Identifying a low-entanglement eigenstate inside a many-body spectrum and preserving it during measurement are distinct tasks. We construct an explicit local ring Hamiltonian with an exact cluster-state eigenvector and study continuous monitoring of its stabilizer defects. For arbitrary mixed inputs, the conditional cluster fidelity is the initial target weight divided by the no-observed-click probability. A positive defect-operator gap gives finite-time bounds that hold for noncommuting Hamiltonian dynamics, nonnormal effective generators and imperfect detection. At fixed total monitoring rate, the guaranteed exponent falls inversely with system size; high conditional fidelity does not remove the preparation cost set by the initial overlap. Exact diagonalization up to eleven qubits gives finite-size evidence for a cluster-state outlier in a chaotic spectral background. Independent matrix and trajectory calculations verify the dynamics and a conservative coherent-error bound. This construction specializes established scar embedding and nondemolition verification frameworks, with explicit measurement assumptions, finite-time guarantees and resource limitations.