{"id":"82a7153a-3356-497c-a027-7aad8ad11525","arxiv_id":"2505.06012","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For large alternating groups, any three conjugacy classes of size at least |G|^(1-δ) multiply to the whole group for a universal small δ.","lead":"This paper proves that in the alternating group Alt(n), the product of any three very large conjugacy classes is the entire group, once n is large and the classes have size at least a small power below the whole group. It settles an open problem from the Kourovka Notebook and improves the previous four-class result to three classes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3.2's split-class case is delegated to a citation and its constructive alternative is circular; Theorem 1.2's large-class half rests on this unverified adaptation.","rationale":"The reader's weakest assumption was Proposition 2.3, but I do not see a concrete failure mode there: Proposition 2.3 supplies arbitrary additive constants, and the reduction chain only needs to choose δ sufficiently small, so the successive inflation factors (2, 2, 2, 2, 10, 7, 3) can be absorbed by taking δ smaller. The genuinely load-bearing gap is Proposition 3.2's split-class case. Theorem 1.2 splits target classes into small classes, handled by Theorem 1.4, and large classes, handled entirely by Proposition 3.2. The non-split part of Proposition 3.2 is fully proved in the text via Proposition 3.1, but the split case is delegated to an assertion that a cited lemma 'works out similarly' for a different m. The circular 'constructive proof' in the same paragraph does not repair this, and its size claim is not justified. This is exactly the kind of missing support that the reviewing rules require flagging. The rest of the reduction chain is detailed and internally coherent, and the explicit solutions in Section 12 are checkable; I found no reason to doubt Theorem 1.4 itself. The correct verdict remains CONDITIONAL: the main theorem is credible but not unconditionally established until the split-class case is either proved in the preprint or shown to follow line-by-line from the cited reference.","tokens_in":43388,"tokens_out":22257,"duration_ms":211333,"concrete_test":"Independently write out the split-case proof of Proposition 3.2 for m ∈ {n−3, n−2}, following [GM21, Lemma 5.3(i)]: compute the relevant character sum N = |G|^{-1} ∑_χ χ(C1)χ(C2)χ(O_m)/χ(1) and verify the nonvanishing bound, replacing the m ∈ {n−1, n} estimates by the corresponding estimates for m = n−3 or n−2. If the dominant-term or error-bound inequality changes sign for split classes with |C1||C2| just above |G|^{1+δ}, the citation does not cover the needed case. As a supporting computational check, use GAP for n = 15,...,25 with δ = 0.1 to enumerate split Alt(n) classes with |C1||C2| ≥ |G|^{1+δ} and test whether C1C2 ⊇ O_m; a counterexample would disprove Proposition 3.2 as stated, while success would be suggestive but not settle the asymptotic claim.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The most load-bearing unproved step is the split-class case of Proposition 3.2. In the proof of Theorem 1.2, Proposition 3.2 is the only mechanism that reaches target classes C with |C| ≥ |Alt(n)|^{2δ/3}: it must give both C1C2 ⊇ O_m and C C3^{-1} ⊇ O_m for m the odd element of {n−3, n−2}. The proof of Proposition 3.2 reduces to the case C1 ≠ (C1)^g and C2 ≠ (C2)^g and then states that a character-theoretic proof is 'essentially contained in [GM21, Lemma 5.3(i)]' with m ∈ {n−1, n}, 'but the technique works out similarly'. The constructive alternative offered in the same paragraph ('apply Theorem 1.4') is circular, because Theorem 1.4 is what Proposition 3.2 is used to prove; moreover, the asserted inference |C1|, |C2|, |O_m| ≥ |Alt(n)|^{1−δ} from |C1||C2| ≥ |Alt(n)|^{1+δ} does not follow from Proposition 2.3, since one class may be small and the other large. Thus Theorem 1.2 rests on an unverified adaptation of a cited character argument. This is not a stylistic gap: if the m = n−3, n−2 case requires different inequalities or a different dominant character, the large-class half of Theorem 1.2 has no proof in this preprint.