Special eccentricities of rational four-dimensional ellipsoids

A striking result of McDuff and Schlenk asserts that in determining when a four-dimensional symplectic ellipsoid can be symplectically embedded into ball, the answer is governed by an "infinite staircase" determined odd-index Fibonacci numbers Golden Mean. Here we study embeddings one another, show if target rational, then infinite staircase phenomenon found quite rare. Specifically, rational case, there precisely three cases -- has "eccentricity" 1, 2, or 3/2; all other given classical volume obstruction except on finitely many compact intervals which it linear. This verifies special case ellipsoids conjecture Holm, Mandini, Pires, author.