{"id":"641eefcf-05a3-4294-864b-782f4a5c8780","arxiv_id":"2506.13749","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves the average size of 2-torsion in class groups is 2 for one 3-adic ramification family of pure cubic fields and 3/2 for the other, and attributes the split to Hecke reciprocity.","lead":"Number theorists compute the average size of 2-torsion in class groups of cubic fields built from cube roots, obtaining 2 in one 3-adic ramification family and 3/2 in the other, with a new Hecke reciprocity law as the explanation. The paper also proposes a systematic heuristic for class groups in larger families of pure and resolvent-field extensions.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main averages depend on an unverified weighting φ=1/m(v) in Theorem 6.2, and Theorem 6.3 (for Theorem 1.1) is only sketched; the printed abstract/Theorem 1.2 also reverse the tame/wild labels.","rationale":"The reader's weakest assumption identifies the same load-bearing issue: the counting theorem is applied to a weighting whose hypothesis is not verified. This is not a stylistic gap. Theorems 7.3 and 7.7 produce the headline averages 2 and 3/2 entirely from the Euler product in Theorem 6.2, and the function φ(v)=1/m(v) is what turns rational orbit counts into Selmer group sizes. If admissibility fails, the local factors ν_p are not well-defined and the final averages are unsupported. The imaginary quadratic case is worse: Theorem 1.1 depends on Theorem 6.3, whose proof is only asserted to follow 'essentially unchanged' from the rational case, and the paper explicitly says the general number-field details have not been worked through. These are specific, checkable gaps rather than known contradictions. I also note the internal inconsistency in the printed statement: the abstract and Theorem 1.2 assign the 3/2 average to wildly ramified fields and 2 to tamely ramified fields, while Section 7 proves the reverse. That must be amended for the theorem statement to match the body, but it is a correction to presentation rather than an additional mathematical obstruction. The numerical tables and the coherence of the Selmer parametrization give genuine independent support, and the missing arguments are likely fillable, so the reader's CONDITIONAL verdict is the right one: the paper should not be accepted in its present form until the admissibility check and the proof of Theorem 6.3 are supplied and the labels are made consistent.","tokens_in":38115,"tokens_out":16459,"duration_ms":182076,"concrete_test":"Verify the admissibility hypothesis directly: for a rational orbit with primitive integral representative v and A3=n, prove an exact local formula m(v)=∏_p m_p(v), where m_p(v) is the number of G(Z_p)-orbits in the G(Q_p)-orbit of v, and show that m_p(v)=1 whenever p∤3n and p^2∤n, while m_3(v) is constant on the classes n≡±1 (mod 9) versus n not congruent to ±1 (mod 9). If this formula holds, φ is acceptable, Theorem 6.2 applies, and the concern does not land; if an exceptional prime p has m_p(v) depending on finer p-adic data, the Euler-product computation in Theorems 7.3 and 7.7 needs revision. Separately, expand the proof of Theorem 6.3 to a full argument following ABS22 and Bro21, including the analogue of [BHB21, Theorem 1.1] over O_K, and list every point where the circle method argument over Z must be changed for K=Q(√−3).