  A phenomenological Hamiltonian of a closed (i.e., unitary) quantum system is
assumed to have an $N$ by $N$ real-matrix form composed of a unperturbed
diagonal-matrix part $H^{(N)}_0$ and of a tridiagonal-matrix perturbation
$\lambda\,W^{(N)}(\lambda)$. The requirement of the unitarity of the evolution
of the system (i.e., of the diagonalizability and of the reality of the
spectrum) restricts, naturally, the variability of the matrix elements to a
"physical" domain ${\cal D}^{[N]} \subset \mathbb{R}^d$. We fix the unperturbed
matrix (simulating a non-equidistant, square-well-type unperturbed spectrum)
and we only admit the maximally non-Hermitian antisymmetric-matrix
perturbations. This yields the hiddenly Hermitian model with the measure of
perturbation $\lambda$ and with the $d=N$ matrix elements which are, inside
${\cal D}^{[N]}$, freely variable. Our aim is to describe the quantum
phase-transition boundary $\partial {\cal D}^{[N]}$ (alias exceptional-point
boundary) at which the unitarity of the system is lost. Our main attention is
paid to the strong-coupling extremes of stability, i.e., to the Kato's
exceptional points of order $N$ (EPN) and to the (sharply spiked) shape of the
boundary $\partial {\cal D}^{[N]}$ in their vicinity. The feasibility of our
constructions is based on the use of the high-precision arithmetics in
combination with the computer-assisted symbolic manipulations (including, in
particular, the Gr\"{o}bner basis elimination technique).
