Zero-error realizable CMI conjecture

Let $\mathcal{X}$ be a domain, let $\mathcal{W}$ be a class of functions $f:\mathcal{X}\to\{0,1\}$ with VC dimension $d\geq 1$, and let $n\geq 1$. For a dataset $(x,y)\in(\mathcal{X}\times\{0,1\})^n$, write $\ell(h,(x,y))$ for the empirical $0$-$1$ loss, and let $\mathsf{CMI}(A)$ denote the conditional mutual information of an algorithm's output. Zero-error realizable CMI conjecture. The realizable conjecture above holds with $c'=0$; equivalently, there exists an algorithm with $\mathsf{CMI}(A)\leq c d$ whose expected empirical loss is zero on every realizable dataset.

This strengthens the realizable conjecture by requiring exact consistency rather than an $O(d/n)$ error bound. It is stated as a further conjectural refinement of the proposed CMI guarantee.