{"id":"4d954b6f-f620-4ced-ae9b-0fce6d0a552d","arxiv_id":"2607.22795","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":9.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"An explicit finite skew brace with soluble additive group and insoluble multiplicative group, with quotient PSL2(7), disproves Byott's conjecture.","lead":"This paper constructs a finite skew brace—an algebraic structure with two group operations obeying a distributive law—whose additive group is soluble while its multiplicative group has quotient PSL2(7), so it is insoluble. It is the first counterexample to Byott's conjecture, which asserted this could not happen for finite soluble groups.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Tables 1–2 are load-bearing and not independently verified, but spot checks support them; residual arithmetic risk does not change the reader's accept.","rationale":"The reader's weakest assumption is precisely the correctness of the finite computations in Tables 1–2 and the matrix identities in Lemma 2.1/2.6. My review agrees that this is the load-bearing point: every subsequent step—σ in F, the cocycle identities (10)–(11), the distinct cosets in Lemma 2.6, and the bijection b in Proposition 2.7—depends on those tables. However, I found no inconsistency in the construction. The mathematical framework is coherent: the finite group computations are explicit, the presentation of PSL2(7) is standard, the semidirect products are built with verified automorphisms, and the opposite group orientation is handled consistently. Spot-checking the least transparent table rows and key matrix identities gave the printed values, which raises confidence that the tables are correct. The paper's omission of full GAP output is a reproducibility weakness, not a demonstrated mathematical flaw. Since the reader already set moderate confidence and explicitly flagged this residual risk, my read does not change the verdict.","tokens_in":10463,"tokens_out":58663,"duration_ms":387515,"concrete_test":"Run an independent GAP/Sage script that (1) defines A,B as in §2, computes the unique lift bg in AGL(E) from the projection, and checks π(t_x)=e_x for all eight words in Table 1; (2) for g=A,B and each x, computes σ(g,x)=t_{g⋆x}^{-1} g t_x, reduces to U^mY^k, and compares with Table 2; (3) verifies bA^2=bY^3=(bA bY)^7=[bA,bY]^4=1, the displayed bY and bA bY, and U^7=Y^3=1, YUY^{-1}=U^2. If all checks pass, the construction is confirmed; if not, the first failing row identifies the error.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim depends on the printed finite data. Table 1's eight words must satisfy π(t_x)=x; Lemma 2.6 uses this for the unique expressions g=t_xU^mY^k and for the distinct triples q(c_g). Table 2 supplies m,k, which define β_W, β_P and the cocycle identities (10)–(11). Lemma 2.1 and Lemma 2.6 contain additional matrix identities: bY, bA bY, the four PSL2(7) relators, and the stabilizer factorization F=⟨U,Y⟩. If any of these values is wrong, π(t_x)=x fails, σ(g,x) need not lie in F, the partition (33) collapses, and the skew-brace construction is invalid. The manuscript says these were checked in GAP but does not display the computations or code, so the correctness of the printed proof rests on unshown arithmetic. I independently recomputed the three least transparent Table 1 entries (t_e3=A B^2 A B^4 A, t_e5=B^3 A B^4 A, t_e7=A B^2 A B^5) from the affine multiplication rule and obtained π=e3,e5,e7 respectively, consistent with the table; I also spot-checked the bY and bB^2 matrices in Lemma 2.1. Thus I have not found an actual error; the concern is the unshown dependence on the full table.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs an explicit finite skew brace (B,+,∘) with soluble additive group and insoluble multiplicative group. The additive group is the opposite of a soluble group N0 = P ⋊ (W ⋊ E), with P ≅ F_7^8, W ≅ F_3^7, E ≅ F_2^3, while the multiplicative group is a semidirect product H ⋊ L with L = GL_3(2) ≅ PSL_2(7), giving an insoluble quotient. The construction proceeds by embedding an affine copy of GL_3(2) into AGL(E), building a transversal and exponent tables, defining cocycles β_W and β_P, forming the group N0, passing to the opposite group, and transporting a semidirect-product law to N. The main theorem is stated as a counterexample to Byott's conjecture, and the paper also derives a regular embedding into Hol(N).","tokens_in":10849,"tokens_out":27566,"duration_ms":207746,"significance":"If correct, this is a significant result: it disproves Byott's conjecture that every regular subgroup of the holomorph of a finite soluble group is soluble. The construction is fully explicit and the main structural steps — the cocycle identities (10)–(11) and (15)–(16), the automorphism property of the ϑ_g, and the coset partition underlying the bijection b — are proved from the definitions. The paper also connects the counterexample to Hopf–Galois structures. A notable strength is that the proof is coordinate-based and largely checkable by direct multiplication. The main weakness is that several load-bearing finite computations are reported as checked in GAP but not fully displayed; I spot-checked some of them and found no error.","major_comments":[{"comment":"The central claim depends on the finite data in Table 1 (the transversal t_x) and Table 2 (the exponent functions m and k for A and B), as well as on the matrix identities in Lemma 2.1 and the stabilizer factorization F=⟨U,Y⟩. These data feed directly into Lemma 2.2, Proposition 2.3, and Lemma 2.6; for example, a single incorrect