= Solution
The <history-subspace propagation Hamiltonian> has diagonal entries $1/2$ at the endpoints and $1$ in the interior, and nearest-neighbour entries $-1/2$. The <spectrum of a path propagation Hamiltonian> is
$$
\boxed{\lambda_k=1-\cos\left(\frac{\pi k}{T+1}\right),
\qquad k=0,\ldots,T.}
$$
To see the indexing directly, set $q_k=\pi k/(T+1)$ and $f_t=\cos[q_k(t+1/2)]$. The interior difference equation gives $Ef=(1-\cos q_k)f$. The endpoint equations correspond to $f_{-1}=f_0$ and $f_{T+1}=f_T$, both satisfied by these $q_k$. The $T+1$ distinct <eigenvalues> and their nonzero <eigenvectors> exhaust the matrix.
The zero <eigenvalue> is simple, so
$$
\boxed{\Delta(E)=1-\cos\left(\frac{\pi}{T+1}\right)
\geq\frac{2}{(T+1)^2}=\Omega(T^{-2}).}
$$
The inequality uses $\sin x\geq2x/\pi$ on $[0,\pi/2]$ and $1-\cos q=2\sin^2(q/2)$.
The supplied path-matrix hint sums through $i=N$ while listing a basis ending at $N$. That last summand would introduce an extra vertex. The matrix derived in (a) has exactly $T$ edges and $T+1$ vertices, as used here; this also proves the needed spectrum without relying on the misindexed summation.
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