{"id":"455c41f8-285a-4781-b27b-0c43c692bad3","arxiv_id":"2504.18390","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper enumerates all non-isomorphic unitals of order 5 with a point-transitive group of order 126 or a 1-rotational group of order 125, yielding 1214 and 97 designs respectively.","lead":"This paper counts and lists highly symmetric block designs called unitals of order 5, which are Steiner systems with 126 points. It presents 1214 point-transitive and 97 one-rotational non-isomorphic unitals, constructed as difference families.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The census counts are unverifiable: the paper never describes the exhaustive search or the isomorphism reduction, so the central claim rests on undocumented computation.","rationale":"The reader's weakest-assumption analysis identifies exactly the same load-bearing gap: the absence of a described search and isomorphism-checking procedure. The paper is a computational census; its headline contribution is not merely the existence of certain unitals (which the listed difference families support) but the completeness and exactness of the counts. The manuscript provides no algorithm, no pseudocode, no code, and no reproducible data for the classification step, and it explicitly acknowledges that the only described invariant is not injective. These facts make the central claim conditional rather than established. I do not see an internal inconsistency in the listed families or a contradiction with the unital block counts, and the difference families are concrete enough to be checked. The primary concern is verifiability, not mathematical plausibility, so the appropriate verdict remains CONDITIONAL rather than REJECT or ACCEPT.","tokens_in":136269,"tokens_out":1929,"duration_ms":26235,"concrete_test":"Independently re-run the enumeration for one nontrivial case, e.g. SmallGroup(126,8): construct the group as a transitive permutation group on 126 points, enumerate all 6-subsets modulo the group action via orbit-stabilizer (e.g., GAP's OrbitsDomain or a documented backtracking search), test each representative for the difference-family/Steiner S(2,6,126) property, and then canonically label the resulting designs with nauty to separate isomorphism classes. Compare the final number and the family list with the paper's 129 designs. If the count is not 129, the census claim is invalid; if it is 129, the main concern is resolved for that group and should be repeated for SmallGroup(126,10).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is a complete census: for each listed group, the displayed difference families produce, up to isomorphism, exactly the tabulated number of unitals (e.g., 900 for SmallGroup(126,10), 129 for SmallGroup(126,8)). To establish this, the authors must have (a) exhaustively generated every candidate difference family in each group and (b) exactly classified those designs up to isomorphism. Neither procedure is described. The text gives only the input families and the final counts, plus a remark in Section 3 that two designs sharing the same hyperbolic frequency fingerprint are 'non-isomorphic' according to 'computer calculations' whose method is not stated. Since the fingerprint is explicitly admitted to collide for non-isomorphic designs, the counts depend entirely on unlisted canonical labelling or equivalence checks. A bug in that code, or an incomplete search over base blocks, would change the counts without being detectable from the paper. The explicit difference families themselves can be verified, but the 'exactly N' claim cannot.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims to give a complete enumeration, up to isomorphism, of unitals of order 5 (Steiner systems S(2,6,126)) whose automorphism group acts point-transitively, together with 1-rotational unitals admitting one fixed point and an orbit of size 125. For each of the 16 groups of order 126 and the 5 groups of order 125, it lists explicit difference families in GAP notation and tabulates the resulting numbers of non-isomorphic unitals, including counts such as 64 for Z126, 129 for SmallGroup(126,8), and 900 for SmallGroup(126,10). The paper also introduces a 'hyperbolic frequency fingerprint' invariant used to separate the designs, while noting that the fingerprint is not injective. The central assertion is that the displayed families produce, up to isomorphism, exactly the tabulated numbers of unitals, thus adding many new examples to the previously known ones.","tokens_in":136423,"tokens_out":6803,"duration_ms":64537,"significance":"If the census is correct, this is a valuable systematic enumeration: it would be the first complete classification of point-transitive and 1-rotational unitals of order 5, with explicit, independently checkable difference families and a nontrivial separating invariant. The listed families and fingerprints are concrete data that can be verified by any reader with a short script, and the paper honestly flags the known collision of its fingerprint. The strength of the contribution is, however, contingent on the completeness and exactness of the counts, which rest entirely on undocumented