entry in Table 2 would break the cocycle identities (10)–(11), the conclusion β_W(g)∈W in Proposition 2.3, and the partition (33). The text verifies only two sample entries of Table 2 and states that 'every other entry is obtained by the same matrix multiplication'; the GAP code is not supplied. I independently recomputed several entries (t_e3, t_e5, t_e7, bY, bB^2) and found them internally consistent, so I do not suspect an error; nevertheless, the audit trail should be completed by including the code or a full computation appendix.","section":"§2, Tables 1–2 and the GAP-checked assertions (pp. 4–8)"}],"minor_comments":[{"comment":"The presentation PSL2(7)=⟨x,y | x²=y³=(xy)⁷=[x,y]⁴=1⟩ is called standard but no reference or justification is given. Since the proof of Lemma 2.1 depends on it, please add a reference or a one-line justification.","section":"Lemma 2.1, Eq. (3)"},{"comment":"The sentence 'Every other entry is obtained by the same matrix multiplication and unique reduction to U^mY^k' would be more useful with at least one additional fully worked example, especially for an entry with k≠0.","section":"§2, after Table 2"},{"comment":"The coordinate-vector identification for functions E→F_q is introduced only after β_W and β_P are defined. It would be clearer to introduce this identification before displaying the vectors (0,0,0,1,2,2,0,1) and (1,2,0,1,0,1,0,1).","section":"§2, after Eq. (14)"},{"comment":"The proof that the 168 triples q(c_g) are pairwise distinct is compressed: it relies on the uniqueness of g=t_xU^mY^k, but that uniqueness is justified only by a counting argument. This is correct, but because the partition (33) is foundational for the bijection b, the counting argument should be spelled out explicitly.","section":"Lemma 2.6, p. 11"},{"comment":"The symbol F is used both for the stabilizer subgroup of L and for the fields F_2,F_3,F_7. This is conventional but potentially confusing; consider using blackboard-bold or a different letter for the stabilizer.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The mathematical construction appears coherent and the proof is likely correct. My recommendation for major revision is driven entirely by the auditability gap: several load-bearing finite computations are asserted rather than displayed, and the paper explicitly says they were checked in GAP but does not supply the code or a full computation appendix. In my view this should be fixed before publication, but the required revisions are local and do not touch the structure of the argument."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is the real thing: a first counterexample to Byott's conjecture. The authors build an explicit finite skew brace whose additive group is soluble and whose multiplicative group has PSL2(7) as a quotient. If the tables are correct, this settles a question that many partial results had made look plausible.\n\nWhat is genuinely good: the construction is not a black box. It is an affine lift of GL3(2), two cocycles in characteristics 3 and 7, and an opposite-group twist. The structural lemmas—cocycle identities, automorphism actions, coset partition—are proved by hand from coordinates. The cited literature is surveyed honestly: affirmative cases are listed, and no prior counterexample exists. The paper also connects back to the holomorph formulation, so the original form of Byott's problem is addressed directly.\n\nThe soft spot is exactly what the stress-test says: Tables 1 and 2 carry the construction, and the full computations are not printed. The paper says GAP checked them. That is a verification gap, not a mistake I can point to. I independently spot-checked three of the less transparent transversal entries and the bY matrix; they match. The arithmetic risk is real but small, and it is the kind of thing a referee with GAP can close in an afternoon.\n\nMinor notes: the presentation is dense, and the opposite-group convention makes formulas easy to trip over, but the paper is consistent. The claim that H is normal in the multiplicative group follows from the semidirect product structure; that is handled correctly.\n\nConclusion: I would accept this for serious refereeing without hesitation. The result is important, the proof is explicit, and the only concerns are checkable. If a referee verifies the printed tables or the authors make the GAP session available, the result should be accepted. I would cite it in my own work on skew braces. Worth a reading group slot.","headline":"First counterexample to Byott's conjecture, with an explicit finite skew brace; the construction is credible and the residual risk is a handful of unprinted GAP-verified tables.","tokens_in":11240,"tokens_out":8794,"would_cite":true,"duration_ms":66498,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16T25","20D10","12F10"],"pacs":[],"model":"deepseek-v4-flash","headline":"There exists a finite skew brace whose additive group is solvable but whose multiplicative group is not; the multiplicative group has a quotient isomorphic to PSL2(7), so Byott's conjecture—that regular subgroups of Hol(N) for solvable N ar","keywords":["skew brace","Byott's conjecture","Hopf–Galois structure","regular subgroup","holomorph","bijective cocycle","solvable group","PSL2(7)"],"falsifier":"Recompute the transversal in Table 1 and the exponents m and k in Table 2 from the displayed matrices A and B, then verify for the generators A and B that β_P(AB) = β_P(A) + ρ_P(A)β_P(B) and similarly for β_W; any mismatch would make the cocycle identities (15) or (16) fail and destroy the brace