The proper hat-guessing number of $K_7-e$

We prove $\mathrm{HG}_P(K_7-e)=12$. One lower-bound proof uses two explicit block-disjoint $S(5,6,12)$ Witt designs. A second uses orbit maps from an explicitly regenerated sharply five-transitive twelve-point permutation group, combining one set-symmetric and one order-sensitive rule. Both satisfy a general coordinate-line twin-completion criterion, and Hall's theorem completes the clique strategy. The release also proves a disjoint completion-design theorem, an even-$n$ obstruction scoped to set-symmetric line-permutation twins in this sufficient framework, and a prime-admissibility theorem for the design parameters. It does not solve the general $K_n-e$ problem or $K_8-e$.

We prove $\mathrm{HG}_P(K_7-e)=12$. One lower-bound proof uses two explicit block-disjoint $S(5,6,12)$ Witt designs. A second uses orbit maps from an explicitly regenerated sharply five-transitive twelve-point permutation group, combining one set-symmetric and one order-sensitive rule. Both satisfy a general coordinate-line twin-completion criterion, and Hall's theorem completes the clique strategy. The release also proves a disjoint completion-design theorem, an even-$n$ obstruction scoped to set-symmetric line-permutation twins in this sufficient framework, and a prime-admissibility theorem for the design parameters. It does not solve the general $K_n-e$ problem or $K_8-e$.