Non-Idempotent Oplax Endofunctors and Factorization Instability in Higher Category Theory

We introduce a class of non-idempotent oplax endofunctors on bicategories equipped with non-invertible 2-cells. Unlike classical modalities in higher topos theory and homotopy type theory, these structures do not stabilize under iteration and do not admit coherent multiplication. We show that such endofunctors are invisible under 1-categorical truncation, cannot be modeled in (∞,1)-categorical semantics, and exhibit a systematic failure of factorization stability. A concrete realization is given via free rewriting 2-categories, providing a canonical example of a non-stabilizing higher-categorical construction.