Hello, I confess, I'm only an undergraduate so please bear with me. How would one formally/mathematically describe the following? I have a binary string which I split into $k$ substrings of equal length. I define $x$ to be the number of $0$s required for a substring to be declared as 'valid'. so if my substring $sub$ has exactly $x$ $0$s, it's valid. I want to look at the sequences of solely invalid substrings, add up their sum of zeroes together, for each sequence of invalid substring. I want to find the first sequence such that this sum is $\geq x$. I am doing all this to find within that…

@AlessandroCodenotti here's some intuition based on Riemann surfaces and algebraic geometry (or just complex analysis if you want)
you can think of elements in $\Bbb Z$ as functions on $\mathrm{Spec}(\Bbb Z)$, similarly you can think of elements in $\Bbb Q$ as meromorphic functions on $\mathrm{Spec}(\Bbb Z)$. a rational number $\frac{a}{b}$ (completely reduced form) has a "pole" at a prime $p$ if $p$ divides $b$, think about it as analogous to rational functions $\Bbb C(z)$ the p-adic valuation just gives you the order of the pole or the zero at $p$