{"id":"133a2f74-fd87-434e-9f52-c2a03b31e595","arxiv_id":"2607.14603","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every almost simple group with socle E7(q) or E8(q), vertex-primitive s-arc-transitive digraphs satisfy s≤2, completing the exceptional-group case of the Giudici–Xia problem.","lead":"This paper proves that if a highly symmetric directed graph is built from an almost simple group of type E7 or E8 acting primitively on vertices, the graph can be at most 2-arc-transitive—it cannot have chains of arcs of length 3 or more. It is the final exceptional-group case in a program to bound this symmetry parameter, confirming the expected bound for all exceptional groups of Lie type.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proof's exhaustiveness depends on unpublished Craven E7(q) classification and unshipped Magma computations; a missing non-almost-simple subgroup could yield s≥3.","rationale":"The reader's weakest_assumption identifies the same load-bearing concern: the completeness of the maximal-subgroup enumeration, especially the unpublished Craven classification [12] for E7(q). This is indeed the most critical dependency because a missing non-almost-simple subgroup would not be handled by Lemma 2.10(a) and could potentially support a homogeneous factorization, undermining the s≤2 conclusion. The heavy reliance on unshipped Magma computations compounds the risk, and Remark 3.9 demonstrates that even published tables in this area have had errors. However, no internal mathematical contradiction was found; the reduction from s≥3 to homogeneous factorizations is sound, and the individual lemmas are argued in detail. The appropriate verdict remains CONDITIONAL, pending verifiability of the external classification and computational artifacts. My read does not change the reader's verdict.","tokens_in":36282,"tokens_out":13017,"duration_ms":126756,"concrete_test":"Obtain the authors' Magma scripts and data and independently recompute the finite-group claims (e.g., the factorizations of PSp6(2) in Lemma 3.7, the q=3 case in Lemma 3.6, and the subgroup searches in Lemmas 3.13–3.14). In parallel, cross-check the E7(q) maximal-subgroup list of §3.3.1 against the published/updated version of Craven's classification (arXiv:2201.07081) or an independent verification; confirm that every maximal subgroup in [12] is either covered by cases (i)-(xi), is almost simple (hence bounded by Lemma 2.10(a)), or falls into cases (a)-(e). The theorem stands only if both checks pass.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The single load-bearing assumption is the exhaustiveness of the maximal-subgroup enumeration for E7(q). Theorem 1.1's s≤2 conclusion rests on proving, for every possible core-free maximal subgroup G_v, that the homogeneous factorization G_v = G_uv G_vw required by (G,2)-arc-transitivity cannot occur (or that s≤2 via Lemma 2.10(a)). For L=E7(q), the non-parabolic, non-maximal-rank maximal subgroups are taken verbatim from [12], an unpublished 2022 arXiv preprint by Craven (cases (i)-(xi) in §3.3.1). If [12] omits a maximal subgroup of product type (e.g., a product of two or more simple groups), that subgroup is not covered by Lemma 2.10(a) (which only bounds almost simple G_v) and could in principle support a factorization G_v = G_uv G_vw, giving s≥3. The same case analysis depends on dozens of Magma computations (Lemmas 3.6, 3.7, 3.13, 3.14, 3.20, et al.) with no scripts, logs, or data files, so errors cannot be audited. The paper's own Remark 3.9 shows that even the published maximal-rank table [20, Table 5.2] contains an incorrect structure, which was only fixed by an unverifiable Magma computation; this underscores the risk of relying on unpublished or computationally-verified sources without artifacts.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that if G is an almost simple group with socle E7(q) or E8(q) and Γ is a G-vertex-primitive (G,s)-arc-transitive digraph, then s≤2, and it lists the possible vertex stabilizers for (G,2)-arc-transitive examples. The main strategy, following the authors' companion paper, is to use the fact that s≥3 forces G_v=G_uvG_vw for a 2-arc u→v→w, and then to rule out, by a case analysis over the maximal subgroups of E7(q) and E8(q), every possible vertex stabilizer. Parabolic cases are handled by a Weyl-group double-coset argument combined with primitive prime divisors; maximal-rank cases use the Liebeck–Saxl–Seitz classification and a series of lemmas; the remaining cases depend on the classifications in [12], [10], [11], [18], together with numerous Magma computations. The paper also presents Table 6 listing the remaining possible vertex stabilizers across all exceptional groups.","tokens_in":36599,"tokens_out":16594,"duration_ms":164620,"significance":"If correct, Theorem 1.1 is a major result: it completes the solution of Giudici–Xia's boundedness question