Discrete stable self-consistent kernel conjecture

Let $\mathcal{K}$ be the space of kernels, let $\mathcal{S}^*(k)$ be the optimized path entropy for a kernel held fixed at $k$, and let a self-consistent kernel $k^*$ be a fixed point of the MaxCal kernel dynamics. A self-consistent kernel is stable when $D^2_k\mathcal{S}^*|_{k^*}[h,h]<0$ for every nonzero tangent direction $h\in T_{k^*}\mathcal{K}$. Discrete stable-kernel conjecture. The set of stable self-consistent kernels forms a discrete, generically zero-dimensional, subset of $\mathcal{K}$, separated by unstable fixed points that act as transition states between basins of attraction. These basins are the mathematical counterparts of niches, paradigms, and mastery domains; the conjecture concerns the global structure of the fixed points beyond the stated local stability criterion.