A 3-regular counterexample to the Bilu--Linial signing conjecture

We construct a finite connected simple cubic graph $F$ such that every signing of its edges yields an adjacency matrix with an eigenvalue outside $[-2\sqrt{2},2\sqrt{2}]$. This disproves the Bilu–Linial signing conjecture for general regular graphs. The proof is elementary, using a four-vertex calculation and a scalar recurrence.