Thanks for the response!

• Regarding Equation G in your paper, it's easy to prove from the limit definition of e I mentioned. Technically your Equation G has another exponent c but this can easily be tacked on due to continuity. In fact, some see the limit of (1+x/n)^n as n approaches infinity to be the very definition of the exponential function, which would make proving Equation G even easier.
• I'm not sure what you mean by p(n) representing average speed for large distances and high fps. I think you mean you can use the squeeze theorem to show that, as the length of the stamina interval approaches 0, both the minimum speed p(n) and the maximum speed approach 0.0261, so the average speed must also approach 0.0261. I agree, but this doesn't mean letting the length of the stamina interval approach 0 gives you the greatest average speed; it just means that it gives you an average speed of 0.0261.
• Regarding the approximation, did you mean to say that the exponentials in the approximation as well as the term (1 − F/s)^n in Equation E will approach 0, not 1?
• Your calculus proofs would still be missing some logic if the differentials were replaced with finite differences using Δ like you suggested. You've gotta use limits somewhere, at least implicitly, to prove things like the chain rule or L'Hôpital's rule. I've never seen these rules proven like this so I'm curious if this is how you learned them.
• I saw the word "near" but you didn't use it for the hypothesis that f(a) = g(a) = 0. Unless you specify that f and g are continuous, their values at a aren't relevant for the proof. When you eventually use the rule, it's true that the functions you use it on are continuous, but your proof of the rule is missing that part of the hypothesis.
• Regarding expressions vs. equations: the only labeled equation that isn't technically an equation is Equation H. I was moreso talking about the unlabeled expressions that you called equations in your writing (e.g. "consider the following equation").

Let me know if I'm coming across as too critical. I have no ill will; I'm just contributing to the discussion.

Hi, I read through your paper. Optimal sprinting was something I looked into experimentally before so it's cool to see someone dive into the math.

I found your paper a little difficult to follow for a few reasons:

• The paper's organization was a bit confusing. You start by talking about CurveValue, then you prove a general result about a type of recurrence relation, then you talk about FPSFactor, then you prove some more general results from calculus in an arbitrary order, then you start an Analysis section where you go back to talking about CurveValue. Your equations are lettered out of order too.
• I understand that you proved those general results for the sake of readers who might not be familiar with them, but I feel like anyone who can understand the rest of your paper is likely already familiar with things like L'Hôpital's rule or geometric series, so I don't think all that explanation is necessary. In particular you spend a lot of time deriving a limit expression for the exponential function. (One common definition of e is the limit of (1+1/n)^n as n approaches infinity. Using this definition could've saved you a page or two.)
• Although you went into detail proving those general results, some of that detail is missing in the actual analysis work of the game. I had to check most of the equations in the Analysis section myself to be able to follow along.
• There's a number of writing errors, inconsistent italicization of variables, use of some variables without introducing them first, use of the same variable for two different purposes, etc.

Nevertheless, after a few hours, I believe I've understood the paper. I agree with the calculations, although I have a couple concerns:

• p(n) doesn't represent the player's average speed across the nth sprinting & walking cycle, but rather the player's speed at the very end of the nth cycle, which is gonna be a local minimum of the player's speed during the process. It may be true that this minimum speed increases as the length of the stamina interval decreases, but what about the average speed? What if you can achieve a greater average speed with a lower minimum speed (i.e. a larger stamina interval)?
• You make a useful approximation by letting the framerate f approach infinity. You say this approximation has negligible error, but this approximation is used twice with every sprinting & walking cycle, and hence infinitely many times as you let the number of cycles approach infinity. How can you be sure this error isn't accumulating with every cycle, throwing off your results?

I also found a handful of inconsequential mistakes:

• In your equations for stamina on pages 12 and 13, you use the variable x without introducing it, but it seems like you meant for it to represent elapsed time. It's really supposed to represent the number of elapsed frames, which you usually express as n, and n = xf, so your equations are missing a factor of f. Consequently, your equation for the depletion/regeneration time of a stamina interval incorrectly includes a factor of f, implying that the depletion/regeneration time is proportional to the framerate, which is clearly false!
• The term p(n) − 2.5 in your first expression for p(n+1) on page 13 should instead be p(n) − 0.045.
• You use differentials like dx quite freely in your proofs of the general calculus results. The use of differentials like this is mostly notational trickery and is not rigorous unless you're working with differential forms. I would instead use the limit definition of a derivative when proving these results.
• One of your hypotheses for your statement of L'Hôpital's rule is that f(a) = g(a) = 0, but this only works if f and g are continuous. What you really need is for the limits of these functions at a to be 0, not the values.
• Things like e^x and f(x)/g(x) are expressions, not equations.

Let me know your thoughts!