A uniform disk bound for diagonal Padé approximants of the wave-kernel function

Let $r_m$ be the diagonal $[m/m]$ Padé approximant at the origin to
$\cosh\sqrt z$. We prove that $r_m$ exists, has no poles in $|z| \le 3$, and
satisfies $|r_m(z) - 1| \le 2$ there, for every $m \ge 0$. The bound is
strict for $m \ge 2$ and is attained for $m = 1$ at $z = 3$. The large-order
argument follows from a Schur-function formula for the denominator: its $j$th
coefficient in absolute value is bounded by the $j$th power of the tail sum
of the reciprocals of the zeros of the entire function. The orders
$1 \le m \le 15$ are covered by Nadukandi and Higham's lemma for $m \le 20$
and, independently, by exact-rational certificates. This proves, for every
order, the disk bound conjectured by Nadukandi and Higham.