The proper hat-guessing number of $K_6-e$

We prove $\mathrm{HG}_P(K_6-e)=10$. The lower bound uses two order-sensitive twin-player rules obtained by deleting and repairing one point of an explicit sharply four-transitive eleven-point permutation group. On every coordinate line the repaired rules are derangement permutations, are pointwise unequal, and have fixed-point-free composition. Hall's theorem completes the strategy on the four clique vertices. The release also classifies all repairable orbit labels and proves an even-$n$ obstruction for set-symmetric line-permutation twin rules. It does not solve the general $K_n-e$ family.

We prove $\mathrm{HG}_P(K_6-e)=10$. The lower bound uses two order-sensitive twin-player rules obtained by deleting and repairing one point of an explicit sharply four-transitive eleven-point permutation group. On every coordinate line the repaired rules are derangement permutations, are pointwise unequal, and have fixed-point-free composition. Hall's theorem completes the strategy on the four clique vertices. The release also classifies all repairable orbit labels and proves an even-$n$ obstruction for set-symmetric line-permutation twin rules. It does not solve the general $K_n-e$ family.