{"id":"264603f9-fc29-4277-a15a-19f65e2eedc8","arxiv_id":"2508.19515","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A block-transitive t-(k^2,k,lambda) design with socle PSL(2,q), q>=4, can exist only for q=8, and then it is a 2-(36,6,lambda) design with lambda in {2,6,9,12,18,36}.","lead":"This paper classifies block-transitive t-designs whose number of points is the square of the block size and whose automorphism group contains the group PSL(2,q). It finds that only q=8 can occur, and in that case the designs are 2-(36,6,lambda) designs; the authors list all such designs computed with GAP.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1 is false under the paper's own definition of non-trivial: the complete 2-(9,3,7) design with PSL(2,8) satisfies all stated hypotheses yet violates the conclusion.","rationale":"I read the full manuscript. The arithmetic reductions in Section 3 are mostly coherent, and the GAP-based constructions in Section 4 are plausible though not independently verifiable. The decisive problem is internal and explicit: the paper defines non-trivial as t<k<v, but in Section 3.3 Case (1) it dismisses the resulting complete design as 'trivial' without reconciling the terminology. That complete design is a direct counterexample to Theorem 1.1 as stated, since it satisfies every hypothesis but not the conclusion. This single point is more load-bearing than the external reliance on [10, Theorem 1] or the unverified computational tables, because it does not depend on outside results. The reader's verdict of CONDITIONAL is exactly right: the central claim is salvageable by adding 'not the complete design' to the definition, but without that clarification the theorem is false. I therefore agree with the reader and recommend no change to the verdict.","tokens_in":19076,"tokens_out":6689,"duration_ms":56843,"concrete_test":"Check the counterexample directly: take P to be a 9-set, B to be all 3-subsets, and G=PSL(2,8) acting naturally. Verify (i) t=2<k=3<v=9, so the design is non-trivial per Section 1; (ii) G is block-transitive (indeed 3-transitive); (iii) X=PSL(2,8) ⊴ G = Aut(X) (with G equal to X or PΓL(2,8)); and (iv) the conclusion of Theorem 1.1 fails because v=9, k=3, and the point stabilizer has order 56. This settles that the theorem as written is false unless the definition of non-trivial is amended to exclude complete designs.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper defines a design as non-trivial when t<k<v (Section 1). In Section 3.3, Case (1), the analysis of the maximal subgroup C_2^3⋊C_7 yields q=8, k=3, v=9 for X=PSL(2,8). The authors then write: 'Thus B includes all 3-subsets of P. Therefore, D is trivial, a contradiction.' This is the load-bearing gap: the design consisting of all 3-subsets of a 9-set is the complete 2-(9,3,7) design. It satisfies t=2<k=3<v=9, hence is non-trivial under the paper's own definition. Since PSL(2,8) is 3-transitive on 9 points, it acts transitively on the set of all 3-subsets, so the design is block-transitive. Every hypothesis of Theorem 1.1 is met, yet the conclusion fails: X_α is a 2-group of order 56, not D14, and D is a 2-(9,3,7) design, not a 2-(36,6,λ) design. The contradiction is obtained only by silently switching to a different meaning of 'trivial' (i.e., 'complete'), which is never introduced. Thus, as stated, Theorem 1.1 is false. The classification may well be correct when complete designs are excluded, but the statement and Section 3.3 Case (1) must be amended to do so explicitly.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper classifies block-transitive t-(k^2,k,lambda) designs whose automorphism group G contains X=PSL(2,q), q>=4, with X normal in G and G<=Aut(X). Theorem 1.1 claims that any such nontrivial design must have q=8, point stabilizer X_alpha isomorphic to D14, and parameters 2-(36,6,lambda); Theorem 1.2 then lists the possible lambda values and explicit base blocks obtained by GAP for PSL(2,8) and PGammaL(2,8). The proof proceeds by a maximal-subgroup case split based on the point-primitivity reduction imported from [10], with arithmetic eliminations in Sections 3.1-3.3 and computational construction in Section 4.","tokens_in":19300,"tokens_out":16937,"duration_ms":159902,"significance":"If the intended classification is repaired to exclude complete designs, this would be a useful contribution: it completes the PSL(2,q) case of the block-transitive t-(k^2,k,lambda) program, gives explicit base blocks and lambda lists, and agrees with the previously known flag-transitive 2-(36,6,lambda) examples. The arithmetic eliminations are mostly coherent, there are no fitted parameters, and the GAP tables are a valuable resource. However, the main theorem as stated is false because of a complete design that satisfies the paper's own definition of nontrivial, so the positive assessment is conditional on a substantial correction to the statement.","major_comments":[{"comment":"The treatment of the q=8, k=3, v=9 case is not a contradiction under the paper's own definition. In Section 1, a design is called non-trivial when t<k<v. Let P be the 9 points of PG(1,8) and let B consist of all 3-subsets of P. This is the complete 