= Gap above a degenerate history ground space
{title2=$\Delta\geq(T+1)^{-3}$}
After undoing <quantum circuit> propagation, the Hamiltonian without output penalty is $Q\otimes|0\rangle\langle0|+I\otimes E$. The common kernel is the valid input space times the uniform clock vector. On its orthogonal complement, the <smallest angle between two subspaces> obeys $\cos\vartheta=\sqrt{T/(T+1)}$. The <Kitaev geometrical lemma> and path gap $2/(T+1)^2$ give $\Delta\geq(T+1)^{-3}$. Degenerate valid <quantum witnesses> must be removed together when computing this angle.
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