply for the extra hits?

[term1nally s1ck]

I am aware of this. As stated below that, that is where my reasoning breaks down. What I am asking for is a simple demonstration of the limit to infinity of that product. I notice that there have been many mentions of sums to infinity in this thread, but I am fairly sure that this is a case for products to infinity, and I have little to no experience with those.

Take a look at my most recent post, I found that product in terms of a sum of sums, and proved that the limit of the product is finite (and in a certain case, 8/9) Your product is of a set of functions with a fairly difficult set of conditions...In general when you're looking at products, you re-arrange it into sums of sums, or take logs. I think logs would work here, but I'm not sure...

@Sharnian

If you don't roll, I don't think you can crit.

[sofawall]

Well, the main point of his post wasn't that Aptitude would bypass the restriction, just an addition. His main point was that one could wield a Crystal Aptitude Rapier in one hand and a Steel Aptitude Rapier in the other so as to gain the benefit of both.

However, that wouldn't be worth it, as it would cause differing threat ranges that don't apply.

Someone mentioned a crysteel material from Ebberon that functioned as both steel and crystal. Source?

Edit: I can't recall, does nonlethal damage do anything besides knock you out? Because if not, just Non-lethal damage to keep it infinite. Also, if you coup-de-grace, then you automatically hit, no need for a roll, would that also apply for the extra hits?

I think it needs to be iron :P

[term1nally s1ck]

But if you get Crysteel, you can give it the Metalline property to convert the metal part to cold iron.

[OracleofWuffing]

@Sharnian

If you don't roll, I don't think you can crit.

Coup de grace scores a crit without rolling.

Coup de Grace
As a full-round action, you can use a melee weapon to deliver a coup de grace to a helpless opponent. You can also use a bow or crossbow, provided you are adjacent to the target.

You automatically hit and score a critical hit. If the defender survives the damage, he must make a Fortitude save (DC 10 + damage dealt) or die. A rogue also gets her extra sneak attack damage against a helpless opponent when delivering a coup de grace.

[term1nally s1ck]

Hmmmmmm I think that works for Roundabout Kick, but I'm not sure about lightning mace..you have to 'roll a threat on an attack roll' which is not happening imo.

Roundabout kick is whenever you hit with a critical though, so does work.

[Fortuna]

Ah well. It is always good to try to do something that you've been told is impossible, if only so that you convince yourself of that. Thank you very much for that clarification, and my apologies for wasting your time.

[Kalirren]

Sorry to those who have been trying to explain advanced mathematics to everyone, I am just curious about the little things (I once had an interesting discussion about philosophers, thought for an hour, and then pointed out very succinctly that Bishop Berkely was a moron and why). At any rate, I would like a clarification of this please.

We have a system defined below. This is meant to be a simplified version which should be infinite if the system which we are working with is infinite (note that I have probably used more axioms than neccessary).

Axiom 1: A system is entirely defined by these axioms and two numbers p and m.

Axiom 2: A system begins containing one instance.

Axiom 3: A system's population is equal to the cardinality of the set of instances in the system.

Axiom 4: A system develops in discrete iterations.

Axiom 5: At any iteration, there is a probability that the system will fail equal to 1/p^population.

Axiom 6: If a system does not fail at a given iteration, its population is multiplied by m.

Axiom 7: If a system fails, no further iterations occur.

Hypothesis: That a system with finite m and 0<p<1 will never satisfy population=aleph-0.

Conceptual Proof: At an iteration i, the chance that the system will fail is equal to p^population, and population=m^(i-1). The probability that the system fails is therefore equal to p^(m^(i-1)).
The probability that a system exists at iteration i+1 is equal to the product of (1-p^(m^(x-1))) for all integer values of x from 1 to i inclusive. Let i equal an arbitrarily high finite number. Then the probability that the system exists is the product of an arbitrary number of values less than 1 but greater than zero. Therefore as i increases, the product of (1-p^(m^(x-1))) for x from 1 to i is strictly decreasing.

Now, this is as far as I can get. From here, i believe that the clincher is the limit of the product of (1-p^(m^(x))) for x from 0 to i as i increases without bound. Please show me how you calculate this limit, or at least outline your method, unless of course I have stuffed up my reasoning in which case please point out how.

Corrected a few errors.

Well, you're looking for limit to infinity of P(x) = PRODUCT_i [1-p^(m^i)] from i = 0 to infinity. Consider ln P(x)= SUM_i [ln(1-p^(m^i))]. Since m>1 and 0<p<1, p^(m^i) is very small for large i, and the approximation

ln(1+u) ~ u - 1/2 u^2 + O(x^3) < u

is valid. Thus SUM_i [ln(1-p^(m^i))] converges iff SUM_i [-(p^(m^i))] converges, by direct comparison. (The negative sign on that last actually doesn't matter, so we'll consider SUM_i [p^(m^i)] instead.)

