The quantum pigeonhole effect as a new form of Bell's theorem without inequality

The quantum pigeonhole effect (QPE) appears to contradict the classical pigeonhole principle by allowing three quantum particles distributed between two boxes to exhibit no pairwise coincidence. We show that this effect does not signal a breakdown of classical counting, but instead arises from quantum contextuality. By deriving Bell-type inequalities directly from the pigeonhole principle and reformulating the weak-measurement protocol within a bipartite density-operator framework, we demonstrate that the QPE is a form of Bell's theorem without inequalities. The apparent paradox reflects the impossibility of non-contextual eigenvalue assignments rather than a violation of classical combinatorial logic.