The sharp height-three cutoff for P-partition uniqueness

For naturally labeled posets of Greene shape (3,2,2,1^k), the P-partition generating function distinguishes all orders up to isomorphism exactly when k is at most four. At k=4 an exhaustive enumeration using a four-element marked core gives 143,069 isomorphism classes, separated by three exact specializations. Common padding by isolated vertices transfers the lower bound to smaller k. A twelve-element pair gives equality at k=5. The cutoff is presented as a candidate theorem.