for all finite almost simple exceptional groups of Lie type, reducing the answer to the finite list in Table 6. The proof is not based on fitting parameters or circular use of the target theorem; it imports prior published lemmas from [31] and [32], uses Zsigmondy's theorem, and gives explicit number-theoretic contradictions in many cases. The explicit Table 6 is a useful falsifiable output. The main structural weakness is that the exhaustiveness of the case analysis rests on external classifications (including an unpublished preprint) and on dozens of unarchived Magma computations, so the proof cannot currently be independently audited. These are correctness-risk concerns rather than internal inconsistencies.","major_comments":[{"comment":"The unconditional statement of Theorem 1.1 depends on the completeness of the maximal-subgroup list for E7(q). The non-parabolic, non-maximal-rank cases are taken verbatim from [12], an unpublished 2022 arXiv preprint. The line “By [12] and [17]” is load-bearing: if [12] omits a maximal subgroup whose F*(L_v) is not almost simple (for instance a product-type subgroup), Lemma 2.10(a) does not apply, and such a subgroup could in principle admit the homogeneous factorization L_v=L_uvL_vw that would give s≥3. Since neither the contents of [12] nor a proof of the needed part is reproduced, the theorem as stated is not checkable from this manuscript. The authors should either use a published version of [12], supply a proof or a verifiable table of the maximal subgroups used, or make the theorem conditional on [12].","section":"§3.3.1"},{"comment":"The proof relies on many Magma computations, with no scripts, logs, or output files included. The computations decide nonexistence of factorisations and irreducibility of modules, which are exactly the steps that would break if a computational error occurred. Because this is a classification-style proof with an exhaustive case analysis, the computations need to be reproducible; otherwise the reader cannot distinguish a verified case from a black box. The concern is amplified by Remark 3.9, where a published structure in [20, Table 5.2] is corrected by a Magma computation: computational artefacts have already changed a group structure used in the proof.","section":"§3.1, Lemmas 3.6–3.27"},{"comment":"The step from “S=PSL2(q)” to “we only need to consider PSL2(23)” is not argued. Lemma 2.10(a) rules out PSL2(q) only when the additional “almost simple” clause applies to G_v; for cases (vi) and (vii) this is not automatic from the displayed group structure. The subsequent discussion of R(G_v)=2 for q=23 shows that non-almost-simple vertex stabilizers are considered possible. The text should explain for which q the extension by Out(L) centralizes the PSL2(q) factor, why all other q are covered by the almost-simple clause, and why no other exceptional q arises. As written, this is a gap in a step that is necessary for the s≤2 conclusion.","section":"§3.3.1, Lemma 3.22"}],"minor_comments":[{"comment":"In the final paragraph, “Now, we deal with case (8)” should be “case (10)”; the lemma covers cases (9)–(12).","section":"Lemma 3.12"},{"comment":"Case (3) is titled “q is even and q is not a Mersenne prime,” but the paragraph immediately treats q=5 and q=9, which are odd. The title should say “q is odd and not a Mersenne prime” (or equivalent).","section":"Lemma 3.23"},{"comment":"The expressions “with the index 24 or 23” and “with the index 24” should be “index 2^4 or 2^3” and “index 2^4” respectively; the current text is confusing.","section":"Lemmas 3.6 and 3.14"},{"comment":"In the q=3 paragraph, “Now H=L” should presumably be “Now G=L” (since q=3 forces G=L); H is a small subgroup SL2(3)^4, not the whole group.","section":"Lemma 3.14"},{"comment":"References [22], [23], and [30] appear in the bibliography but do not seem to be cited in the text; please check and either cite them or remove them.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"This is a strong paper that, if fully checked, would complete a significant programme. My concern is not the internal logic but the unverifiable dependence on an unpublished classification and unshipped Magma computations. I would be willing to accept after the authors provide a reproducible computational appendix and either update the status of [12] or prove the needed classification cases."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is the last brick in the Giudici–Xia exceptional-group program, and the argument looks right in broad strokes. But the E7(q) branch is built on an unpublished classification, and the proof leans on dozens of Magma checks that no one can audit. That is fixable, but it has to be fixed before I would bet on the result.