{"id":"c037b22a-722f-4288-89e6-061ecb1e0ef6","arxiv_id":"2504.14931","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Explicit difference families and block lists are given for new point-transitive Steiner systems S(2,6,111), S(2,6,121), S(2,6,126), S(2,7,169), and S(2,7,175), including an exhaustive count of 30 systems on 111 points.","lead":"This preprint reports new examples of Steiner systems, set systems where every pair of points appears together in exactly one block, for five parameter families. It lists explicit block and difference family data, including 30 point-transitive designs on 111 points, although the search algorithm is left for a future paper.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Enumeration and novelty claims rest on an unpublished algorithm and a private nauty reimplementation; without an independent check, the '30 non-isomorphic' count for S(2,6,111) and the assertion that designs are new remain unverified.","rationale":"The reader's weakest-assumption analysis already identifies the unpublished algorithm and private nauty reimplementation as the load-bearing gap. I agree: the printed block lists may well be correct, but the exact count of 30 and the claim that the other designs are previously unknown cannot be checked from the manuscript alone. The paper itself acknowledges the algorithm will be published later, and Section 2 explicitly relies on a private Java version of nauty and a heuristic fingerprint that is not a complete isomorphism invariant. These concerns do not invalidate the data, but they do justify the CONDITIONAL verdict already assigned.","tokens_in":135019,"tokens_out":9360,"duration_ms":93183,"concrete_test":"Independently re-run the v=111 enumeration: fix a concrete action of Z37⋊Z3 on 111 points, implement a separate exhaustive search over all unions of group orbits totalling 407 blocks, verify each candidate with a standard S(2,6,111) checker, and canonical-label the results with standard nauty; compare with the claimed 30 and with Mills's design. Also parse every printed v=111 block list, verify the Steiner property, compute automorphism groups with GAP/nauty, and test pairwise isomorphism to known designs. If the independent count is not 30, or a printed design is absent or isomorphic to a known one, the enumeration and novelty claims should be revised.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is an exact enumeration and a novelty classification. Section 3 states 'All this results are enumerations, so they give exhaustive list of difference families that generate non-isomorphic Steiner systems,' and Section 4 says the S(2,6,111) count was 'confirmed by generalized algorithm which will be published later.' Section 2 describes the isomorphism filter as a 'naive copy of nauty' plus the hyperbolic-frequency fingerprint, with no code or formal specification shipped. The fingerprint is only an invariant, not a complete invariant: the author says same-frequency pairs are 'almost sure' isomorphic and must be checked by nauty, and Example 3.1 even displays two same-fingerprint designs that are non-isomorphic. Thus any pairwise non-isomorphism or 'new with respect to the literature' statement ultimately leans on a private implementation. If the search is incomplete or the isomorphism test miscertifies, the count of 30 and the novelty of designs collapses, even if every printed block list is a valid Steiner system. The manuscript itself flags this by deferring the algorithm description to a future paper.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents explicit difference families and block lists for point-transitive Steiner systems S(2,k,v) with (k,v) = (6,111), (6,121), (6,126), (7,169), and (7,175), together with several smaller examples for k = 3,4,5. The central claim is that for v = 111 there are exactly 30 non-isomorphic S(2,6,111) systems with automorphism group of order 111 isomorphic to Z37⋊Z3, one due to Mills and the remaining ones asserted to be new. The constructions are given as raw data, and the paper also introduces a 'hyperbolic frequency' fingerprint used to distinguish designs. The enumeration is attributed to a generalized difference-family algorithm whose description is deferred to a later paper.","tokens_in":135245,"tokens_out":5260,"duration_ms":53749,"significance":"If the enumeration and isomorphism claims are correct, the paper provides new explicit point-transitive Steiner systems for five parameter sets and determines the exact number of S(2,6,111) systems with a point-transitive automorphism group of order 111. The printed block lists are explicit and machine-checkable, and the frequency fingerprint is a legitimate isomorphism invariant rather than a fitted parameter; these are real strengths. However, the exact-count and novelty claims currently rest on unpublished software and a private reimplementation of nauty, so the significance of the paper depends on an independent reproducibility check.","major_comments":[{"comment":"The central claim of exactly 30 non-isomorphic systems is not independently verifiable from the manuscript. The text says the count