is the inverse of the lowest effect along the prefixes of $\gamma$. We call $1\le d\le k$ a *nadir* of $\gamma$ if it is the index of a prefix that attains the depth of $\gamma$. That is, $\effect{s_1,\ldots, s_d}=-\depth{\gamma}$. We say that $\gamma$ is *positive* if $\effect{\gamma}$ is positive (and similarly for negative, non-negative, zero, etc.). We call $\gamma$ *good* if it a simple, non-negative cycle, and $\depth{\gamma}=0$.

\[obs:shifted cycle\] If $\gamma$ is non-negative and it has a nadir $d$, then the *shifted cycle* $\gamma^{\leftarrow d}\eqdef s_d\, s_{d+1},\cdots , s_k,s_2,\cdots ,s_d$ is good. Similarly, if $\gamma$ is negative, then $\effect{\gamma^{\leftarrow d}}=-\depth{\gamma^{\leftarrow d}}$.

For a state $r\in\states$ and an initial configuration $\initialstate,c_0$, let $\LangVia{\initialstate,c_0}{r} \subseteq \Lang{\initialstate,c_0}$ be the language of words accepted by a run that visits $r$.

The first tool we use in studying the universality problem is a canonical form for accepting runs, akin to *linear path schemes* of [@LS2004; @BFGHM15].

\[def:linear path scheme\] A path $\pi$ is in *linear form* if there exist simple cycles $\gamma_1,\ldots,\gamma_k$ and paths $\tau_0,\ldots,\tau_{k}$ such that $\pi=\tau_0\gamma_1^{e_1}\tau_1\cdots \tau_{k-1}\gamma_k^{e_k}\tau_k$ for some numbers $e_1,\ldots,e_k\in\N$, and such that every non-negative cycle $\gamma_i$, is taken from a nadir, and so is executable with any counter value.

We 