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The quantitative results hinge on applying Theorem 6.2 to the G(Z)-invariant function φ(v)=1/m(v), where m(v) is the number of G(Z)-orbits inside the G(Q)-orbit of v. Theorem 6.2 requires φ to be acceptable and defined by congruence conditions, with an Euler product; the paper never proves this for m(v). If m(v) is not locally constant in the p-adic topology — for example, if the number of G(Z_p)-orbits in a G(Q_p)-orbit depends on more than the valuations of A3 and the class of n modulo 9 — then the integrals ν_p are not defined and the Euler product formula (6.3) cannot be invoked. The same missing verification enters the proof of Theorem 7.7, where the 3-adic factor ν3=1/2 is the entire source of the 3/2 average. For Theorem 1.1, the paper relies instead on Theorem 6.3, whose proof is explicitly only a sketch ('proof goes through essentially unchanged'), and the preceding paragraph admits the general number-field adaptation is not worked out. These are load-bearing because every numerical average in the abstract is produced by these counts. The separate reversal of the tame/wild labels in the abstract and Theorem 1.2 relative to Section 7 is also present and must be corrected, but the outstanding mathematical question is the admissibility of φ and the completed proof of Theorem 6.3.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the average size of the 2-torsion of class groups in families of pure cubic fields F_n = Q(∛n), split according to whether the field is wildly or tamely ramified at 3, and in the analogous Kummer family over K = Q(√-3). The method is to identify Cl_{F/K}[2]^∨ with an unramified Selmer group for the Galois module M = ker(Res_{F/K} μ_2 → μ_2), to introduce a global 'Hecke reciprocity' constraint on the number of inert Hecke primes, and to parameterize Selmer elements by orbits of pairs of binary cubic forms with vanishing A_1-invariant. The counting of such orbits is imported from [ABS22]. The main numerical conclusions as printed in Section 7 are that the average of |Cl_F[2]| is 2 for wildly ramified n ≢ ±1 (mod 9) and 3/2 for tamely ramified n ≡ ±1 (mod 9); the abstract and Theorem 1.2 print the opposite labels. For K = Q(√-3), the paper claims average 3/2. Sections 9–11 propose heuristics for 2-torsion in families of Γ-extensions, including explicit predictions for pure cubic, quintic, and septic fields, with extensive Magma tables in Appendix A.","tokens_in":38431,"tokens_out":5457,"duration_ms":56552,"significance":"If the counting ingredients are fully supplied, the results would be a significant advance: Theorem 1.1 would be the first proven instance of the Sawin–Wood conjectural moment formula (1.1) in the case (G,K,V) = (C3, Q(√-3), F4), and Theorems 7.3/7.7 would give the first rigorous explanation of the Cohen–Martinet–Williams–Shanks dichotomy for pure cubic fields. The paper also contains a genuinely new structural idea, Hecke reciprocity, and proposes a concrete framework (gyroscopic families, aberrant Γ-groups) with falsifiable numerical predictions. The appendix supplies reproducible-looking data and explicit code availability statements. However, as printed the central average theorems depend on two unproven counting inputs—the admissibility of the weight φ = 1/m(v) in Theorem 6.2 and the sketched adaptation in Theorem 6.3—and the abstract/Theorem 1.2 are inconsistent with Section 7. The underlying algebraic framework is coherent, but the paper is not yet in a form where the advertised theorems are established.","major_comments":[{"comment":"The proof of Theorem 7.3 applies Theorem 6.2 to the weight φ(v) = 1/m(v), where m(v) is the number of G(Z)-orbits in the G(Q)-orbit of v. Theorem 6.2 requires φ to be an acceptable G(Z)-invariant function defined by congruence conditions, with an Euler product expansion (6.3). The paper never proves that 1/m(v) satisfies the required local conditions: in particular, it does not verify that m(v) is locally constant on Y(Z_p) in the relevant p-adic topology, nor that φ_p(y)=1 for all sufficiently large p when p^2 ∤ A_3(y). The Euler factors ν_p in Section 7 are obtained from this factorization, so the final averages in Theorems 7.3 and 7.7 are conditional on an unstated hypothesis about m(v). This is load-bearing because every numerical average in the paper is produced