But since i is very large and m>1, m*i < m^i, which means that SUM_i [p^(m^i)] < SUM_i [p^(m*i)] (since 0<p<1). But SUM_i [p^mi] is just the sum of a geometric series with r=p^m, which is finite since 0<p<1 and m>0, so SUM_i [p^mi] converges. Thus SUM_i [ln(1-p^(m^i))] also converges to a finite negative number by direct comparison, which means that our original PRODUCT_i [1-p^(m^i)] must also converge to a finite positive number greater than 0.

[term1nally s1ck]

^^Yeah, that's muuuch easier. Should have done logs to start, but wasn't sure it'd sum nicely, using the approximation is very clever (though you really should use inequalities for proofs like this, and the < means it still works)

[lvl 1 sharnian]

I think it needs to be iron :P

Not exactly...

Iron Power (Ex): When using an iron or steel weapon, a
4th-level disciple of Dispater gains a +1 insight bonus on
attack and damage rolls. Furthermore, his threat range is
doubled as if he were using a keen weapon. At 8th level, the
insight bonus improves to +2, and the threat range triples.
This ability does not stack with the keen weapon quality, but
it does stack with the Improved Critical feat.

[SensFan]

Because if I only hit on 11s, and you're blinking, I only hit 1/4 of the time.

And yes, the build can go infinite, I spent a while writing out some of the maths to prove it. the 7-20 threat range which can hit on a 2 has a 60/133 chance to never run out of attacks, with each attack it makes. It doesn't hit with every attack, and it's not a guarantee that it'll go infinite....but if you get 3 attacks, and you hit with a 2 on all of them (Blood on the Water, hit rats. a lot.) then you get an 83.5% chance to go infinite that turn.

You seem to not understand the concept of infinite.

Does your build keep attacking even if you roll one hundred billion 1s in a row? If not, you won't go infinite.
In fact, if you roll long enough, you will have n rolls, and will then roll 9999n 1s in a row.

Haven't read past this post, so if you learnt what infinite means in the meantime, great.

[Kalirren]

^^Yeah, that's muuuch easier. Should have done logs to start, but wasn't sure it'd sum nicely, using the approximation is very clever (though you really should use inequalities for proofs like this, and the < means it still works)

Yeah, you're right, I should have made the error of the approximation more explicit...as you said, the next-higher order term in the approximation is

ln(1+u) = u - 1/2 u^2 + O(u^3) < u

so it happens to work. I was relying on the fact that as u-> 0, it's really just u +/- O(u^2), but I guess that's not really sound, is it?

[sofawall]

Not exactly...

Probably only allows steel because it contains iron... Stupid Dispater, not sticking to his guns...

[Kalirren]

You seem to not understand the concept of infinite.

Does your build keep attacking even if you roll one hundred billion 1s in a row? If not, you won't go infinite.
In fact, if you roll long enough, you will have n rolls, and will then roll 9999n 1s in a row.

Haven't read past this post, so if you learnt what infinite means in the meantime, great.

On the contrary, -you- seem not to understand the concept of infinite. Specifically. you're confusing "finite" with "non-deterministically infinite".

As it stands, the sequence of partial sums representing the probability that the attack sequence terminates converges to a number less than 1. This build therefore has a finite, nonzero probability of an attack sequence not terminating. It has a -chance- of going infinite in the sense that it does not terminate. It -also- has a chance of terminating. This is thus "non-deterministically infinite", and the expected value of the number of attacks you make is undefined.

Suppose that a revised version of this build had a luck feat or something attached to it that allowed the rerolling of natural 1s, and one was attacking on a 2 to hit. Then the probability that the attack sequence does not terminate is 1. This is -also- infinite: it differs from the first case in that it is deterministically infinite.

[OracleofWuffing]

Not exactly...

Hm... Any way we can abuse the wording on this, given that the text says you need to be "using" the weapon, as opposed to "wielding" or "attacking"? I guess you could set up a rube machine mouse trap back at your home that uses an iron or steel weapon, thus freeing up a hand...

"... Since I'm using the steel weapon as a lever back home, my crystal weapon's threat range is increased, and I'll just two-handed power attack for full..."

(Except not power attacking, since we're already taking TWF anyways.)

*Gets Curmudgeoned*

[AgentPaper]

On the contrary, -you- seem not to understand the concept of infinite. Specifically. you're confusing "finite" with "non-deterministically infinite".

As it stands, the sequence of partial sums representing the probability that the attack sequence terminates converges to a number less than 1. This build therefore has a finite, nonzero probability of an attack sequence not terminating. It has a -chance- of going infinite in the sense that it does not terminate. It -also- has a chance of terminating. This is thus "non-deterministically infinite", and the expected value of the number of attacks you make is undefined.

Suppose that a revised version of this build had a luck feat or something attached to it that allowed the rerolling of natural 1s, and one was attacking on a 2 to hit. Then the probability that the attack sequence does not terminate is 1. This is -also- infinite: it differs from the first case in that it is deterministically infinite.

