KKR restriction to correct two-qubit propagation

Let ${\mathrm{\Pi }}_1$ project onto $S_{prop1}$ and order the physical work sectors as $00,01,10,11$. The two inserted $Z$ gates after the controlled-$Z$ gate give forward snapshot sign vectors $s_b=(1,1,1),(1,1,-1),(1,-1,-1),(1,-1,1)$ on clocks $t,t+1,t+2$, up to a common phase; the two preceding $Z$ gates give the same four vectors on $t-3,t-2,t-1$. On legal clock words, the forward part of $H_{time,t}$ has matrix $M=\frac{1}{8}\left(\begin{array}{ccc}1& 1& 2\\ 1& 6& 1\\ 2& 1& 1\end{array}\right)$ on its three snapshots, and the backward part is its reflected copy. Direct multiplication gives $s_b^{\ast }M{s}_b=a_b$ with $a_b=(2,1\mathrm{/}2,1\mathrm{/}2,1)$, so the two parts contribute $a_b$ to the two endpoint diagonal entries in physical clock coordinates. There are no cross-endpoint entries from $H_{time,t}$. The work coefficient of $H_{qubit,t}$ is $d_b=(-2,-1\mathrm{/}2,-1\mathrm{/}2,1)$, and its clock bit-flip contributes that coefficient directly to the off-diagonal entries in physical coordinates. The resulting matrices are $a_b$ on each diagonal and $d_b$ off diagonal: they are $4D_{-},D_{-},D_{-},2D_{+}$, respectively. In the block-history basis, the state at the later endpoint includes the controlled-$Z$ phase; in sector $11$ this changes its sign, conjugating $2D_{+}$ to $2D_{-}$. The common block normalization is $1\mathrm{/}L$, so after this basis change every sector dominates one copy of $D_{-}\mathrm{/}L$.

At boundary $t=lL$, the computational basis vectors in $S_{prop1}$ are the normalized block histories $\mathrm{\mid }{\eta }_{l-1,i}⟩,\mathrm{\mid }{\eta }_{l,i}⟩$. The clock endpoint projectors compress as ${\mathrm{\Pi }}_1(I\otimes \mathrm{\mid }{c}_{t-1}⟩⟨c_{t-1}\mathrm{\mid }){\mathrm{\Pi }}_1=L^{-1}{\sum }_i\mathrm{\mid }{\eta }_{l-1,i}⟩⟨{\eta }_{l-1,i}\mathrm{\mid }$ and ${\mathrm{\Pi }}_1(I\otimes \mathrm{\mid }{c}_t⟩⟨c_t\mathrm{\mid }){\mathrm{\Pi }}_1=L^{-1}{\sum }_i\mathrm{\mid }{\eta }_{l,i}⟩⟨{\eta }_{l,i}\mathrm{\mid }$. Since $U_t=CZ$, its cross term compresses to ${\mathrm{\Pi }}_1(CZ\otimes \mathrm{\mid }{c}_t⟩⟨c_{t-1}\mathrm{\mid }){\mathrm{\Pi }}_1=L^{-1}{\sum }_i\mathrm{\mid }{\eta }_{l,i}⟩⟨{\eta }_{l-1,i}\mathrm{\mid }$, with the adjoint giving the reverse block. Substitution into the half-weight propagation edge gives $L^{-1}{\sum }_i{D}_{-}({\eta }_{l-1,i},{\eta }_{l,i})$, where $D_{-}(u,v)=\frac{1}{2}(\mathrm{\mid }u⟩-\mathrm{\mid }v⟩)(⟨u\mathrm{\mid }-⟨v\mathrm{\mid })$. Summing over boundaries gives the displayed $H^{\mathrm{\prime }}$. Its kernel consists exactly of equal adjacent block-history coefficients, i.e. the full history space with every controlled phase correctly propagated. It is a half-weight path Laplacian on $T_2+1$ blocks scaled by $1\mathrm{/}L$, so [F3] gives gap at least $2\mathrm{/}[L(T_2+1{)}^2]$.

Since $H_{prop2}\mathrm{\mid }{S}_{prop1}\ge H^{\mathrm{\prime }}$, it suffices for a lower bound to retain $J_2{H}^{\mathrm{\prime }}$ on $S_{prop1}$. Apply [F2] within that subspace using kernel $S_{prop}$ and penalty gap $J_2{g}_2$. The inequality $J_2{g}_2>2K_2+K_2^2\mathrm{/}{ϵ}_2$ makes the projection error less than ${ϵ}_2$. All quantities are polynomially bounded and $g_2$ is inverse polynomial, so an integer $J_2$ of polynomial size exists.