2-(9,3,7) design, and t=2<k=3<v=9, so it is non-trivial by the paper's definition. Since PSL(2,8) (and every group between it and PGammaL(2,8)) is 3-transitive on the 9 points, the group is transitive on all 3-subsets, so the design is block-transitive; it also satisfies X=PSL(2,8) <= G <= Aut(X) with q=8>=4. Thus all hypotheses of Theorem 1.1 hold. The conclusion fails: the point stabilizer has order 504/9=56, namely 2^3*7, not D14, and the design is not a 2-(36,6,lambda) design. The sentence 'Therefore D is trivial, a contradiction' silently uses a second, undefined notion of triviality meaning 'complete design'. Theorem 1.1 is therefore false as stated; either the definition must explicitly exclude complete designs or Theorem 1.1 must include this exceptional complete 2-(9,3,7) design.","section":"Section 3.3, Case (1); Section 1"},{"comment":"In Lemma 2.6(3), the subgroup D_{q+1} is listed for all odd prime powers q=p^f with q not equal to 7 or 9. However, the Case (3) paragraph begins 'Since q is an odd prime' and then uses the subdegree data from [7, Table 2]. If that table applies to all odd prime powers, the word 'prime' should be replaced by 'prime power' and the source should be quoted for that range; if the table applies only to prime q, then q=25,27,49,... are not eliminated and the exhaustive case split is incomplete. Please state the exact validity range of the subdegree data and adjust the hypothesis of this case accordingly.","section":"Section 3.2, Case (3); Lemma 2.6"}],"minor_comments":[{"comment":"The point-primitive reduction is imported from [10, Theorem 1], and the entire maximal-subgroup split rests on it; please state the precise hypotheses of that theorem and confirm that its notion of nontrivial design matches the definition used in this paper.","section":"Section 3, first paragraph"},{"comment":"The GAP computation is described by commands and tables, but no GAP code, log files, or verification scripts are included, and the data availability statement says only that data are available on reasonable request; for verifiability, please attach the code and output as ancillary files.","section":"Section 4"},{"comment":"In Table 3, row number 74 is duplicated and row 75 is skipped; please renumber the rows.","section":"Appendix, Table 3"},{"comment":"There are several typos and notational issues: the title lacks a space ('Block-transitivet'), 'Magam' in Lemmas 4.1 and 4.3 should be 'Magma', and the notation D14:2 in Lemma 4.3 should be defined or accompanied by its order.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The main theorem as printed is false, but the intended result is likely recoverable by explicitly excluding complete designs; the repair is local, so I recommend major revision rather than rejection. The GAP constructions and explicit tables are a substantive strength, but the authors should supply the scripts and should correct the erroneous treatment of the complete 2-(9,3,7) design. The heavy reliance on [10], which is by one of the present authors, should also be checked editorially for overlap and for exact hypothesis matching."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Thanks for the stress-test note. I read the paper carefully, and the reader's take is essentially right: the classification of non-complete block-transitive designs with socle PSL(2,q) is a solid piece of work, but the main theorem as stated is false.\n\nWhat is new: Montinaro-Francot and Guan-Zhou handled flag-transitive cases and alternating/sporadic socles; this paper closes the PSL(2,q) family for block-transitive actions and all lambda. The arithmetic eliminations in Section 3 are coherent: the case split follows the maximal subgroup classification, and the subdegree divisibility arguments check out. The GAP constructions in Section 4 are a real contribution: 46 designs for PSL(2,8) and 330 for PΓL(2,8), with the four known flag-transitive designs reappearing. No code or certificates are included, so I couldn't independently verify the enumeration, but the tables are plausible and match prior results where they overlap.\n\nThe soft spot is exactly where the stress-test lands. The paper defines non-trivial as t<k<v. In Section 3.3, Case (1), the analysis yields q=8, k=3, v=9, and the design contains all 3-subsets of the 9 points. The authors call this 'trivial' and discard it. But under their own definition, 2<3<9, and PSL(2,8) is 3-transitive on 9 points, so the complete 2-(9,3,7) design is a block-transitive, non-trivial design satisfying every hypothesis of Theorem 1.1. Its point stabilizer has order 56, not D14, and the design is 2-(9,3,7), not 2-(36,6,λ). So Theorem 1.1 is false as written. The fix is easy: either explicitly exclude complete designs in the definition, or state the convention that 'trivial' means complete and adjust Theorem 1.1 accordingly. The classification of non-complete designs probably survives unchanged.\n\nA minor second concern: the reduction to point-primitivity is imported from Guan-Zhou [10] and not re-proved here. That is a legitimate reliance on prior work, but it means the whole argument rests on that theorem. The self-citation is not a problem per se.