I'm not sure what you're talking about here. To make sure we're talking the same case, we're talking about a situation where every time you make an attack, you have a 40% chance of getting two additional attacks, a 55% chance of getting one additional attack, and a 5% chance of getting no additional attacks. This essentially creates a "chain" of attacks, rolling a certain amount of attacks per step, adding an attack to the next step if you get two attacks, and removing an attack from the next step if you get no attacks.

So, first attack, you have a 5% chance of the chain ending, a 55% chance of continuing with no change, and a 40% chance of adding another attack to the chain. Now, let's say you got average results for the next few hundred rolls, and so for each step of the chain, you are now rolling 200 dice. The chance for each of those rolls to terminate is still 5%. The chance for ALL of them to terminate right now and end the chain is about 6.22x10^-261. That's ridiculously small, but it's still there, so it CAN happen.

Now continue another few hundred thousand steps in this chain. Let's say now you're rolling a few million dice. Each of those rolls STILL has a 5% chance of terminating, so there is STILL a chance that all of them will do so and the chain will suddenly end. That chance is even more ridiculously small, but it IS there, and no matter how high your numbers get, it will always be above 0. For such a chain to result in infinite damage, it would need to continue for an infinite amount of steps. And since each of those steps will always have some chance greater than 0 to stop right there, you have to multiply that chance by infinity. And since even 1x10^-googleplex multiplied by infinity is still infinity, you have a 100% chance of the chain terminating before you reach infinity.

Yes, you can and likely will reach a number so absurdly high that the number of atoms in the universe looks laughable next to it, but that's NOT infinity, and it never will be.

[SilverSheriff]

Don't know if anyone else has covered this or if it is relevent to the discussion, but the 20th level fighter ability 'Weapon Mastery' in Pathfinder gives an auto-confirm on criticals.

[Gpope]

That chance is even more ridiculously small, but it IS there, and no matter how high your numbers get, it will always be above 0. For such a chain to result in infinite damage, it would need to continue for an infinite amount of steps. And since each of those steps will always have some chance greater than 0 to stop right there, you have to multiply that chance by infinity. And since even 1x10^-googleplex multiplied by infinity is still infinity, you have a 100% chance of the chain terminating before you reach infinity.

The problem is that you're trying to multiply an arbitrarily small number by infinity, when the actual situation involves multiplying an infinitely small number by infinity. You're not rolling an infinite number of attempts at 1 in 10^(10^100) odds, you're rolling e.g. an attempt at 1 in 10^(10^101), then an attempt at 1 in 10^(10^102), then at 10^(10^103), and so on. Obviously that's not the exact math, but it illustrates the point.

Intuitively, you might say "So what? There's still always a chance to fail." But because that chance to fail is dropping at such a rapid rate, if you add up the chance to fail for all attacks from 1 to infinity, you end up with a total chance of failure that is less than 100%, which means there is a chance that the chain will truly never end.

[AgentPaper]

The problem is that you're trying to multiply an arbitrarily small number by infinity, when the actual situation involves multiplying an infinitely small number by infinity. You're not rolling an infinite number of attempts at 1 in 10^(10^100) odds, you're rolling e.g. an attempt at 1 in 10^(10^101), then an attempt at 1 in 10^(10^102), then at 10^(10^103), and so on. Obviously that's not the exact math, but it illustrates the point.

Intuitively, you might say "So what? There's still always a chance to fail." But because that chance to fail is dropping at such a rapid rate, if you add up the chance to fail for all attacks from 1 to infinity, you end up with a total chance of failure that is less than 100%, which means there is a chance that the chain will truly never end.

Except it's NOT an infinitely small number. That would be 0, IE no chance of failing. If there's any chance of it failing, it cannot reach infinity. The fact that the chance of failing is decreasing at an exponential rate is irrelevant. There is a chance to fail, so it WILL fail eventually. The only way for it to have a chance at reaching infinity would be for that chance to fail to reach 0%, which it never does.

[Gpope]

Except it's NOT an infinitely small number.

Yes it is, unless you care to show that the chance of X attacks in a row failing has a lower bound other than 0.

[Gan The Grey]

Okay, I'm going to say this about infinity without dropping math equation bombs all over the place.

1. The basic premise of infinity is something that never ends. If there is a problem with this definition, webster it.
2. Infinity is NOT a specific number. It is an idea, and sometimes a state.
3. In order to achieve the ability to continue on for an infinite amount of time, there must be a 100% chance or greater to do so.
4. Infinity either IS or IS NOT. There is no middle ground. Either you have a 100% chance of being infinite, or you are finite.

Fancy math doesn't change this. The only way for this build to gain infinite attacks, is if it never suffers a miss on a natural 1. Even with luck rerolls, this is not possible. It doesn't matter how many attacks it generates, there is still a chance that you will roll 1 freakin godmillion 1's in a row and end your series. As long as that chance exists, you are not infinite.

[AgentPaper]

Yes it is, unless you care to show that the chance of X attacks in a row failing has a lower bound other than 0.

Yes, it approaches 0. It WOULD hit 0, AFTER reaching infinity, except it doesn't reach infinity, because it fails an infinite amount of times before reaching infinity.