by these orbit counts.","section":"§6, Theorem 6.2; §7, proofs of Theorems 7.3 and 7.7"},{"comment":"Theorem 1.1 depends on Theorem 6.3, whose proof is explicitly only a sketch: the text says the proof of [ABS22, Theorem 4.1] 'goes through essentially unchanged' and the preceding paragraph states that the general number-field adaptation has not been worked through. Theorem 8.3 uses Theorem 6.3 to obtain the main term X·(...) and the Euler product over primes of K = Q(√-3). As printed, the average 3/2 over the Kummer family over Q(√-3) is therefore not a proved theorem but a conditional statement pending a complete proof of Theorem 6.3 (or a precise reference establishing the analogue of the counting theorem over imaginary quadratic fields of class number one).","section":"§6, Theorem 6.3; §8, Theorem 8.3"},{"comment":"The tame/wild terminology is reversed in the abstract and in Theorem 1.2 relative to Section 7. In Theorem 1.2, W(X) is defined as n ≡ ±1 (mod 9) and is assigned average 2, while T(X) is n ≢ ±1 (mod 9) with average 3/2. But in Section 7 the wild family is n ≢ ±1 (mod 9) (Theorem 7.3, average 2) and the tame family is n ≡ ±1 (mod 9) (Theorem 7.7, average 3/2). The abstract's 'wildly (resp. tamely) ramified fields ... average 3/2 (resp. 2)' prints the opposite assignment. This inconsistency affects the central claims exactly as stated in the abstract and theorem, and must be corrected.","section":"Abstract and Theorem 1.2"}],"minor_comments":[{"comment":"The sentence 'because of the obstruction coming from Theorem 5.8' appears to refer to Proposition 7.4; Theorem 5.8 actually states that all classes in Sel^un_2(F_n) lie in ker(q_n), so the local obstruction at 3 is not coming from Theorem 5.8 as written.","section":"§7.2, after Proposition 7.4"},{"comment":"The notation 'p = p' in 'the only possible Hecke prime is p = p' is confusing because p denotes both a prime of K and the rational prime 3; please use a distinguishing notation such as p_3.","section":"§8.1, Proposition 8.1 and proof of Theorem 8.3"},{"comment":"In the proof of Theorem 5.5 the step 'v(p) is odd, by Lemma 5.6' is terse: Lemma 5.6 gives b ≡ -p (mod K×2), and one needs to spell out that in the totally ramified case the valuation of b is odd. Adding one sentence would make the local computation easier to check.","section":"§5.1, Lemma 5.6"},{"comment":"The tables report observed moments to four decimal places, but the text does not state the standard error or the exact number of fields used for each individual moment. Since the agreement for large |H| is visibly imperfect, reporting the error bars or field counts would strengthen the numerical evidence.","section":"Appendix A"},{"comment":"The statement of Theorem 6.2 counts 'irreducible' G(Z)-orbits, but the application in Section 7 uses all orbits corresponding to unramified vectors; the paper should clarify that every orbit arising from Sel^un_2(F_n) is irreducible, or that the counting theorem applies to the appropriate subset.","section":"§6, Theorem 6.2"}],"recommendation":"major_revision","confidential_remarks":"The central counting input is [ABS22], which is coauthored by the first author. This is a legitimate citation of prior work, but the paper does not highlight the overlap in the text; editors may wish to confirm that the contribution of the present paper is appropriately separated from the counting theorem. Additionally, the numerical data are described as reproducible with code on GitHub, but no URL is given in the manuscript; including a link or repository identifier would make the reproducibility claim verifiable. The label reversal in the abstract and Theorem 1.2 is the kind of error that must be fixed before any further editorial decision, but the more substantive issue is the unverified admissibility of φ = 1/m(v) and the sketched proof of Theorem 6.3, which are