A Bicategorical Theory of Non-Idempotent Negation

This preprint introduces a bicategorical framework in which negation is modeled as a non-idempotent oplax modality. We define directed negation as a non-invertible 2-cell from identity to an endofunctor and study its coherence failure. A concrete rewriting-based model is given, and we show that such structures cannot be reduced to classical idempotent or monadic semantics.