Existence of a codimension epsilon bifurcation set conjecture

Let $f$ be a mapping (neural network) with sufficiently high dimension $d$, and let $U$ be a bifurcation chain set as in the hyperbolicity violation conjecture. The existence of a codimension-$\epsilon$ bifurcation set conjecture asserts the following equivalent statements: (i) in the infinite-dimensional limit, the cardinality of $U$ tends to infinity and $\max_i|a_{i+1}-a_i|$ tends to zero on a one-dimensional parameter interval, so that $U$ is $a$-dense in its closure $\overline{U}$; (ii) in the asymptotic high-dimensional limit, for every $s\in U$ and every $f$ at $s$, an arbitrarily small perturbation $\delta_s$ of $s$ produces a topological change corresponding to a different number of global stable and unstable manifolds for $f$ at $s$ and at $s+\delta$. The claim is presented as a framework for computational verification of hyperbolicity violation, while the supplied text gives no resolution.