A linear lower bound for generalized doubly nonnegative critical exponents

For every $n\ge3$, the conventional critical exponent of the $n\times n$
generalized doubly nonnegative matrices is at least $2n-4$. The proof uses a
rational, nonnegative cyclic-bidiagonal family with two eigenvalues of order
one and $n-2$ positive eigenvalues of order $\epsilon$. A particular entry of
its real powers has a negative leading coefficient throughout the interval
$(2n-5,2n-4)$. Together with the published upper bound, this proves
$\mathrm{CE}_4=4$, which is Conjecture 4.6 of Han, Johnson and Paparella.
Equality $\mathrm{CE}_n=2n-4$ for all $n\ge5$ is not proved. Exact rational
interval certificates for eight finite examples accompany the analytic proof.