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In the history basis, $M(s)=(1-s)D+sE$ is a Hermitian tridiagonal matrix. Its first diagonal entry is $s/2$, its interior entries are $1$, its last entry is $1-s/2$, and each off-diagonal entry is $-s/2$.

The Gershgorin disc theorem confines the first row's eigenvalue to $[0,s]$, the interior rows to $[1-s,1+s]$, and the last row to $[1-s,1]$. For $0\le s\le 1/3$, the first disc is disjoint from all the others and therefore contains exactly one eigenvalue. Since $M(s)$ is positive semidefinite, that eigenvalue is the ground state energy. All other eigenvalues are at least $1-s$, while the lowest is at most $s$. Hence the Gershgorin gap bound for an adiabatic history path givesThis argument also covers $s=0$, where the first disc is the isolated point zero.