computation.","major_comments":[{"comment":"The central claim that the displayed difference families produce, up to isomorphism, exactly the tabulated numbers of unitals (e.g., 900 for SmallGroup(126,10), 129 for SmallGroup(126,8), 32 for Z5 x Z25) is not supported by any description of the enumeration algorithm, the search for base blocks, or the isomorphism reduction. The only hint in Section 3 says 'computer calculations show' that two designs with identical fingerprints are non-isomorphic, but no method, code, or certificate is supplied. Since the hyperbolic frequency fingerprint is explicitly admitted to collide, the 'exactly N' claims cannot be checked from the manuscript. Please provide a full algorithmic description of the exhaustive search, the invariant-based filtering, and the exact isomorphism check (or an independently runnable program), or else weaken the claims to 'at least N' and state that completeness is conjectural.","section":"Section 1, Table 1; Sections 2 and 3"},{"comment":"For each listed difference family, the paper does not state how the S(2,6,126) property was verified, nor how the development of the family under a nonabelian group is defined beyond a brief remark about using left multiplication and the Cayley table. A difference family in a nonabelian group does not automatically give a Steiner system under the standard orbit development; one must prove that the union of the group-orbits of the base blocks has exactly 525 blocks and that every pair of points occurs in exactly one block. If any listed family fails this check, the corresponding count is wrong. Please include the verification method (e.g., a computer check with available code, or a short argument that the orbit sizes sum correctly) for every family in the tables.","section":"Sections 2 and 3 (difference families)"},{"comment":"The paper does not compare the newly constructed unitals with the 938 unitals of order 5 found by Stoichev and Gezek as subdesigns of projective planes of order 25, even though the introduction explicitly notes that three unitals from the Desarguesian plane could appear in the lists. For a claim of an up-to-isomorphism census, such a comparison is necessary; otherwise the statement that the paper 'add[s] additional unitals' may double-count known designs. Please either perform the comparison and report which of the new families are isomorphic to the known subdesign unitals, or explicitly restrict the classification claim to unitals with the specified automorphism groups and state that the intersection with previously known unitals has not been determined.","section":"Section 1, Introduction"}],"minor_comments":[{"comment":"The manuscript contains many typos and misspellings (e.g., 'authomorphism', 'nodoby', 'se ems', '1-rotat ional') and should be carefully proofread before submission.","section":"Throughout"},{"comment":"The hyperbolic frequency fingerprint is defined only informally, and the example shows frequencies for counts 1 to 4, but many later fingerprints include a 0-frequency component (e.g., {0=1250, 1=42000, ...}). Please clarify the exact parameter range and how the value 0 is treated in the definition.","section":"Section 1, fingerprint definition"},{"comment":"The references are not visible in the manuscript text provided; in particular, the source for the classification of groups of order 126 and 125 (cited as [7]) and the exact contents of [1], [2], [3], and [6] should be fully specified.","section":"References"},{"comment":"The labels 'Classical unital' for SmallGroup(125,3) and 'resolvable unital' for SmallGroup(125,5) are asserted without explanation or citation; please provide a justification or reference for these identifications.","section":"Section 1, Table 1"},{"comment":"For SmallGroup(126,6) and SmallGroup(126,16) the paper says 'we don't include them and just point to [1] and [2]', but the table gives counts for these groups. For a self-contained census, either reproduce the corresponding difference families or give precise statements of the cited results.","section":"Section 3"}],"recommendation":"major_revision","confidential_remarks":"The paper is essentially a data appendix of difference families and counts. The central completeness claim is unverifiable without the missing algorithmic description, but the data itself is explicit and can be checked by independent computation. If the authors can supply the search and isomorphism-checking details (or make code available), the paper could become a valuable resource for the design theory community. The current form is not a complete research paper for a journal, and the massive tables might be better placed in a supplementary file or an online repository."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper gives the first systematic sweep for unitals of order 5 over all groups of orders 125 and 126, together with explicit difference families and per-group counts. That is a real contribution: the construction is standard difference-family stuff, the left-action convention for non-commutative groups is sensible, and the listed families are concrete enough for a reader to verify that each produces an S(2,6,126). If the counts are right, this is a big jump from the 938 subdesign unitals of Stoichev–Gezek to 1311 additional designs, and the authors are appropriately careful about the small overlap with earlier work.