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that for a sufficiently small constant δ and all sufficiently large n, every product C1C2C3 of three conjugacy classes of Alt(n) with |Ci| ≥ |Alt(n)|^{1−δ} equals the whole group (Theorem 1.2), and it extends this to three normal subsets of every non-abelian finite simple group (Theorem 1.3). The main combinatorial work establishes Theorem 1.4, which asserts that two large classes cover a third, through seven reductions on 'class strings'; each reduction is accompanied by an injective forward map and an explicit inverse walk-back procedure, and the final reduced problem is solved by explicit local solutions in §12.","tokens_in":43709,"tokens_out":10201,"duration_ms":103111,"significance":"If the proof is completed, the paper answers Kourovka Problem 20.23, improves the four-class result of Garonzi and Maróti to three classes, and complements the Lie-type results. The class-string machinery is original and substantial: the seven reductions are individually equipped with explicit inverse constructions, and the overall argument has no apparent circularity in the combinatorial core. The main weakness is the reliance of Proposition 3.2 on an unverified adaptation of a character-theoretic lemma; this proposition is load-bearing for the large-class half of Theorem 1.2. I was not able to certify every long cycle manipulation in §§5–12 by hand, but the structure of the reductions is coherent and the local solutions in §12 are directly checkable.","major_comments":[{"comment":"The split-class case of Proposition 3.2 is load-bearing for Theorem 1.2: it is the only mechanism that reaches classes C with |C| ≥ |Alt(n)|^{2δ/3}. In the case where both C1 and C2 are split classes, the proof states that a character-theoretic argument is 'essentially contained in [GM21, Lemma 5.3(i)]' with m ∈ {n−1, n}, and that 'the technique works out similarly' for m ∈ {n−3, n−2}. This is an unverified adaptation: changing m changes the size of the class O_m and can affect which inequalities or which dominant character are used. The constructive alternative offered in the same paragraph is circular, because it applies Theorem 1.4, the very statement that Proposition 3.2 is used to prove. As written, the large-class half of Theorem 1.2 is therefore unsupported unless the split-class case for m ∈ {n−3, n−2} is proved in full.","section":"§3, Proposition 3.2"},{"comment":"The proof of Proposition 3.2 derives m ≥ k1 + k2 from the hypothesis |C1||C2| ≥ |Alt(n)|^{1+δ} by applying Proposition 2.3(a) to each class. But Proposition 2.3(a) requires |C_j| ≥ |Alt(n)|^{1−δ1} for the individual class, and the product hypothesis does not imply this: one factor may be much smaller than |Alt(n)|^{1−δ} while the other is large. In the application inside Theorem 1.2 the three classes are individually large, so the statement can be repaired by adding that assumption, but the proposition as stated is not proved by the given argument.","section":"§3, Proposition 3.2, first paragraph"}],"minor_comments":[{"comment":"In the paragraph on special classes, 'the union of two 1 distinct classes' contains a stray '1'; it should read 'two distinct classes'.","section":"§1.1"},{"comment":"The claims that 'no character theory is involved' and that the proof is constructive are overstated: Proposition 3.2 relies on the character-theoretic [GM21, Lemma 5.3(i)] for its split-class case. The statement should be qualified to the combinatorial proof of Theorem 1.4, or Proposition 3.2 should be made elementary.","section":"Abstract and §1.2"},{"comment":"The sentence 'By [MP21, Thm. 1.3], the result holds for every non-abelian finite simple group G ≠ Alt(n)' should be clarified, since the introduction presents [MP21] as a Lie-type result; if the cited theorem covers sporadic groups as well, a brief explanation would help the reader.","section":"Proof of Theorem 1.3"},{"comment":"The notation µ is used both for the map from class strings to classes and for the map from element strings to permutations; although the context makes the meaning clear, a sentence flagging the overloaded notation would prevent confusion in the long reduction sections.","section":"§4"},{"comment":"In the small-class argument, the variable c3 in 'the class C′ ∋ c3c^{-1}' appears to mean γ3; renaming it would improve readability.","section":"§3, proof of Theorem 1.2"}],"recommendation":"major_revision","confidential_remarks":"The main obstruction is Proposition 3.2. If the authors can supply a complete proof of the split-class case for m ∈ {n−3, n−2}, or restructure the proof of Theorem 1.2 to avoid this lemma, the paper would be a strong contribution. I would also recommend softening the 'no character theory' claim in the abstract until that point is resolved. The combinatorial core §§5–12 appears carefully constructed, but it is lengthy and I could not certify it line by line."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper answers Kourovka 20.23: for n large and δ small, any three conjugacy classes of Alt(n) of size at least |G|^{1−δ} multiply to all of G. That is a genuine advance over Garonzi–Maróti's four-class theorem and Rodgers's special cycles, and the proof is elementary throughout, no character theory. The seven reductions in §§5–12 are the real work: each step comes with an explicit map and an inverse walk-back, and the final solutions in §12 are explicit products. I read through the reduction chain and it looks internally coherent; I did not certify every displayed inequality by hand.