was 'confirmed by generalized algorithm which will be published later', and Section 2 describes the isomorphism tool only as 'a naive copy of nauty' in Java, with no code or formal specification. Since the hyperbolic frequency fingerprint is not a complete invariant—Example 3.1 exhibits two same-fingerprint designs that are non-isomorphic—the pairwise non-isomorphism and completeness of the 30-item list depend entirely on this private software. Please provide a reproducible certificate, a complete algorithm description, or an independent verification (for example machine-checked canonical forms) for the S(2,6,111) list.","section":"Section 4, second paragraph"},{"comment":"The blanket sentence 'All this results are enumerations, so they give exhaustive list of difference families that generate non-isomorphic Steiner systems' is contradicted by the manuscript's own Example 3.2, which says that isomorphism was not checked and that some listed designs could be isomorphic. The statement should be restricted to the cases where the fingerprint is proved to be a complete invariant or where a nauty-level isomorphism test is actually run; otherwise the published lists in Sections 3.1–3.4 cannot be read as exhaustive non-isomorphic classifications.","section":"Section 3, opening paragraph"},{"comment":"The assertion 'It is obvious that group Z37 × Z3 can't produce any' is load-bearing for the exhaustiveness of the search, because it is the step that eliminates the abelian group of order 111 and leaves only Z37 ⋊ Z3. Since the paper's headline count claims to be exhaustive, please include a proof or a precise computational argument for this elimination, for example a counting or orbit argument showing that no difference family with the required parameters exists in that group.","section":"Section 4, first paragraph"}],"minor_comments":[{"comment":"The claims connecting fingerprint keys to projective spaces, affine spaces, and unitals are stated without proof or precise definitions; since they are not needed for the main results, please either remove them or provide a reference and a precise statement.","section":"Section 2"},{"comment":"The term 'multiplier-nonisomorphic' is not defined; please define it or rephrase to avoid confusion with non-isomorphic designs, especially because the same passage states that isomorphism was not checked.","section":"Example 3.2"},{"comment":"Citations to [1], [7], and the 'future paper' describing the algorithm are informal; please add precise references or explicitly state that the algorithm is not yet available.","section":"References and citations"},{"comment":"The raw block lists are very long and no validation command, checksum, or verification script is provided; adding a short machine-readable verification script would substantially improve reproducibility.","section":"Section 4"},{"comment":"There are numerous typographical and grammatical errors, such as 'All this results are enumerations', 'others seems to be unknown', 'freequency', and 'alrogithm'; a careful copyedit is needed.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The main obstacle is verifiability rather than an evident mathematical error. The explicit lists may well be correct, but the exact-count claim cannot be accepted without a reproducible certificate or a complete description of the enumeration and isomorphism software."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing you should know: this is a raw data paper, not a conceptual one. Hetman lists explicit difference families / block lists for dozens of point-transitive Steiner systems in five parameter families, including 30 S(2,6,111) designs (one from Mills, 29 claimed new) and what looks like a previously missing S(2,7,175). The printed block lists are the real evidence, and for v=111 they are complete as printed—anyone can check that they define Steiner systems.\n\nWhat the paper does well: it is honest about provenance. Known designs are bolded and cited; the author explicitly asks readers to email if they spot an earlier occurrence. The hyperbolic frequency fingerprint is a legitimate isomorphism invariant, and using it to prefilter before an nauty-style check is sensible engineering. The negative result for Z3×Z3×Z9 and the careful treatment of multiplier groups are useful details.\n\nThe soft spot is the classification claims. The 'exactly 30 non-isomorphic systems' count and the 'new with respect to the literature' assertions rest on an unpublished generalized algorithm and a private Java reimplementation of nauty. The fingerprint is not a complete invariant—Example 3.1 shows two same-fingerprint designs that are non-isomorphic—so the final pairwise checks necessarily rely on code the reader never sees. The author says the algorithm will be published later and that the count was confirmed by it. That is a promissory note, not evidence. If the enumeration missed a design or miscertified an isomorphism, the count of 30 and the novelty of some entries collapse, even though every printed list would still be a valid Steiner system.