Expression for $g(k)$ Related to Waring's Problem

Waring's Problem asks whether, for each positive integer $k$, there exists an integer $s$ such that every positive integer is a sum of at most $k$th powers. While Hilbert proved the existence of such $s$, Waring's Problem has lead to areas of related work, namely the function $g(k)$, which denotes the least such $s$. There is no known general closed form for $g(k)$, though for $g(k)$ has been evaluated for small $k$. Prior work has reduced the problem to verifying a particular condition, which if never occurs, implies an expression for $g(k)$. In this paper, I present a proof the condition never occurs, thus fixing the value of $g(k)$.