Shannon meets Gödel-Tarski-Löb: Undecidability of Shannon Feedback Capacity for Finite-State Channels

We study the exact decision problem for feedback capacity of finite-state channels (FSCs). Given an encoding $e$ of a binary-input binary-output rational unifilar FSC with specified rational initial distribution, and a rational threshold $q$, we ask whether the feedback capacity satisfies $C_{fb}(W_e, π_{1,e}) \ge q$. We prove that this exact threshold problem is undecidable, even when restricted to a severely constrained class of rational unifilar FSCs with bounded state space. The reduction is effective and preserves rationality of all channel parameters. As a structural consequence, the exact threshold predicate does not lie in the existential theory of the reals ($\exists\mathbb{R}$), and therefore cannot admit a universal reduction to finite systems of polynomial equalities and inequalities over the real numbers. In particular, there is no algorithm deciding all instances of the exact feedback-capacity threshold problem within this class. These results do not preclude approximation schemes or solvability for special subclasses; rather, they establish a fundamental limitation for exact feedback-capacity reasoning in general finite-state settings. At the metatheoretic level, the undecidability result entails corresponding Gödel-Tarski-Löb incompleteness phenomena for sufficiently expressive formal theories capable of representing the threshold predicate.