\n\nThe soft spot is Proposition 3.2, the only step that reaches classes of size ≥ |G|^{2δ/3} in the proof of Theorem 1.2. In the split-class case (both C_i ≠ (C_i)^g), the proof says the argument is 'essentially contained' in [GM21, Lemma 5.3(i)], which handles m = n−1,n, and 'the technique works out similarly' for m = n−3,n−2. That is a load-bearing delegation. The offered constructive alternative—apply Theorem 1.4—is not circular, since Thm 1.4 is proven independently later, but it is invalid as written: from |C1||C2| ≥ |G|^{1+δ} you cannot infer that |C1|,|C2|,|O_m| each exceed |G|^{1−δ}; one class might be small. So the split case currently has no proof in this preprint. A referee needs to see the character argument written out for the actual m, or a precise reference.\n\nMinor: in the small-n part of Theorem 1.3, the displayed bound |G|^{1−(n2 log n2)^{-1}} > |G|^{−1} does not force S_i = G; the exponent should be much smaller. Fixable.\n\nCitation pattern is fine: the only imported quantitative tool is Proposition 2.3 from [DMP24]/[GM21], and it is used as a lemma, not a restatement of the result.\n\nBottom line: this deserves a serious referee. The main theorem is almost certainly true, and the combinatorial machinery is worth publishing once Prop 3.2 is fixed. I would not cite it in my own work until the gap is closed, but I would encourage the editor to send it to review and ask for that revision.","headline":"Genuine advance with a load-bearing gap: the split-class case of Proposition 3.2 is delegated to a citation, and the constructive alternative offered there does not work as written.","tokens_in":44243,"tokens_out":7640,"would_cite":false,"duration_ms":65249,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20B30","20E45"],"pacs":[],"model":"deepseek-v4-flash","headline":"Three conjugacy classes of size at least |Alt(n)|^{1−δ} always multiply to the entire alternating group.","keywords":["alternating group","conjugacy classes","products of conjugacy classes","Kourovka Notebook problem","normal subsets","finite simple groups","class strings","constructive proof"],"falsifier":"Find n and three conjugacy classes C1,C2,C3 of Alt(n) with |Ci| ≥ |Alt(n)|^{1−δ} and a single permutation g outside C1C2C3; such a triple would refute Theorem 1.2 directly. A cheaper partial check is to test Proposition 2.3 numerically on the extremal class of one (1−δ)n-cycle plus δn fixed points, the class used throughout, to see whether its cycle count stays within (δ+o(1))n; failure there would break the reduction chain.","tokens_in":43175,"feed_emoji":"🔀","tokens_out":7827,"duration_ms":81240,"temperature":0.7,"pith_summary":"The paper proves that, for sufficiently large n and small δ>0, any three conjugacy classes of Alt(n) of size at least |Alt(n)|^{1−δ} satisfy C1C2C3=Alt(n). This answers Problem 20.23 of the Kourovka Notebook, improves the earlier four-class theorem of Garonzi and Maróti, and combines with known Lie-type results to extend the statement to three normal subsets of every nonabelian finite simple group. The proof avoids character theory entirely: it encodes cycle structures as class strings and reduces the equation α1α2=α3 to a short list of explicitly solved cycle products. Because the reductions are injective and constructive, the argument in principle outputs the three permutations c1,c2,c3 for any given target element.","feed_headline":"Three large conjugacy classes cover the alternating group","feed_subtitle":"An elementary proof answers Kourovka Problem 20.23 and extends to every finite simple group.","key_machinery":"The central mechanism is a chain of seven injective reductions θ1,…,θ7 acting on string triples: three class strings, whose ■ symbols mark cycle boundaries, together with a label function that records the information needed to undo each step. Each reduction simplifies the cycle structure while preserving the existence of an aligned solution, meaning a solution α1α2=α3 in which triples of cycles sharing positions also share prescribed common values. The final reduction splits the problem into small subproblems, each solved by one of the explicit templates (12.1)–(12.4), and the solutions are concatenated and pulled back through the inverses of the reductions. The quantitative glue is Proposition 2.3, which translates class size into a bound on the number of cycles and is invoked at every stage to guarantee that shortened classes remain large.","core_discovery":"The paper establishes Theorem 1.2: there exist constants δ>0 and n0 such that for all n≥n0 and any three conjugacy classes C1,C2,C3 of G=Alt(n) with |Ci| ≥ |G|^{1−δ}, the product set C1C2C3 equals all of G. The engine is the stronger intermediate Theorem 1.4, which says the same hypotheses imply C1C2⊇C3. The proof works in Sym(n), replacing each class by a class string