= Solution
For an arbitrary vector $|\Psi\rangle=\sum_{t=0}^T|\eta_t\rangle|t\rangle$, the preceding positive <quadratic forms> show that zero energy requires
$$
\eta_t=U_t\eta_{t-1}\quad(1\leq t\leq T),\qquad Q\eta_0=0.
$$
Thus all its clock components are generated by one correctly initialized input. If
$$
|a\rangle=|0\rangle_{A_0}\otimes|+\rangle_{A_+},
$$
then $\ker Q$ consists of $|\zeta\rangle\otimes|a\rangle$, with unrestricted <quantum witness> $\zeta$. Normalization gives the <computational history state>
$$
\boxed{|\operatorname{hist}(\zeta)\rangle=\frac1{\sqrt{T+1}}\sum_{t=0}^TV_t(|\zeta\rangle\otimes|a\rangle)|t\rangle.}
$$
Conversely, every such <computational history state> annihilates every propagation term and the input term, hence has zero energy. \b[These states are exactly the <ground space>], not merely examples of its vectors. In particular, histories of <computational basis> <quantum witnesses> span the <ground space>; their coherent superpositions are also zero-energy states.
Another way to see the propagation constraint is to conjugate by $W=\sum_tV_t\otimes|t\rangle\langle t|$. Then $W^\dagger H_{\rm prop}W=I\otimes E$, where
$$
E=\frac12\sum_{t=0}^{T-1}(|t\rangle-|t+1\rangle)(\langle t|-\langle t+1|).
$$
The <spectrum of a path propagation Hamiltonian> has a unique uniform clock zero mode. This is the path with $T+1$ vertices. The upper summation index in the printed auxiliary formula must be $T-1$, rather than $T$, to agree with its stated dimension and spectrum; using $T$ would introduce an extra vertex.
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