load-bearing for the main theorems as printed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a genuine contribution, but the printed paper garbles its own main theorem and has a gap in the counting step that supports every numerical average. The body of §7 gives wild average 2 and tame average 3/2 for Q(∛n), and the tables match that. The abstract and Theorem 1.2 print the reverse. That's confusing but harmless once you decode it. The deeper problem is that Theorem 6.2 is applied to φ(v)=1/m(v) without checking that this function is acceptable and defined by congruence conditions in the sense of the theorem. The local factor ν3=1/2 in the tame case is exactly what produces the 3/2 average, so this is not cosmetic. Theorem 6.3, which carries the counting over Q(√-3), is only sketched. Both gaps are probably fixable, but as written they are load-bearing.\n\nWhat's genuinely new: Hecke reciprocity as a constraint on Cl_F/K[2] is a nice idea, and the Γ-extension heuristics in §10 are a sensible way to organize the dichotomy. The first exact averages in these Kummer families are a real step beyond Ruth's upper bound. The paper is honest about which parts are conjectural.\n\nI would send this to a serious referee. The referee should demand a proof that the weighting φ satisfies the hypotheses of Theorem 6.2, and a complete proof of Theorem 6.3 (or a statement that Theorem 1.1 depends on a conjecture). The label swap should be fixed in the abstract and introduction. After those changes, the central claims look solid to me.","headline":"Strong paper with a real new idea and first exact averages, but the abstract and Theorem 1.2 swap the tame/wild averages, and the counting step in §7 needs an unverified weighting hypothesis fixed before I'd trust the numbers.","tokens_in":38964,"tokens_out":3998,"would_cite":true,"duration_ms":36700,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11R16","11R29","11R45"],"pacs":[],"model":"deepseek-v4-flash","headline":"A reciprocity law for Hecke primes fixes the average 2-torsion class group sizes of pure cubic fields at 2 and 3/2.","keywords":["class group 2-torsion","pure cubic fields","Hecke reciprocity","Kummer extensions","Cohen-Martinet heuristics","arithmetic invariant theory","quadratic refinement","orbit counting"],"falsifier":"For cubefree integers $n\\leq X$, split the family by $n\\equiv\\pm 1 \\pmod{9}$ versus $n\\not\\equiv\\pm 1 \\pmod{9}$, compute the average of $|\\mathrm{Cl}_{\\mathbb{Q}(\\sqrt[3]{n})}[2]|$, and check whether the two averages tend to $3/2$ and $2$ respectively as $X\\to\\infty$; the paper's central claim fails if either limit differs. Equivalently, verifying the acceptability condition for $\\phi(v)=1/m(v)$ would settle the completeness of the orbit-counting step.","tokens_in":37904,"feed_emoji":"🔢","tokens_out":11938,"duration_ms":104271,"temperature":0.7,"pith_summary":"This paper claims that the average size of the 2-torsion subgroup of the class group of pure cubic fields $F_n=\\mathbb{Q}(\\sqrt[3]{n})$ is $2$ in the wild family ($n\\not\\equiv\\pm 1 \\pmod{9}$) and $3/2$ in the tame family ($n\\equiv\\pm 1 \\pmod{9}$), when the fields are ordered by $n$. It derives both averages from a single parity law: an everywhere unramified quadratic extension of an odd-degree field must split in an even number of Hecke primes. That law explains why the tame family, which has exactly one Hecke prime above $3$, loses one local square-class and therefore has the smaller average. The same machinery gives average $3/2$ over $K=\\mathbb{Q}(\\sqrt{-3})$, matching the Cohen–Martinet prediction, and supplies new heuristics for families of $\\Gamma$-extensions. The abstract prints the tame/wild labels in the opposite order from §7; the body's theorems are the ones summarized here.","feed_headline":"Hecke primes fix 2-torsion averages at 2 and 3/2","feed_subtitle":"A