\n\nWho is this for? Design theorists working on block-transitive t-design classifications. It deserves a serious referee: the methods are standard, the gap is localized and fixable, and the computational data is worth having. I would send it to review with a request for a revised statement and, ideally, GAP code or verification certificates.\n\nRecommendation: accept with major revision, conditional on the complete-design issue being resolved.","headline":"A useful classification for a subfield, but Theorem 1.1 is false as stated: the complete 2-(9,3,7) design is non-trivial under the paper's own definition and violates the conclusion.","tokens_in":19926,"tokens_out":3714,"would_cite":false,"duration_ms":30493,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05B05","05B25","20B25"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that every non-trivial block-transitive t-(k^2,k,λ) design whose automorphism group contains PSL(2,q) must have q=8 and be a 2-(36,6,λ) design.","keywords":["t-designs","block-transitive automorphism groups","projective special linear groups PSL(2,q)","point-primitive reductions","2-(36,6,λ) designs","maximal subgroups","coset space design construction"],"falsifier":"A disproof would be a single block-transitive t-($k^{2}$,k,λ) design with automorphism group G satisfying PSL(2,q) ≤ G ≤ Aut(PSL(2,q)) for some q≠8, or with q=8 and λ outside the listed sets. A reader could run an exhaustive orbit search on the coset action of PSL(2,16) or PSL(2,32), taking all k-subsets whose size squared equals the index, and check whether any yields a design; if one exists, Theorem 1.1 is false.","tokens_in":18778,"feed_emoji":"📐","tokens_out":9679,"duration_ms":76734,"temperature":0.7,"pith_summary":"This paper aims to classify non-trivial block-transitive t-($k^{2}$,k,λ) designs whose automorphism group G contains the projective special linear group PSL(2,q) with G inside Aut(PSL(2,q)), for q≥4. The main theorem asserts that in every such design the group parameter must be q=8, the point stabilizer is the dihedral group D14, and the design is a 2-(36,6,λ) design. If the theorem is correct, the infinite family collapses to a finite, explicit list: 46 designs for G=PSL(2,8) with λ∈{2,6,12}, and 330 designs for G=PΓL(2,8) with λ∈{2,6,9,12,18,36}. The interest is that this gives a complete structural description of one whole infinite family of block-transitive t-designs, with every possibility written down in tables.","feed_headline":"Every block-transitive t-(k^2,k,λ) design with PSL(2,q) has q=8","feed_subtitle":"Forces q=8, reducing the infinite family to 2-(36,6,λ) designs and listing all possible λ values.","key_machinery":"The load-bearing mechanism is the point-primitive reduction: an imported theorem ensures that a block-transitive automorphism group of a t-($k^{2}$,k,λ) design acts primitively on points, so each point stabilizer is a maximal subgroup. The point set is identified with the coset space G/G_α, giving v=|G:G_α|, and the design parameter equations together with a subdegree divisibility lemma force k+1 to divide certain subdegrees. The case analysis then runs through the complete list of maximal subgroups of an almost simple group with socle PSL(2,q), using index computations and divisibility to eliminate every q except 8; for q=8, a computer search over all 6-subsets of a 36-point set up to the group action produces the full design lists.","core_discovery":"The paper's central claim is that the purely numerical constraints `k+1 | n` on subdegrees and `k+1` divisibility involving the outer automorphism group, combined with the maximal-subgroup classification for groups with socle PSL(2,q), leave only one admissible case. Concretely, Theorem 1.1 states that a non-trivial t-($k^{2}$,k,λ) design with a block-transitive group G satisfying X=PSL(2,q) ≤ G ≤ Aut(X), q≥4, must have q=8, X_α ≅ D14, and the design must be a 2-(36,6,λ) design. The proof rules out every other maximal subgroup of PSL(2,q), including the projective line stabilizer, the dihedral subgroups, the subfield subgroups, and the exceptional A4, S4, and A5 cases. For the surviving case q=8, the authors carry out a computational orbit search and list all designs up to isomorphism: the λ values are {2,6,12} for G=PSL(2,8) and {2,6,9,12,18,36} for G=PΓL(2,8), with four flag-transitive examples matching an earlier construction.","pith_inferences":["If the same point-primitive reduction holds for other simple socles, the identical divisibility machinery could be applied to families such as classical groups of Lie type; the main missing ingredient would be the analogous maximal-subgroup lists.","The explicit tables suggest a testable extension: an independent orbit-counting verification of the 46 and 330 designs could confirm that the computer search was exhaustive, since the paper does not display the search code.","The forced value q=8 may hint at a general phenomenon that block-transitive t-(k^2,k,λ) designs with simple socle concentrate at very small prime powers; testing the next admissible groups directly would show whether the divisibility constraints alone already force this concentration."],"forward_implications":["Every non-trivial block-transitive t-(k^2,k,λ) design