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\begin{definition}  Fix $n,m \in \Nats$. $n$.  A \emph{growth space} $\Gamma$ of type $(n,m)$ rank $n$  is a set of functions $\gamma: \Nats^n \rightarrow \Nats^m$ \Nats$  s.t. \begin{enumerate}[(i)]  \item If $\gamma_1, \gamma_2 \in \Gamma$ then $\gamma_1 + \gamma_2 \in \Gamma$.  \item If $\gamma_1 \in \Gamma$, $\gamma_2: \Nats^n \rightarrow \Nats^m$ \Nats$  and $\forall K \in \Nats^n: \gamma_2(K) \leq \gamma_1(K)$ then $\gamma_2 \in \Gamma$. \end{enumerate}