  The ellipsoid embedding function of a symplectic manifold gives the smallest
amount by which the symplectic form must be scaled in order for a standard
ellipsoid of the given eccentricity to embed symplectically into the manifold.
It was first computed for the standard four-ball (or equivalently, the complex
projective plane) by McDuff and Schlenk, and found to contain the unexpected
structure of an "infinite staircase," that is, an infinite sequence of
nonsmooth points arranged in a piecewise linear stair-step pattern. Later work
of Usher and Cristofaro-Gardiner--Holm--Mandini--Pires suggested that while
four-dimensional symplectic toric manifolds with infinite staircases are
plentiful, they are highly non-generic. This paper concludes the systematic
study of one-point blowups of the complex projective plane, building on
previous work of Bertozzi-Holm-Maw-McDuff-Mwakyoma-Pires-Weiler, Magill-McDuff,
Magill-McDuff-Weiler, and Magill on these Hirzebruch surfaces.
  We prove a conjecture of Cristofaro-Gardiner--Holm--Mandini--Pires for this
family: that if the blowup is of rational weight and the embedding function has
an infinite staircase then that weight must be $1/3$. We show also that the
function for this manifold does not have a descending staircase. Furthermore,
we give a sufficient and necessary condition for the existence of an infinite
staircase in this family which boils down to solving a quadratic equation and
computing the function at one specific value. Many of our intermediate results
also apply to the case of the polydisk (or equivalently, the symplectic product
of two spheres).