with PSL(2,q) ≤ G ≤ Aut(PSL(2,q)) and q≥4 has v=36, k=6, and t=2, so higher-strength designs do not arise in this family.","For G=PSL(2,8), the only possible λ values are 2, 6, and 12, realized by exactly 46 designs up to isomorphism.","For G=PΓL(2,8), λ can only be 2, 6, 9, 12, 18, or 36, realized by exactly 330 designs up to isomorphism.","No non-trivial design of this type exists for any q≠8, so the full automorphism groups PSL(2,q) contribute only the single parameter value q=8.","The four flag-transitive 2-(36,6,λ) designs admitted by PGL(2,8) reappear in the new lists, confirming the earlier construction and embedding it in the complete classification."],"supporting_citations":[{"why":"Supplies the reduction theorem that block-transitive automorphism groups of t-(k^2,k,λ) designs are point-primitive, and the divisibility lemmas (k+1 | n and the outer automorphism bound) used throughout.","marker":"[10]"},{"why":"Provides the classification of maximal subgroups of almost simple groups with socle PSL(2,q), which is the backbone of the case analysis in Section 3.","marker":"[9]"},{"why":"Gives the subdegrees of the permutation representations of these groups, used to apply the k+1 divisibility condition.","marker":"[7]"},{"why":"Documents the 3-transitive action of PSL(2,8) on 9 points, used to eliminate the complete-design case q=8, k=3.","marker":"[5]"},{"why":"The computer algebra system used to generate the permutation representation and enumerate all base-block designs for q=8.","marker":"[8]"},{"why":"Earlier construction of four flag-transitive 2-(36,6,λ) designs for PGL(2,8), which the present classification recovers.","marker":"[13]"},{"why":"Block's theorem that block-transitive automorphism groups are also point-transitive, a preliminary used to set up the reduction.","marker":"[2]"},{"why":"Used to compute the index |G:G_α| as |X|/|X∩G_α| in the odd-characteristic case analysis.","marker":"[1]"}],"fun_headline_variants":["PSL(2,q) block-transitive t-designs forced to q=8","q=8 is the only admissible case for these t-designs","Theorem: All block-transitive t-(k^2,k,λ) designs have q=8","Classification reduces to 2-(36,6,λ) designs with listed λ","Only q=8: PSL(2,q) block-transitive designs classified"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the imported theorem that a block-transitive automorphism group of a t-($k^{2}$,k,λ) design is point-primitive, because the exhaustive maximal-subgroup case split in Section 3 covers all possibilities only if that theorem holds; the argument also relies on the convention that 'non-trivial' means t<k<v, which is what discards the complete 2-(9,3,7) design in the q=8, k=3 case.","fun_headline_variants_meta":{"raw":{"variants":["PSL(2,q) block-transitive t-designs forced to q=8","q=8 is the only admissible case for these t-designs","Theorem: All block-transitive t-(k^2,k,λ) designs have q=8","Classification reduces to 2-(36,6,λ) designs with listed λ","Only q=8: PSL(2,q) block-transitive designs classified"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000311,"raw_usage":{"total_tokens":1757,"prompt_tokens":919,"completion_tokens":838,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":535,"completion_tokens_details":{"reasoning_tokens":730}},"tokens_in":535,"tokens_out":838,"duration_ms":6835,"temperature":1.0,"reasoning_tokens":730,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:51:54.219135+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A disproof would be a single block-transitive t-($k^{2}$,k,λ) design with automorphism group G satisfying PSL(2,q) ≤ G ≤ Aut(PSL(2,q)) for some q≠8, or with q=8 and λ outside the listed sets. A reader could run an exhaustive orbit search on the coset action of PSL(2,16) or PSL(2,32), taking all k-subsets whose size squared equals the index, and check whether any yields a design; if one exists, Theorem 1.1 is false.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the reduction theorem that block-transitive automorphism groups of t-(k^2,k,λ) designs are point-primitive, and the divisibility lemmas (k+1 | n and the outer automorphism bound) used throughout."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the subdegrees of the permutation representations of these groups, used to apply the k+1 divisibility condition."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Documents the 3-transitive action of PSL(2,8) on 9 points, used to eliminate the complete-design case q=8, k=3."},{"cited_title":"https://www.gap-system.org","cited_arxiv_id":null,"evidence_quote":"The computer algebra system used to generate the permutation representation and enumerate all base-block designs for q=8."},{"cited_title":"Montinaro, E","cited_arxiv_id":null,"evidence_quote":"Earlier construction of four flag-transitive 2-(36,6,λ) designs for PGL(2,8), which the present classification recovers."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Block's theorem that block-transitive automorphism groups are also point-transitive, a preliminary used to set up the reduction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Used to compute the index |G:G_α| as |X|/|X∩G_α| in the odd-characteristic case analysis."}],"review_version":2}