Deterministic separation on two explicit tridiagonal classes

For a Hermitian tridiagonal $T$ and $0<\delta<1/2$, the note gives
deterministic constructions, on two explicit classes, of a real diagonal $D$
with $\|D\|_2\le\delta$ that separates the eigenvalues of $T+D$. Write
$T=\operatorname{diag}(t_i)+R$. If the absolute off-diagonal row sums of $R$ are at
most $\delta/[2(n-1)]$, sorting the diagonal and adding an arithmetic ramp of
displacements gives $\operatorname{gap}(T+D)\ge\delta/(n-1)$ in
$O(n\log n)$ operations, by a quadratic-form eigenvalue perturbation bound.
The second class has a constant diagonal and off-diagonal entries of common
modulus with arbitrary complex phases, and so contains every Hermitian
tridiagonal Toeplitz matrix. For it, an $O(n)$ algorithm compares the
coupling with $\delta/(4n)$: weak coupling uses the ramp, and strong coupling
returns $D=0$, relying on the explicit sine spectrum. This gives
$\operatorname{gap}(T+D)\ge\delta/n^3$ at every coupling strength. Every
$2\times2$ input is also treated. General tridiagonal inputs remain outside
these classes.