A graph can be studied through the eigenvalues of its adjacency matrix. Negative square energy adds the squares of just the negative eigenvalues. An AIM problem asks whether this quantity must rise to one peak and then fall as all missing edges are added in a listed order. This anonymous, unrefereed candidate finds the smallest strict failure. A connected graph on five vertices starts with negative square energy five. Adding one edge lowers the energy to about four point six seven six. Adding a second edge raises it exactly back to five. Rational root bounds, not decimal approximations, prove both inequalities. A structural argument rules out every graph on four or fewer vertices. An exact census then finds eight hundred and forty labelled five-vertex valleys in nine relabelling classes. The order quantifier is crucial. The same starting tree also has a different full edge order whose energies rise to six and then decrease steadily. So the candidate refutes the statement that every prescribed order works, but it does not refute the possibility that every graph has some favourable order. Python and JavaScript replay, four mutation tests, internal review, and the supplied full review support the candidate package. They do not establish historical priority, unaffiliated reproduction, formal verification, specialist review, or journal peer review. The scholarly creator is Anonymous. This synthetic-voice briefing is a communication aid, not additional mathematical evidence.