Eigenvalue spectrum conjecture for type $j,k$ saddle-node ghosts

Let $\dot{x}=f(x,\rho)$, with $x\in\mathbb{R}^n$, $f\in\mathcal{C}^\omega$, and parameters $\rho=(\rho_1,\dots,\rho_m)^T\in\mathbb{R}^m$. Suppose that $x_g\in\mathcal{A}$ is a ghost of a type $j,k$ saddle-node $x_{sn}$ at $\rho_{\textnormal{crit}}$. For each $i$ with $1\leq i\leq j$, there is a unit vector $u_i\in\mathbb{R}^n$ such that, in a neighborhood of $x_g$, the eigenvalues $\lambda_i$ of $D_xf(x,\rho)$ along the line

$$\{x\in\mathcal{A}\mid x_g+tu_i,\ t\in\mathbb{R}\}$$

vary continuously and change sign from negative to positive, with

$$\lambda_i(t)\approx ct,$$

where $c\in\mathbb{R}$, to leading order. Eigenvalue spectrum conjecture. Every type $j,k$ saddle-node ghost has, along each of $j$ suitable directions, a continuously varying eigenvalue that changes sign from negative to positive and is linear in the direction parameter to leading order. This conjecture formalizes the observed linear spatial distribution of instantaneous eigenvalues along the former center directions of the parental saddle-node. It is proposed as an algorithmically identifiable criterion for detecting and characterizing higher-dimensional ghosts; the supplied text does not establish its resolution.