KKR restriction to correctly initialized witness histories

By [F4], orthogonal clock sectors give $R^{\ast }R=(T+1{)}^{-1}{\sum }_t{V}_t^{\ast }{V}_t=I$, so $R$ is an isometry onto $S_{prop}$. The clock factors in $H_{in}$ and $H_{out}$ select only times $0$ and $T$, respectively. Direct multiplication gives $R^{\ast }{H}_{in}R=(T+1{)}^{-1}{\sum }_{a\in A}\mathrm{\mid }1⟩⟨1{\mathrm{\mid }}_a$ and $R^{\ast }{H}_{out}R=V^{\ast }(\mathrm{\mid }0⟩⟨0{\mathrm{\mid }}_{out}\otimes I)V$, agreeing with the energy identities of [F3] and requiring no padding. Under the padding hypotheses, [F1] identifies this same image with the preceding KKR restriction space. The input compression has eigenvalues $r\mathrm{/}(T+1)$, where $r$ counts workspace bits equal to one, so its kernel is exactly the zero-workspace subspace. If $A\mathrm{\ne }\mathrm{\varnothing }$, every positive eigenvalue is at least $1\mathrm{/}(T+1)$; if $A=\mathrm{\varnothing }$, the input operator is zero on all of $S_{prop}$.

If $A\mathrm{\ne }\mathrm{\varnothing }$, the weighted input penalty has nonzero gap $J_{in}\mathrm{/}(T+1)$. Apply [F2] with the compressed output term as $H_1$ and this input penalty as $H_2$; the chosen inequality makes the correction strictly less than ${ϵ}_{in}$. If $A=\mathrm{\varnothing }$, the input penalty is zero, its kernel is all of $S_{prop}$, and no projection error is incurred. On the restricted kernel, [F3] identifies the least output energy with minimum rejection probability over witness states. Completeness $3\mathrm{/}4$ gives energy at most $1\mathrm{/}4$; soundness $1\mathrm{/}4$ gives rejection at least $3\mathrm{/}4$ for every clean witness, so the restricted NO minimum is at least $3\mathrm{/}4$.