Rowmotion is a dynamical operation on labels attached to a partially ordered set. Many important families are periodic: after finitely many steps, every generic labelling returns to where it started. This anonymous, unrefereed candidate gives a smallest possible obstruction for two standard lifted versions of rowmotion. The poset has four elements. Two elements sit below a top element, and one of those has a further element below it. Its two maximal chains have different lengths. For birational rowmotion, the proof constructs a positive algebraic fixed point using the positive root t of t to the ninth minus t minus one. If the map had finite order, its derivative at that fixed point would also have finite order, forcing a coefficient field into a totally real cyclotomic field. Exact irreducibility and real-root calculations show that the field instead has eight nonreal embeddings, a contradiction. Piecewise-linear rowmotion has a separate rational fixed point. Its local derivative contains a nontrivial Jordan block at minus one, which also cannot occur for a finite-order map. Minimality is global, not just a search among rooted trees. The verifier enumerates all labelled strict partial orders on zero through three elements, quotients them by relabelling, and finds exact birational periods for all nine isomorphism classes. Subtraction-free tropicalisation transfers those finite-order identities to the piecewise-linear maps. Five deliberately corrupted certificates are rejected. The underlying four-element Hasse diagram is not claimed as new: it already appears, with the opposite orientation, in earlier work by Grinberg and Roby. A supplied review independently recalculated the central argument, but the reviewer's identity and unaffiliated status were not authenticated. Producer replay, public availability and review repair are not independent validation, formal proof, specialist review or peer review. This synthetic-voice briefing is a communication aid, not additional mathematical evidence.