\n\nThe soft spot is exactly where the stress-test note lands. The paper's central claim is not just existence of these unitals but completeness: for each group, exactly N designs up to isomorphism. The search over candidate difference families is never described, and the isomorphism reduction is never described. The fingerprint from the authors' earlier paper is explicitly admitted to collide, yet the only evidence for separating the two designs that share a fingerprint is a remark that \"computer calculations\" show non-isomorphism. No code, no algorithm, no canonical labelling procedure, no data file. A bug in that unstated computation, or an incomplete enumeration of base blocks, would change the reported counts without being detectable from the paper. The explicit families themselves are checkable, but the \"exactly N\" claim is not.\n\nI do not think this is a case of a load-bearing flaw in the math; the construction itself is sound and the parameter checks pass. The flaw is in the evidence for exhaustiveness. That is fixable by releasing code or writing a precise description of the search and the isomorphism test, and the authors should be asked to do that before acceptance. The paper would also benefit from a cleaner write-up; the current text is rough, with typos and asides that do not help.\n\nFor a reader in finite geometry or design theory, the explicit difference families are useful regardless of the census counts, and the counts are plausible. But I would not rely on the exact numbers until the computation is reproducible. A serious referee should see this, and the right recommendation is to send it to review with a request for code or a full algorithm description, not to desk reject it.","headline":"Useful cache of explicit unitals of order 5 with prescribed automorphism groups, but the census counts rest on undocumented computation and need code or a detailed algorithm before the completeness claim can be trusted.","tokens_in":136967,"tokens_out":1987,"would_cite":true,"duration_ms":25988,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05B05","05B30","20B25","51E20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims a complete census of point-transitive and one-rotational unitals of order 5, with explicit difference families for all 1,311 designs.","keywords":["unital","Steiner systems","difference families","point-transitive designs","1-rotational designs","order 5","finite geometry","hyperbolic frequency fingerprint"],"falsifier":"Re-run the difference-family search independently for one group, such as SmallGroup(126,10): enumerate all candidate six-element base blocks, develop each family into an $S(2,6,126)$, and compare the isomorphism classes with the 900 listed; finding one additional class, or identifying two listed classes as isomorphic, would refute the count.","tokens_in":136058,"feed_emoji":"🧩","tokens_out":10554,"duration_ms":90329,"temperature":0.7,"pith_summary":"The paper sets out to enumerate all unitals of order 5, meaning Steiner systems $S(2,6,126)$, whose automorphism groups either act transitively and effectively on the 126 points or fix one point and act transitively on the other 125. For every one of the 16 groups of order 126 and the 5 groups of order 125, it lists explicit difference families and reports how many non-isomorphic unitals each group produces. Summing the tables gives 1,214 point-transitive designs and 97 one-rotational designs, for a total of 1,311 unitals, including the classical and resolvable unitals previously known. If the enumeration is accepted, this is the first systematic census of these symmetry classes and by far the largest explicit algebraic collection of order-5 unitals.","feed_headline":"Census counts 1,311 highly symmetric unitals of order 5","feed_subtitle":"Explicit difference families in the 21 groups of orders 125 and 126 reconstruct every such design up to isomorphism.","key_machinery":"The load-bearing object is a difference family in a finite group: a collection of six-element base blocks such that every nonzero group element appears exactly once as a difference of two points inside some base block. In a group of order 126 acting regularly on the 126 points, or of order 125 acting with one singleton orbit $\\{\\infty\\}$ and one regular orbit, such a family develops, through the left action and a fixed 0-based Cayley table, into the full block set of a Steiner system $S(2,6,126)$. The paper pairs each family with a hyperbolic frequency fingerprint, a count distribution over certain quadruples used as a quick distinguishing statistic, while recording cases where two non-isomorphic designs share the same fingerprint.","core_discovery":"On its own terms, the paper's discovery is a complete difference-family census: up to isomorphism, every point-transitive unital of order 5 arises from one of the listed base-block collections in a group of order 126, and every one-rotational unital from a listed family in a group of order 125 with one