reciprocity law splits pure cubic fields into tame and wild families with different class-group averages.","key_machinery":"The load-bearing object is the Hecke ideal $H_{F/K}=\\mathrm{Disc}_{F/K}\\mathrm{Diff}_{F/K}^{-1}$, together with the associated Hecke primes, and the quadratic refinement $q_n\\colon H^1(K,M_n)\\to \\mathrm{Br}(K)[2]$ attached to the Kummer extension $F_n=K(\\sqrt[3]{n})$. The refinement sends a square-class $t$ to the class of the quadratic form $\\frac{1}{3}\\mathrm{Tr}_{F/K}(tx^2)$ in $H^1(K,\\mathrm{SO}(3))\\cong \\mathrm{Br}(K)[2]$; its kernel selects the classes that correspond, via arithmetic invariant theory, to $G(K)$-orbits of pairs of binary cubic forms with $A_1=0$ and $A_3=n$, where $G=\\mathrm{SL}_2^2/\\mu_2$. The counting theorem of [ABS22] then turns local orbit counts into Euler products: the average is $1+2\\prod_p \\nu_p$, with Tamagawa number $2$, $\\nu_p=1$ at $p\\neq 3,\\infty$, $\\nu_\\infty=1/2$, and $\\nu_3=1$ in the wild family but $\\nu_3=1/2$ in the tame family, because Hecke reciprocity keeps the nontrivial class inside the global Selmer group while placing it outside the local kernel of $q_{n,3}$.","core_discovery":"On the paper's own terms, the central discovery is a parity constraint on 2-torsion classes. For an odd-degree extension $F/K$, define the Hecke ideal $H_{F/K}=\\mathrm{Disc}_{F/K}\\mathrm{Diff}_{F/K}^{-1}$; a Hecke prime is a prime whose exponent in this ideal is odd and that is locally inert in an unramified quadratic square-norm extension. The Hecke reciprocity theorem says that any unramified quadratic extension $F(\\sqrt{t})/F$ with square norm has an even number of inert Hecke primes. In the family $\\mathbb{Q}(\\sqrt[3]{n})$, a tame prime above $3$ has splitting type $(121)$ and supplies exactly one Hecke prime, so reciprocity forbids the nontrivial unramified local square-class from occurring globally; this cuts the local factor at $3$ from $1$ to $1/2$ and turns the wild-family average $1+2\\cdot(1/2)\\cdot 1=2$ into the tame-family average $1+2\\cdot(1/2)\\cdot(1/2)=3/2$. Over $K=\\mathbb{Q}(\\sqrt{-3})$, odd degree Galois extensions are Hecke unramified, no obstruction appears, and the average is $3/2$, confirming the Cohen–Martinet prediction. The abstract states the correspondence with the wild/tame labels reversed; throughout §7 the wild family is $n\\not\\equiv\\pm 1 \\pmod{9}$ and the tame family is $n\\equiv\\pm 1 \\pmod{9}$.","pith_inferences":["If the abstract's reversed labels are a typo, the paper's headline claim should be quoted from §7; the appendix tables labeled type I (wild) and type II (tame) are consistent with the body's values $2$ and $3/2$.","The same mechanism should be visible in higher moments: if Hecke reciprocity acts as an extra relation on the alternating-matrix model of §9, then the full distribution of 2-ranks in the tame family should match a $u$-shifted Cohen–Lenstra–Martinet distribution, not just the first moment.","The Hecke-ideal parity constraint is not limited to Kummer families: any odd-degree family whose resolvent field is $\\mathbb{Q}(i)$, $\\mathbb{Q}(\\sqrt{\\pm 3})$, or another aberrant twist should display a 2-torsion boost, and the one-parameter family $x^3-3ax^2-3x+a$ with resolvent $\\mathbb{Q}(\\sqrt{3})$ provides a direct test.","Corollary 3.10 suggests that families with exactly one Hecke prime are the extremal case: one could test whether any transitive permutation group whose members generically have a single Hecke prime reproduces the tame-family average $3/2$ rather than the wild-family average $2$."],"forward_implications":["In the wild family $n\\not\\equiv\\pm 1 \\pmod{9}$, the average of $|\\mathrm{Cl}_{\\mathbb{Q}(\\sqrt[3]{n})}[2]|$ over cubefree $n$ up to $X$ is $2$.","In the tame family $n\\equiv\\pm 1 \\pmod{9}$, the average is $3/2$, so at least 50% of these fields have