\n\nThis is a real limitation, but it is not fatal to the paper's value. The constructions are checkable, and the catalogue of small designs is exactly where such raw data belongs. The paper is not going to reshape the field, but it is honest archival progress.\n\nWho benefits: design theorists and anyone maintaining databases of small Steiner systems. I would not bring it to a general reading group, but a specialist would want the data.\n\nRecommendation: it deserves a serious referee, not a desk reject. An editor should send it, but the referee should be asked to verify a sample of the printed constructions and to require the algorithm/code (or at least an independent implementation) before the enumeration count is accepted.","headline":"Honest raw data with checkable constructions, but the headline enumeration claim leans on unpublished code; worth refereeing if the code ships.","tokens_in":135765,"tokens_out":2139,"would_cite":true,"duration_ms":24718,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05B05","05B10","05E18"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims a complete census of 30 point-transitive S(2,6,111) Steiner systems, and new difference families for four further parameter sets.","keywords":["Steiner systems","difference families","point-transitive designs","S(2,6,111)","combinatorial design enumeration","automorphism groups","block designs","isomorphism fingerprints"],"falsifier":"Independently enumerate all difference families in $\\mathbb{Z}_{37}\\rtimes\\mathbb{Z}_3$ for block size 6 using an independent exhaustive method, or run an independent graph-isomorphism check on the thirty printed systems; finding more than 30 non-isomorphic systems, or finding two of the claimed new systems to be isomorphic, would refute the central claim. A cheaper partial check is to verify that every 111-point block list satisfies the Steiner property and that each of the 29 non-first designs is non-isomorphic to the first.","tokens_in":1797,"feed_emoji":"📐","tokens_out":2016,"duration_ms":70705,"temperature":0.7,"pith_summary":"A Steiner system $S(2,k,v)$ is a collection of $k$-element blocks on $v$ points such that every pair of points lies in exactly one block. This paper claims to introduce new Steiner systems $S(2,6,111)$, $S(2,6,121)$, $S(2,6,126)$, $S(2,7,169)$, and $S(2,7,175)$ by writing them as difference families in finite groups, mostly commutative except for the $S(2,6,111)$ case. Most entries are asserted to be new, and the lists are presented as exhaustive enumerations for the searched groups. The headline claim is that for $v=111$ there are exactly 30 non-isomorphic point-transitive Steiner systems with automorphism group $\\mathbb{Z}_{37}\\rtimes\\mathbb{Z}_3$, one previously known and 29 apparently unknown. A reader should care because complete point-transitive systems with block size 6 or 7 are sparse, and these results replace isolated examples with a census.","feed_headline":"Exactly 30 point-transitive S(2,6,111) designs found","feed_subtitle":"New difference families also give S(2,6,121/126) and S(2,7,169/175), filling gaps in known catalogues.","key_machinery":"The load-bearing object is a difference family: a collection of $k$-element blocks inside a finite group such that every non-identity group element occurs the same number of times as a difference of two entries in a block, so that translating by the group makes the resulting Steiner system point-transitive. The search is carried out by a generalization of an earlier cyclic difference-family algorithm to commutative groups, plus a first attempt for the non-commutative case. To certify non-isomorphism, the paper computes a hyperbolic-frequency fingerprint, counting, for non-collinear triples $oxy$ and points $p$ on $xy$, the points $u$ on $oy$ for which the line $pu$ does not meet $ox$; equal fingerprints are followed up with a private reimplementation of graph isomorphism. The multiplier of a difference family supplies a lower bound on the automorphism group of the design.","core_discovery":"On the paper's own terms, the central discovery is that the listed difference families and block lists are valid constructions of Steiner systems with the stated parameters, that they are pairwise non-isomorphic, and that the lists are exhaustive for the groups searched. For $S(2,6,111)$, the paper argues that $\\mathbb{Z}_{37}\\times\\mathbb{Z}_3$ cannot produce a design and that the semidirect product $\\mathbb{Z}_{37}\\rtimes\\mathbb{Z}_3$ is the only remaining point-transitive group; exhaustive search then yields 30 non-isomorphic systems whose full automorphism groups have order exactly 111. The first of these 30 is the known design, and the paper states that the others appear to be unknown. For the remaining parameters, the examples are given as multiplier, hyperbolic-frequency fingerprint, and difference family, with a few marked as already known (for instance a $S(2,7,175)$ design with multiplier 24, which forces $4200$ to divide the automorphism-group order). The paper is explicit that the generalized search algorithm proving exhaustiveness will be published later, so the constructions themselves are the content of this