KKR restriction to legal unary clocks

The sum is diagonal in the computational clock basis. It has eigenvalue equal to the number of pairs $i<j$ with bit $i=0$ and bit $j=1$. This number is zero exactly when no zero precedes a one, which is equivalent to a word $1^t{0}^{T-t}$; every other word has at least one such pair and hence eigenvalue at least one. Thus its zero space and gap are as stated.

Apply [F1] with $S=S_{legal}$, $H_1=H_0$ and $H_2=J_{clock}{H}_{clock}^{KKR}$. The spectral penalty on $S^{\perp }$ is at least $J_{clock}$, while it vanishes on $S$. The chosen strict bound gives $J_{clock}-2K>4K^2\mathrm{/}g$, and so the correction $K^2\mathrm{/}(J_{clock}-2K)<g\mathrm{/}4$. For polynomial resource bounds, choose specifically $J_{clock}=\lfloor 2K+4K^2\mathrm{/}g\rfloor +1$, using a supplied rational norm bound $K$. This is strictly above the threshold and at most $2K+4K^2\mathrm{/}g+1$. If $K$ and $1\mathrm{/}g$ are polynomially bounded and their encodings have polynomial length, this choice has polynomial magnitude and bit length, and the $O(T^2)$ term list has polynomial size; no size bound is claimed for arbitrary larger admissible integers.