**TLDR:** The max is 2,147,483,647 based on how computers store numbers and deduction based on when the Boxing Clever score goes negative. Also, the whole scoring system on that stage is broken and stupid and the game wasn't test played enough to figure that out.

Numbers in a computer are based on increasing powers of two, represented as 1 or 0, instead of the powers of ten based system we use in most other areas of life. Computers need to know how much memory to set aside when storing a number, such as the Boxing Clever score. It is mostly up to the programmer to decide how much memory the computer should allocate, but it also depends on limitations of the system itself and by limitations within a given programming language.

255 is a number that pops up in a lot in games, for example. It can be represented in 8 binary digits as 1111 1111, at least when you don't need to use negative numbers. This is the same as 2^0 + 2^1 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7.

If you need to represent negative numbers, while still being constrained by only having a maximum of 8 digits, the what you do is use the first digit to tell you if the number is positive or negative (0 is positive, 1 is negative). Then you use the remaining 7 digits to give the actual value of the number(the process of negative numbers is a bit more complicated, but that's the gist). Since you are using one of the digits to signify positive/negative, you also cut in half the maximum value you can represent from 255 down to 127, but you can now also represent numbers all the way down to -128 instead of just 0. That's where the 127 comes from with the maximum wave numbers in TS2.

Now, if instead of using a maximum of 8 binary digits, you used 32 binary digits, then you can represent all of the numbers from 0 to 4,294,967,295. If you want to do negative numbers also, that gets cut to -2,147,483,648 to 2,147,483,647. This is where the max value for Boxing Clever comes from.

So what in the world is happening with the crazy negative numbers and positive numbers switching back and forth? Let's say that your currently have an awesome score of exactly 2,147,483,647. In binary this would be: 0111 1111 1111 1111 1111 1111 1111 1111. Notice the first number is a 0, indicating that we have a positive number. Then, let's say you get 1 more point. Thus we add 1 to that binary number and the result will be: 1000 0000 0000 0000 0000 0000 0000 0000. Now, if we weren't using the first digit to indicate positive/negative, then this would be 2,147,483,648, or 2^31. But if we are using the first digit to indicate positive/negative, then by adding 1, we just caused our score to become a negative number. This is exactly what happens in Boxing Clever. The score was programmed as a signed binary number, thus it is able to store negative numbers. Those huge bonuses are causing the signed bit to end up getting flipped back and forth.

The fact that we know exactly what the max is is based on how computers store numbers and based on when we see the scores on boxing clever flip from positive to negative.

Also, the max is (2^31 - 1), or 2,147,483,647, not exactly 2^31. 0 counts as a positive number, so the counting starts from there.

So, the programmers should have either used more memory when storing the score, or gotten rid of a stupid bonus system that increases exponentially per hit.