\n\nThe genuinely new content is real. E7(q) and E8(q) were the two open exceptional socles, and the paper closes them with a uniform reduction: s≥3 forces a homogeneous factorization of the vertex stabilizer, and then each maximal subgroup class is eliminated. The parabolic argument via Levi-factor factorizations and primitive prime divisors is a clear improvement over Part I, and the maximal-rank analysis for E8 is substantially deeper. Table 6, listing possible 2-arc-transitive vertex stabilizers across exceptional groups, is a useful reference on its own. No circularity: the reused lemmas from Part I are published, and the paper is honest about what it inherits.\n\nThe soft spots are about provenance, not logic. For E7(q), the non-parabolic non-maximal-rank maximal subgroups come verbatim from Craven's 2022 arXiv preprint [12]. The authors do not hide this, but the preprint is unpublished and unversioned. If that list misses a class — say a product-type subgroup that is not almost simple — Lemma 2.10(a) no longer controls it, and the proof of s≤2 has a hole. The stress-test worry here is not manufactured; it is exactly what a referee should press.\n\nThe second issue is computational auditability. Dozens of lemmas rely on “computation in Magma shows...” with no scripts, logs, or data files. I do not doubt the authors ran the computations, but a referee cannot verify them. Remark 3.9 makes the problem vivid: the published Liebeck–Saxl–Seitz maximum-rank table has a wrong structure, and the fix is a Magma computation the reader cannot reproduce. That is not an error in the paper, but it is a warning that trust is doing real work.\n\nThere are minor internal slips—for example, Lemma 3.23 Case (3) says “q is even” and then treats q=5,9—cosmetic, but worth fixing.\n\nI would send this to a serious referee, with instructions that the referee may condition acceptance on public access to Craven's classification and on the authors posting Magma scripts or logs. As it stands, the result is important and probably true, but it is not independently checkable.","headline":"Last two exceptional socles handled, and the proof is probably right, but the E7(q) branch rests on an unpublished classification and unshipped Magma checks.","tokens_in":37112,"tokens_out":3367,"would_cite":true,"duration_ms":35089,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20B25","20D06","05C25"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that every vertex-primitive s-arc-transitive digraph admitting an almost simple automorphism group with socle E7(q) or E8(q) has s ≤ 2, completing the boundedness question for all exceptional groups of Lie type.","keywords":["s-arc-transitive digraph","vertex-primitive digraph","almost simple group","exceptional group of Lie type","E7(q)","E8(q)","maximal subgroups","homogeneous factorization"],"falsifier":"Find a core-free maximal subgroup H of E7(q) or E8(q) outside the lists used in the paper that admits a homogeneous factorization H = AB with A ≅ B and |H : A| ≥ 3, where the element g with (u,v)^g = (v,w) normalizes no nontrivial normal subgroup of H; such a factorization would produce a G-vertex-primitive (G,2)-arc-transitive digraph and could lift to s ≥ 3. Equivalently, constructing any connected G-vertex-primitive (G,3)-arc-transitive digraph with socle E7(q) or E8(q) would refute Theorem 1.1.","tokens_in":36136,"feed_emoji":"🕸️","tokens_out":7798,"duration_ms":69682,"temperature":0.7,"pith_summary":"For digraphs, unlike graphs, the arc-transitivity parameter s can be arbitrarily large; the open question is whether vertex-primitive examples—digraphs whose automorphism group acts primitively on the vertex set—force s to be small. This paper settles the last open exceptional-group cases: if an almost simple group with socle E7(q) or E8(q) acts vertex-primitively on a connected s-arc-transitive digraph, then s ≤ 2, and any 2-arc-transitive example must have a vertex stabilizer taken from an explicit list. Together with earlier work on the other exceptional Lie-type groups, this answers the boundedness question for every exceptional group. A sympathetic reader would care because it sharply limits where highly symmetric primitive digraphs can exist.","feed_headline":"E7 and E8 socles cap arc-transitivity at 2","feed_subtitle":"The theorem completes the boundedness question for vertex-primitive digraphs across all exceptional Lie-type groups.","key_machinery":"The load-bearing object is the homogeneous factorization condition G_v = G_uv G_vw—a factorization by two isomorphic subgroups—which 2-arc-transitivity imposes on the vertex stabilizer. The argument couples this with a reduction lemma bounding the 2-part of |L_v| by |L_uv|^3 times the order of the outer automorphism group, and uses primitive prime divisors (primes dividing q^n − 1 but no earlier q^i − 1) to isolate a normal simple factor. The arc-shift element g satisfies (u,v)^g = (v,w); Lemma 2.5 says g normalizes no proper nontrivial normal subgroup of G_v, which is what turns 'one factor contains a whole simple factor' into a