A directed fixed-entry criterion

Let the checkerboard vertex matrices $A^-$ and $A^+$ of a real interval
matrix $[L,U]$ be nonsingular sign regular with a common signature. The note
proves that every matrix in $[L,U]$ is nonsingular sign regular with that
signature whenever the fixed nonzero entries ($L_{ij}=U_{ij}\ne0$), oriented
by sign and checkerboard parity, form an acyclic directed bipartite graph
and the fixed zero entries all have the same parity of $i+j$. The proof
scales one vertex by positive diagonal matrices to open the fixed nonzero
entries and applies the same-parity theorem of Adm and Garloff. An
inverse-positive box lemma gives nonsingularity and the signs of minors of
orders $1$, $n-1$ and $n$ for every interval. Exact five-dimensional
examples with zero middle minors illustrate the criterion. Companion notes
control all order-two minors and, for $n=5$, give the full conclusion when
there are no fixed zeros and every fully fixed adjacent $2\times2$ block is
nonsingular.