We Do What We Must Because We Can · 8:59pm Mar 20th, 2020
After reading RQK's Divergence I decided to write a math paper. The result of this endeavor is Time Manipulations in the Crystal Ball Trilogy. I hereby license it under CC BY-SA 4.0. It should really go without saying, but be wary of spoilers both in the paper and possibly even in whatever comments to the blog post others or I may make.
The monograph is primarily intended for those who like analyzing the underlying mathematics of a fanfiction, verifying for consistency, exploring all the consequences that must pertain, and generalizing it as far as can be consistently done. Most of the math is concerned with infinite series and finite products, so anyone familiar with those should have little trouble understanding the paper, but if you have any questions I will try to answer them in the comments.
There is one notational issue I should bring up. Most people who learn about summation notation usually learn it like so
\begin{align}\sum_{i = 0}^{n - 1} a_i = a_0 + a_1 + a_2 + \ldots + a_{n - 1} && \sum_{i = 0}^\infty a_i = a_0 + a_1 + a_2 + \ldots\end{align}with a starting index underneath the capital sigma and an ending index or a lemniscate above the capital sigma. But as a matter of fact, sums can be written like so
\begin{align}\sum_{i < n}a_i = a_0 + a_1 + a_2 + \ldots + a_{n - 1} && \sum_{i < \omega}a_i = a_0 + a_1 + a_2 + \ldots\end{align} by indicating a limit value our index must be less than. In the second case, the \omega is an ordinal representing the least position that comes after every natural.