contradiction. The proof is an exhaustive case analysis over t","core_discovery":"The central claim is Theorem 1.1: let G be an almost simple group with socle E7(q) or E8(q), and let Γ be a G-vertex-primitive (G,s)-arc-transitive digraph; then s ≤ 2. Moreover, if Γ is (G,2)-arc-transitive and the socle is any simple exceptional group of Lie type, the vertex stabilizer L_v must be one of the groups assembled in Table 6, ranging from small groups such as A6 and M12 to product and torus-type subgroups. The proof assumes s ≥ 2 and exploits the homogeneous factorization G_v = G_uv G_vw that 2-arc-transitivity forces, then runs through every core-free maximal subgroup of E7(q) and E8(q), organized into parabolic, maximal-rank, and remaining almost simple cases. In each family,","pith_inferences":["A natural next test is to run the same factorization machinery against the remaining almost simple classical groups, where the boundedness question is not yet settled by this method; the E7/E8 proof supplies reusable prime-divisor arguments.","The explicit Table 6 stabilizer list suggests a targeted computational search: for each listed L_v, test whether a homogeneous factorization L_v = AB exists with index at least 3 and with the normalizer-freeness condition; existence would produce the first primitive 2-arc-transitive digraphs with E7/E8 socle, while nonexistence would strengthen the conclusion to full non-existence.","Because the proof leans on an unpublished E7 maximal-subgroup classification and on many computer checks without reproducible scripts, an independent verification of those inputs is the main thing that would change confidence in the theorem—though this is an editorial observation, not a flaw identified by the paper.","The parabolic-subgroup argument appears to be new and more direct than earlier approaches; it may apply to other Lie-type groups where parabolic cases were previously handled only by lengthy computation."],"forward_implications":["For every connected vertex-primitive digraph admitting an almost simple exceptional group of Lie type, s-arc-transitivity stops at s = 2; in particular, no such digraph is 3-arc-transitive.","Any 2-arc-transitive example with E7(q) or E8(q) socle has its vertex stabilizer in Table 6, so the search for actual examples is reduced to a finite list of stabilizer groups.","The boundedness question for vertex-primitive digraphs is answered affirmatively for all exceptional groups of Lie type when combined with earlier work.","Maximal parabolic vertex stabilizers are eliminated entirely for 2-arc-transitivity: in that case s ≤ 1.","Whether any 2-arc-transitive digraphs with E7(q) or E8(q) socle actually exist remains open; the result constrains but does not construct them."],"fun_headline_variants":["E7 and E8 socles bound s-arc-transitivity to 2","Exceptional groups: vertex-primitive digraphs have s ≤ 2","All exceptional groups now answered: s ≤ 2 for primitive digraphs","Vertex-primitive digraphs: E7(q)/E8(q) socles limit s to 2","Complete exceptional group result: s ≤ 2 for vertex-primitive digraphs"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The argument depends on the completeness of the maximal-subgroup classifications it cites—including an unpublished E7(q) list and one characteristic-3 E8(q) exceptional subgroup—and on the correctness of many computer-algebra factorization checks; if any maximal subgroup class were missing, a vertex stabilizer could exist that the proof never considers, and the s ≤ 2 conclusion would not follow.","fun_headline_variants_meta":{"raw":{"variants":["E7 and E8 socles bound s-arc-transitivity to 2","Exceptional groups: vertex-primitive digraphs have s ≤ 2","All exceptional groups now answered: s ≤ 2 for primitive digraphs","Vertex-primitive digraphs: E7(q)/E8(q) socles limit s to 2","Complete exceptional group result: s ≤ 2 for vertex-primitive digraphs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001949,"raw_usage":{"total_tokens":7468,"prompt_tokens":762,"completion_tokens":6706,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":506,"completion_tokens_details":{"reasoning_tokens":6603}},"tokens_in":506,"tokens_out":6706,"duration_ms":47113,"temperature":1.0,"reasoning_tokens":6603,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T01:36:31.512416+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a core-free maximal subgroup H of E7(q) or E8(q) outside the lists used in the paper that admits a homogeneous factorization H = AB with A ≅ B and |H : A| ≥ 3, where the element g with (u,v)^g = (v,w) normalizes no nontrivial normal subgroup of H; such a factorization would produce a G-vertex-primitive (G,2)-arc-transitive digraph and could lift to s ≥ 3. Equivalently, constructing any connected G-vertex-primitive (G,3)-arc-transitive digraph with socle E7(q) or E8(q) would refute Theorem 1.1.","supporting_citations":[],"review_version":1}