Epistemic Decay Rates in Iterated Knowledge Structures: A Category-Theoretic Perspective with Applications to Cybersecurity

Iterated statements of knowledge-constructions of the form "an agent knows that another agent knows"-arise naturally in modal logic, distributed systems, and game-theoretic security. While such constructions are formally iterable within standard logical frameworks, their semantic clarity and operational usefulness often degrade rapidly with increasing depth. In this paper, we investigate this phenomenon through a category-theoretic lens, introducing the notion of an epistemic decay rate (DK rate) to characterize the loss of coherence in higher-order knowledge structures. By modeling epistemic states as objects in a category of pointed coalgebras, we demonstrate that for finite state spaces, the filtration of epistemic distinctions must stabilize, leading to inevitable saturation. Focusing on four canonical configurations involving observers and objects, we show that beyond a small, structurally determined number of iterations, additional layers cease to contribute actionable information. Although our analysis is mathematical rather than psychological, it yields a rigorous justification for the "bounded horizons" observed in adversarial settings such as cybersecurity.