The proper hat-guessing number of $K_8-e$

We prove $\mathrm{HG}_P(K_8-e)=14$ using an explicit $\mathrm{PGL}(2,13)$-equivariant strategy. A compact formula fixes 990 normalized twin decisions, and a 48,510-edge residual-orbit matching induces consistent rules on 53,460 labelled clique-view orbits. Independent implementations regenerate the certificate and check all 138,378,240 proper fourteen-colourings with zero failures. Unlike the $n=5,6,7$ constructions, residual right degrees reach eight, so the explicit global matching is load-bearing. A separately labelled companion exhibits one fixed clique completion compatible with $2^{380}$ equivariant twin-rule pairs and proves 380 optimal within the stated independent whole-tail reversal model; that companion's independent review is pending. The release does not solve the general $K_n-e$ family or $K_9-e$.

We prove $\mathrm{HG}_P(K_8-e)=14$ using an explicit $\mathrm{PGL}(2,13)$-equivariant strategy. A compact formula fixes 990 normalized twin decisions, and a 48,510-edge residual-orbit matching induces consistent rules on 53,460 labelled clique-view orbits. Independent implementations regenerate the certificate and check all 138,378,240 proper fourteen-colourings with zero failures. Unlike the $n=5,6,7$ constructions, residual right degrees reach eight, so the explicit global matching is load-bearing. A separately labelled companion exhibits one fixed clique completion compatible with $2^{380}$ equivariant twin-rule pairs and proves 380 optimal within the stated independent whole-tail reversal model; that companion's independent review is pending. The release does not solve the general $K_n-e$ family or $K_9-e$.