On the recognition problem for limits of entropy functions

We prove that there is no algorithm to decide whether a given integer vector is in the closure of the entropic cone $\overline{\Gamma_{n}^{*}}$. Equivalently, there is no decision procedure to determine whether a given integer-valued function $h:\mathcal{P}(\{1,\ldots,n\})\rightarrow\mathbb{Z}_{\ge 0}$ is a pointwise limit of joint entropy functions. In other words, given such an $h$, it is undecidable whether for all $\varepsilon > 0$ there exists a finite probability space $(\Omega,P)$ with random variables $X_{1},\ldots,X_{n}$ such that their joint entropy $H$ satisfies $\max_{I\subseteq\{1,\ldots,n\}}\left|H\left(X_{I}\right)-h\left(I\right)\right|<\varepsilon$. This settles the last open case in a sequence of related undecidability results proved by L. K\"{u}hne and the author, with applications in algorithmic information theory. The main new tool is a Desargues'-type theorem for almost entropic polymatroids.