paper.","pith_inferences":["A natural testable extension is to rerun the same enumeration on the next admissible parameter sets with $k=6$ or $k=7$; the paper's method suggests that many more point-transitive systems will appear than currently catalogued.","The hyperbolic-frequency invariant deserves independent analysis: if its distinguishing power can be characterized theoretically, it may serve as a structural certificate rather than merely a practical filter.","If the count of 30 for $S(2,6,111)$ is independently verified, that family becomes a rare complete classification of point-transitive Steiner systems with a small non-abelian automorphism group, and a useful source of examples for questions about group realizability.","Because the paper marks certain entries as possibly known, a careful reconciliation with published catalogues could change which of the 30 designs are considered new, though the count of isomorphism classes itself would not change."],"forward_implications":["Every listed difference family yields an explicit Steiner system with the stated parameters whose translation group acts point-transitively, giving constructions that others can copy and test directly.","If the enumeration for $S(2,6,111)$ is sound, the previously isolated example is completed by a full census of 30 non-isomorphic systems, settling the point-transitive case for that parameter set and group.","The hyperbolic-frequency fingerprints give a practical isomorphism filter for Steiner systems with $k\\ge 5$, and the printed lists provide concrete test cases for stronger isomorphism invariants.","Multiplier data yield automorphism-group lower bounds such as $24\\cdot 175=4200$ for the flagged $S(2,7,175)$ design, linking the constructions to questions about which groups arise as full automorphism groups.","The extension from cyclic to general finite groups means the pipeline can be rerun on other admissible parameters, so additional point-transitive Steiner systems are likely to appear once the full algorithm is published."],"supporting_citations":[{"why":"Supplies the original cyclic difference-family algorithm that this paper generalizes to commutative and non-commutative groups.","marker":"[1]"},{"why":"Provides the background theory of difference families and the known-results catalogue against which newness is judged.","marker":"[2]"},{"why":"Records the known S(2,7,175) design whose multiplier 24 gives the automorphism-group lower bound used in Section 2.","marker":"[3]"},{"why":"The graph-isomorphism method whose reimplementation is used to certify non-isomorphism when fingerprints agree.","marker":"[6]"},{"why":"Attributes the first S(2,6,111) design, the one previously known example that the enumeration extends to 30.","marker":"[11]"}],"fun_headline_variants":["Exhaustive search yields exactly 30 non-isomorphic S(2,6,111) designs","Thirty point-transitive Steiner systems from one automorphism group","New difference families fill gaps: five Steiner parameters covered","Point-transitive Steiner systems: exhaustive count for S(2,6,111)","Group theory finds 30 S(2,6,111) designs and four more families"],"cache_read_input_tokens":137984,"weakest_assumption_plain":"The paper's load-bearing premise is that the not-yet-published generalized search algorithm really is exhaustive and that the private reimplementation of graph isomorphism classifies designs correctly; if either fails, the claimed count of 30 for $S(2,6,111)$ and the novelty of the remaining designs collapse, even if the printed blocks themselves are valid.","fun_headline_variants_meta":{"raw":{"variants":["Exhaustive search yields exactly 30 non-isomorphic S(2,6,111) designs","Thirty point-transitive Steiner systems from one automorphism group","New difference families fill gaps: five Steiner parameters covered","Point-transitive Steiner systems: exhaustive count for S(2,6,111)","Group theory finds 30 S(2,6,111) designs and four more families"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001546,"raw_usage":{"total_tokens":6140,"prompt_tokens":859,"completion_tokens":5281,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":475,"completion_tokens_details":{"reasoning_tokens":5180}},"tokens_in":475,"tokens_out":5281,"duration_ms":35897,"temperature":1.0,"reasoning_tokens":5180,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T11:36:41.156496+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Independently enumerate all difference families in $\\mathbb{Z}_{37}\\rtimes\\mathbb{Z}_3$ for block size 6 using an independent exhaustive method, or run an independent graph-isomorphism check on the thirty printed systems; finding more than 30 non-isomorphic systems, or finding two of the claimed new systems to be isomorphic, would refute the central claim. A cheaper partial check is to verify that every 111-point block list satisfies the Steiner property and that each of the 29 non-first designs is non-isomorphic to the first.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Attributes the first S(2,6,111) design, the one previously known example that the enumeration extends to 30."}],"review_version":1}