Probabilities of causation (PoCs) have evolved from Pearl's original binary measures to the recent multi‑valued framework studied by Shu et al., finding uses in personalized medicine, marketing and legal attribution. Shu et al. derived closed‑form bounds for multi‑valued treatments and outcomes and proved their validity, yet left open whether these bounds are sharp in high dimensions. This work supplies a rigorous mathematical proof that the bounds are indeed sharp for any dimension. Moreover, we show that $\text{PNS}$, $\text{PSub}$, $\text{PRep}$ and $\text{PN}$ are all special cases of a single, more general formula, and we exploit their equivalence classes to unify them. Consequently, sharp bounds for discrete PoCs from experimental and observational data are given in closed form by one theorem, applicable to arbitrary numbers of treatment and outcome levels.$$\text{SharpBound}=g(\text{experimental data},\text{observational data})$$
Review: The paper closes a key theoretical gap by proving the optimality of multi‑valued causation bounds and presents a unified expression that streamlines inference across diverse applications.