point fixed. The counts allocate the designs across the 21 groups, with $C_6\\times(C_7:C_3)$ (SmallGroup(126,10)) alone giving 900, $S_3\\times(C_7:C_3)$ giving 129, and $\\mathbb{Z}_{126}$ giving 64; seven groups of order 126 yield no designs. Two prior designs from the literature are reproduced inside the lists, and the paper records for each design a hyperbolic frequency fingerprint, which it uses as a distinguishing statistic while noting that the fingerprint is not by itself a complete invariant.","pith_inferences":["Because the paper's own Section 3 reports two non-isomorphic designs with identical fingerprints, the hyperbolic frequency fingerprint is not a complete invariant, and future automated extensions should combine it with a canonical-label or graph-isomorphism check.","The same difference-family recipe should transfer to other unital orders $k$, using groups of order $k^3+1$ for transitive designs and $k^3$ for one-rotational designs, with the bottleneck being a certified enumeration rather than block generation.","Most of the 1,311 designs probably do not embed in the Desarguesian plane of order 25; checking embeddability of the listed families would link this algebraic census to the earlier subdesign census.","The seven zero groups of order 126 invite a structural characterization of when a group admits a unital difference family, for example by comparing normal Sylow structure or abelianization across the zero and nonzero groups in the table."],"forward_implications":["If the census is exhaustive, the number of non-isomorphic unitals of order 5 in these two symmetry classes is exactly 1,214 + 97 = 1,311, recovering the previously known explicit examples and the earlier cyclic families.","Each tabulated difference family is a complete reconstruction recipe: applying the left action of the group to the listed base blocks yields the full block set of a Steiner system $S(2,6,126)$ without further search.","Seven groups of order 126 admit no such design, while $C_6\\times(C_7:C_3)$ admits 900, so the table gives a sharp numerical profile of which group structures can carry an order-5 unital.","The overlap between the algebraic designs and the earlier projective-plane subdesign census is limited to at most three unitals from the Desarguesian plane of order 25, so almost all 1,311 designs are new relative to that census."],"supporting_citations":[{"why":"Supplies the 64 cyclic ($\\mathbb{Z}_{126}$) unitals of order 5 already known and recovered by the new census.","marker":"[1]"},{"why":"Supplies the 8 previously found unitals for $\\mathbb{Z}_2\\times\\mathbb{Z}_3\\times\\mathbb{Z}_3\\times\\mathbb{Z}_7$ and introduces the hyperbolic frequency fingerprint used as a distinguishing invariant.","marker":"[2]"},{"why":"Contains the two previously known explicit order-5 unitals, the classical and resolvable designs, which appear in bold in the listed families.","marker":"[3]"},{"why":"Documents the earlier 938 unitals found as subdesigns of projective planes of order 25, against which the paper measures possible overlap.","marker":"[6]"},{"why":"Classifies the 16 groups of order 126 and the 5 groups of order 125 that serve as the ambient groups for all difference families.","marker":"[7]"},{"why":"Provides the computational tool used to convert each group representation into a 0-based Cayley table, from which all blocks are generated by left action.","marker":"[8]"}],"fun_headline_variants":["Census: 1,311 unitals from 21 groups","Difference families build all 1,311 unitals","All point-transitive unitals of order 5 counted","1,311 unitals: complete difference-family census"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The census counts are only as reliable as the unstated computer search and isomorphism check that found no missing base block and no false merges among the listed designs.","fun_headline_variants_meta":{"raw":{"variants":["Census: 1,311 unitals from 21 groups","Difference families build all 1,311 unitals","All point-transitive unitals of order 5 counted","1,311 unitals: complete difference-family census"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001352,"raw_usage":{"total_tokens":5391,"prompt_tokens":751,"completion_tokens":4640,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":367,"completion_tokens_details":{"reasoning_tokens":4573}},"tokens_in":367,"tokens_out":4640,"duration_ms":34187,"temperature":1.0,"reasoning_tokens":4573,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:45:43.283969+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Re-run the difference-family search independently for one group, such as SmallGroup(126,10): enumerate all candidate six-element base blocks, develop each family into an $S(2,6,126)$, and compare the isomorphism classes with the 900 listed; finding one additional class, or identifying two listed classes as isomorphic, would refute the count.","supporting_citations":[{"cited_title":"Unitals in Projective Planes o f Order 25","cited_arxiv_id":null,"evidence_quote":"Documents the earlier 938 unitals found as subdesigns of projective planes of order 25, against which the paper measures possible overlap."}],"review_version":1}