 Input penalty of a history Hamiltonian

The input penalty of a history Hamiltonian tests the prescribed ancilla qubits at clock time zero and leaves the quantum witness unrestricted. Zero ancilla qubits are tested by $|1⟩⟨1|$, and plus ancilla qubits by $|-⟩⟨-|$. Their sum $Q$ is positive and has positive integer eigenvalues. Together with propagation, its kernel selects histories of valid initial data.