that fixes an ordering of its cycles, then applying seven injective reductions that progressively simplify the cycle structures, and finally solving the reduced problem by concatenating explicit solutions for four basic types of cycle configurations. Inverting the seven reductions yields the required permutations. Theorem 1.3 follows by combining Theorem 1.2 with the known Lie-type case, giving the same statement for three normal subsets of every nonabelian finite simple group.","pith_inferences":["One could extract explicit, though likely far from optimal, values of δ and n0 by tracking constants through the seven reductions, turning the in-principle algorithm into a runnable one.","The class-string reduction scheme is a general device and may apply to other product questions in Sym(n), such as deciding when C1C2 contains a prescribed class for classes of intermediate size, or to two-class covering results for special cycle types.","The author leaves open whether a character-theoretic proof of Theorem 1.2 exists; the constructive nature of this proof suggests that such a proof, if found, would need different quantitative estimates from the usual character-sum bounds.","The tightness discussion implies that the true maximal δ is unknown and probably smaller than 1/2; a natural next step is a numerical study for moderate n to see how large δ can be before three-class coverage fails."],"forward_implications":["Kourovka Problem 20.23 is answered: three large conjugacy classes suffice to cover Alt(n), where previously four classes were needed.","Theorem 1.3 follows for every nonabelian finite simple group, since the Lie-type case was already known and the alternating case is supplied here, with normal subsets in place of conjugacy classes.","The four-class threshold |G|^{1/2+ε} from Garonzi and Maróti is improved to three classes with exponent 1−δ for Alt(n).","Rodgers' result requiring at most six cycles across three classes is superseded for classes above the size threshold, since the new proof needs only large class size, not a small number of cycles.","Because no character theory is used, the proof yields a constructive algorithm: given g and the three classes, it can output c1,c2,c3 with c1c2c3=g."],"supporting_citations":[{"why":"Supplies Proposition 2.3(a), the bound that large classes have few cycles, and the four-class theorem that this paper improves.","marker":"[GM21]"},{"why":"Supplies Proposition 2.3(b), the converse bound used to keep shortened intermediate classes large.","marker":"[DMP24]"},{"why":"Theorem 5.1(iii), quoted as Proposition 3.1, covers m-cycles and is the tool that upgrades Theorem 1.4 to Theorem 1.2.","marker":"[Dvi85]"},{"why":"The Kourovka Notebook is the source of Problem 20.23, the open question that Theorem 1.2 answers.","marker":"[KM22]"},{"why":"Theorem 1.3 for finite simple groups of Lie type is combined with Theorem 1.2 to obtain Theorem 1.3 for all nonabelian finite simple groups.","marker":"[MP21]"},{"why":"Theorem 7.4 gives the classical-group case used as an alternative input for Theorem 1.3.","marker":"[LST24]"}],"fun_headline_variants":["Three large classes multiply to full Alt(n)","Alt(n) from products of three large classes","Three big classes cover Alt(n) without character theory","Kourovka 20.23 solved for Alt(n)","Three classes, one product: whole Alt(n)"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole chain rests on Proposition 2.3, the quantitative link between class size and number of cycles: if that estimate fails to keep every shortened class nearly maximal after seven rounds of shortening, the argument no longer closes.","fun_headline_variants_meta":{"raw":{"variants":["Three large classes multiply to full Alt(n)","Alt(n) from products of three large classes","Three big classes cover Alt(n) without character theory","Kourovka 20.23 solved for Alt(n)","Three classes, one product: whole Alt(n)"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000227,"raw_usage":{"total_tokens":1458,"prompt_tokens":915,"completion_tokens":543,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":531,"completion_tokens_details":{"reasoning_tokens":469}},"tokens_in":531,"tokens_out":543,"duration_ms":6044,"temperature":1.0,"reasoning_tokens":469,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:52:28.373451+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find n and three conjugacy classes C1,C2,C3 of Alt(n) with |Ci| ≥ |Alt(n)|^{1−δ} and a single permutation g outside C1C2C3; such a triple would refute Theorem 1.2 directly. A cheaper partial check is to test Proposition 2.3 numerically on the extremal class of one (1−δ)n-cycle plus δn fixed points, the class used throughout, to see whether its cycle count stays within (δ+o(1))n; failure there would break the reduction chain.","supporting_citations":[],"review_version":1}