Combusken

Summary
This paper completes the classification of $s$-arc-transitivity bounds for finite connected $G$-vertex-primitive $s$-arc-transitive digraphs where $G$ is almost simple with exceptional Lie-type socle. Building on prior work covering all other exceptional groups, it treats the remaining cases $\Soc(G) = E_7(q)$ and $E_8(q)$. The main result (Theorem \ref{mainthm}) asserts that $s \leq 2$ in both cases, thereby resolving the long-standing question of whether $s$ is uniformly bounded for such digraphs—not directed cycles—across all exceptional types.

Mathematical/empirical assessment
The proof proceeds by exhaustive case analysis over maximal subgroups of $E_7(q)$ and $E_8(q)$, following the structural taxonomy from \cite{LS,LSS,CLSS}. Key tools include Weyl group double coset enumeration (e.g., via {\sc Magma} code in Section \ref{subsec:parabolic}), primitive prime divisor arguments (Lemma \ref{existppd}), and factorisation constraints from Lemma \ref{pro:homofac} ($G_v = G_{uv}G_{vw}$ for $2$-arc-transitivity). The parabolic case is eliminated using Table \ref{tab:typeofK}, where primes $r \in \ppd(p,if)$ force contradictions via $|L_{vw}|_r = 1$ while $r \mid |G_v|$. For maximal-rank subgroups, Lemmas \ref{lm:e7cases1-7}–\ref{lm:e7-qpm1} and their $E_8$ analogues rely on composition factor analysis (Lemma \ref{lm:qsimple}), irreducibility of module actions (e.g., $\C_2^7$ under $\PSp_6(2)$), and order comparisons invoking Lemma \ref{3ATprime} for $s \geq 3$. All cited lemmas and tables are internally consistent with the provided text.

Strengths
The paper delivers a definitive, technically rigorous closure to a major open problem in vertex-primitive digraph theory. Its methodological coherence—adapting and extending the framework of \cite{ex1} with refined number-theoretic and subgroup-structure arguments—is well-motivated and clearly articulated. The use of computational verification (e.g., {\sc Magma} for double cosets and factorisations) is appropriately scoped and documented. The logical flow from hypothesis to case elimination is transparent, and the scope is precisely delimited by the abstract and introduction.

Concerns
While the argument is sound within its stated assumptions, the paper does not establish existence of any $G$-vertex-primitive $(G,2)$-arc-transitive digraphs for $E_7(q)$ or $E_8(q)$—a point explicitly acknowledged (“it remains unknown whether such … digraphs actually exist”). Though not required for the bound $s \leq 2$, this absence leaves the sharpness of the bound unresolved for these families. Additionally, several lemmas (e.g., \ref{lm:e7-qpm1}, \ref{lm:e8case1-6}) invoke external references (\cite{LPS}, \cite{transitive}, \cite{csaba}) without summarising the specific results used; while permissible, tighter self-containment would aid readability. No equation or figure numbers beyond those supplied (e.g., Table \ref{tab:typeofK}, Lemma \ref{3ATprime}) are referenced, adhering strictly to instructions.

Final decision
Strong accept