Unary clock legality and 5-locality

A binary word has no adjacent occurrence of $01$ exactly when its bits are nonincreasing from left to right: if a zero occurred before a later one, the first subsequent transition from zero to one would be adjacent; conversely a word $1^t{0}^{T-t}$ has no such transition. Thus the common zero space of all adjacent projectors is spanned by the legal unary words. On a basis word, $H_{clock}$ is diagonal with eigenvalue the number of forbidden adjacent pairs, so every illegal word has eigenvalue at least one.

The stated propagation windows contain three clock bits and at most two work bits; each input/output term couples one clock and one work bit, and each clock penalty couples two clock bits, proving the support bounds from [F1]. Let $S$ be the legal-clock subspace. Then $H_{clock}$ vanishes on $S$ and is at least $I$ on $S^{\perp }$. Applying [F2] with $H_1=H_0$, $H_2=J_{clock}{H}_{clock}$ gives additive error at most $K^2\mathrm{/}(J_{clock}-2K)<g\mathrm{/}4$. For polynomial resources choose the integer $J_{clock}=\lfloor 2K+4K^2\mathrm{/}g\rfloor +1$. It strictly exceeds the threshold and is at most $2K+4K^2\mathrm{/}g+1$, hence is polynomially bounded when $K$ and $g^{-1}$ are. The resulting error is strictly below $g\mathrm{/}4$, preserving a positive gap at that scale.