The proper hat-guessing number of $K_5-e$

The exact value is $\mathrm{HG}_P(K_5-e)=8$. The lower bound uses explicit legal twin-player rules over $\mathbb F_2^3$. After those rules cover 3,024 of the 8,400 proper colorings, the residual-coloring/local-view incidence graph has left degree three and right degree at most three; Hall's theorem supplies consistent guesses for the three clique players. The release also proves a general sufficient twin-completion lemma for $K_n-e$. It does not solve the full $K_n-e$ family or determine $\mathrm{HG}_P(C_5)$.

The exact value is $\mathrm{HG}_P(K_5-e)=8$. The lower bound uses explicit legal twin-player rules over $\mathbb F_2^3$. After those rules cover 3,024 of the 8,400 proper colorings, the residual-coloring/local-view incidence graph has left degree three and right degree at most three; Hall's theorem supplies consistent guesses for the three clique players. The release also proves a general sufficient twin-completion lemma for $K_n-e$. It does not solve the full $K_n-e$ family or determine $\mathrm{HG}_P(C_5)$.