structure.","tokens_in":10385,"feed_emoji":"🧩","tokens_out":7774,"duration_ms":64583,"temperature":0.7,"pith_summary":"The paper proves Byott's conjecture false by constructing an explicit finite skew brace (B,+,∘) with (B,+) solvable and (B,∘) insoluble. Specifically, (B,+) is the opposite of an iterated semidirect product F_7^8 ⋊ (F_3^7 ⋊ F_2^3), while (B,∘) contains a normal solvable subgroup H whose quotient is GL3(2) ≃ PSL2(7), a simple group of order 168. Because a skew brace encodes the same data as a regular subgroup of the holomorph and as a Hopf–Galois structure, the example also disproves the conjecture in those formulations. The construction is fully explicit: every map is given by matrices and tables, and the finite verifications were done computationally.","feed_headline":"Byott's conjecture is false: skew brace found","feed_subtitle":"An explicit brace has a solvable additive group but a multiplicative group with an insoluble PSL2(7) quotient.","key_machinery":"The construction rests on the cocycle c_g = (β_P(g), β_W(g), π(g)) in the opposite group N = N0^op, together with λ_g = Inn_{−c_g} ϑ_g. Here π(g) is the translation part of the unique affine lift bg = (g, π(g)) of g in an affine copy bL ≃ PSL2(7) of GL3(2); the functions β_P and β_W are defined from the stabilizer decomposition σ(g,x) = t_{g⋆x}^{−1} g t_x = U^{m(g,x)} Y^{k(g,x)} of the affine action g⋆x = gx + π(g), where F = ⟨U,Y⟩ ≃ C7⋊C3 is the stabilizer of e0. The spaces P = F7^E and W = {zero-sum functions E→F3} carry twisted actions with multiplier 2^{k(g,x)}, chosen so that β_P and β_W are 1-cocycles; the opposite-group law reverses the semidirect order so that c_gh = c_g + λ_g(c_h) h","core_discovery":"The central discovery is one finite skew brace with solvable additive group and non-solvable multiplicative group. The additive group N is the opposite of N0 = P ⋊ (W ⋊ E), where P = F7^{E} is the vector space of all functions on E = F2^3, W is the zero-sum hyperplane of F3^{E}, and E is the underlying vector space. The multiplicative group is a semidirect product G = H ⋊ L with L = GL3(2) ≃ PSL2(7) and H a subgroup of N of index 168. The cocycle c_g = (β_P(g), β_W(g), π(g)) is built from two 1-cocycles induced by an affine copy of PSL2(7) in AGL(E), and the opposite-group twist is what orients the cocycle identity correctly.","pith_inferences":["The opposite-group trick may generalize: any affine action of a non-solvable group on a finite vector space with a suitable flag of stabilizers could yield similar cocycles, and the minimal counterexample might be far smaller than the ~10^11-order example presented here.","Since the additive group is a semidirect product of vector spaces, this construction suggests that restricting the additive group to be a p-group or a direct product of elementary abelian groups will not recover Byott's conjecture.","One could test whether the quotient PSL2(7) can be replaced by other small non-abelian simple groups, e.g., by finding analogous stabilizer subgroups C_q ⋊ C_p inside an affine general linear group with the right exponent relations.","A useful public artifact would be a complete verification script (or expanded tables) for all 168 cocycle identities; the paper's computation is convincing but not independently checkable from the printed text alone."],"forward_implications":["The regular-subgroup version of Byott's conjecture fails: there is a regular embedding of the insoluble group G into Hol(N) for a finite solvable group N.","In Hopf–Galois theory, an insoluble finite Galois group can admit a Hopf–Galois structure of solvable type.","The known affirmative cases — nilpotent additive group, orders not divisible by 3, two-sided braces, orders up to 2000 — remain true, but none of them extends to all solvable additive groups.","The same skew brace gives a finite set-theoretic solution of the Yang–Baxter equation whose 'base' group is solvable and whose 'derived' group is not."],"fun_headline_variants":["Solvable additive, insoluble multiplicative: brace found","Brace with solvable additive, insoluble multiplicative groups","Skew brace disproves Byott via PSL2(7) quotient","Brace with PSL2(7) quotient refutes Byott","Byott's conjecture false: explicit skew brace"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is that the finite computations listed in Tables 1 and 2 and the matrix identities in Lemma 2.1 are error-free: the paper says they were verified computationally but does not display all the arithmetic.","fun_headline_variants_meta":{"raw":{"variants":["Solvable additive, insoluble multiplicative: brace found","Brace with solvable additive, insoluble multiplicative groups","Skew brace disproves Byott via PSL2(7) quotient","Brace with PSL2(7) quotient refutes Byott","Byott's conjecture false: explicit skew brace"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000933,"raw_usage":{"total_tokens":3751,"prompt_tokens":590,"completion_tokens":3161,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":334,"completion_tokens_details":{"reasoning_tokens":3078}},"tokens_in":334,"tokens_out":3161,"duration_ms":22637,"temperature":1.0,"reasoning_tokens":3078,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T05:09:51.577326+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the transversal in Table 1 and the exponents m and k in Table 2 from the displayed matrices A and B, then verify for the generators A and B that β_P(AB) = β_P(A) + ρ_P(A)β_P(B) and similarly for β_W; any mismatch would make the cocycle identities (15) or (16) fail and destroy the brace structure.","supporting_citations":[],"review_version":1}