odd class number.","Over $K=\\mathbb{Q}(\\sqrt{-3})$, cubic Kummer extensions $K(\\sqrt[3]{n})$ ordered by norm have average $|\\mathrm{Cl}_{F/K}[2]| = 3/2$, confirming the predicted moment for $C_3$-extensions of $K$.","The Hecke-unramified versus Hecke-ramified split becomes a general heuristic: for $\\Gamma$-extensions with even $|G|$, the predicted distribution on Hecke-unramified subfamilies shifts by one extra relation, so the pure $p$-th power families $\\mathbb{Q}(\\sqrt[p]{n})$ should repeat the dichotomy for every odd prime $p$.","Conjecture 1.7 makes explicit probabilities for class number one: about $0.5662$ for primes $n\\equiv 8 \\pmod{9}$ and about $0.3775$ for $n\\equiv 2,5 \\pmod{9}$, refining the old observed distinction in pure cubic fields."],"supporting_citations":[{"why":"Supplies the counting theorem for integral $G(\\mathbb{Z})$-orbits on the invariant quadric with bounded $A_3$, the Euler-product estimate behind Theorems 7.3 and 7.7.","marker":"[ABS22]"},{"why":"Identifies the map from $H^1(K,\\mathrm{Stab}(v_n))$ to $H^1(K,G)\\simeq \\mathrm{Br}(K)[2]$ with the quadratic refinement $q_n$.","marker":"[BG13]"},{"why":"Provides the arithmetic invariant theory bijection between $G(K)$-orbits with invariants $A_1=0$, $A_3=n$ and the kernel of $q_n$ used in Proposition 6.1.","marker":"[BG14]"},{"why":"Hecke's theorem that the different is a square in the class group underlies the Hecke ideal and the reciprocity law.","marker":"[Wei74]"},{"why":"Supplies the Sawin–Wood moment prediction that Theorem 1.1 verifies in the $(C_3,\\mathbb{Q}(\\sqrt{-3}),F_4)$ case.","marker":"[SW23]"},{"why":"Malle's heuristic for class group 2-parts in odd-degree extensions is the distribution the paper recovers and then modifies for Hecke-unramified subfamilies.","marker":"[Mal10]"}],"fun_headline_variants":["Hecke reciprocity splits cubic fields: tame vs wild 2-torsion","Tame and wild cubic fields have different class group averages","Hecke primes set average 2-torsion: 2 for wild, 3/2 for tame","Reciprocity law explains tame-wild gap in 2-torsion averages"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The computation assumes that the orbit-weighting $\\phi(v)=1/m(v)$, where $m(v)$ is the number of $G(\\mathbb{Z})$-orbits inside a $G(\\mathbb{Q})$-orbit, is an 'acceptable' weighting defined by congruence conditions, so that the counting theorem's Euler-product formula applies; the paper states this hypothesis but does not verify it, and Theorem 1.1 also rests on a sketched generalization of the counting theorem to imaginary quadratic fields.","fun_headline_variants_meta":{"raw":{"variants":["Hecke reciprocity splits cubic fields: tame vs wild 2-torsion","Tame and wild cubic fields have different class group averages","Hecke primes set average 2-torsion: 2 for wild, 3/2 for tame","Reciprocity law explains tame-wild gap in 2-torsion averages"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000424,"raw_usage":{"total_tokens":2295,"prompt_tokens":1188,"completion_tokens":1107,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":804,"completion_tokens_details":{"reasoning_tokens":1019}},"tokens_in":804,"tokens_out":1107,"duration_ms":9672,"temperature":1.0,"reasoning_tokens":1019,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:58:29.504851+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For cubefree integers $n\\leq X$, split the family by $n\\equiv\\pm 1 \\pmod{9}$ versus $n\\not\\equiv\\pm 1 \\pmod{9}$, compute the average of $|\\mathrm{Cl}_{\\mathbb{Q}(\\sqrt[3]{n})}[2]|$, and check whether the two averages tend to $3/2$ and $2$ respectively as $X\\to\\infty$; the paper's central claim fails if either limit differs. Equivalently, verifying the acceptability condition for $\\phi(v)=1/m(v)$ would settle the completeness of the orbit-counting step.","supporting_citations":[],"review_version":2}