{"id":"b702f56a-dca9-42c8-98b9-dfb2a07c1f30","arxiv_id":"2608.00387","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"The 4-ranks of the class groups of Q(sqrt(-d)) and Q(sqrt(-d0 d)) are asymptotically independent, each with the Cohen-Lenstra-Gerth distribution.","lead":"This paper proves that two related number fields, obtained by taking square roots of -d and of -d0 d, have class-group 4-ranks that become statistically independent as d grows. A smart generalist should read it because it gives the first joint distribution result for this correlated family and a corrected Cohen-Lenstra prediction for biquadratic fields.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1 is not affected by the flagged Lemma 5.5, but the advertised biquadratic results depend on a sketched sign correction to [16] that should be independently verified before full acceptance.","rationale":"The reader identified Lemma 5.5 as the weakest assumption, and I agree. The concern is precise, concrete, and directly tied to the secondary but prominently advertised biquadratic results: Theorem 1.3, Conjecture 1.4, and Corollary 1.5 all depend on the corrected 4-rank formula via Proposition 5.6. The proof of Lemma 5.5 is not a full derivation; it asserts a sign-dependent change to an unpublished preprint and does not reproduce the Selmer computation. Since the formula's constant enters lambda_K additively, a one-unit error would change the direct-sum defect and invalidate the 8-rank corollary. This warrants the reader's CONDITIONAL verdict. I did not find a comparable flaw in the proof of the central Theorem 1.1: the reduction to residue assignments, the box-method transfer, and the bordered random-matrix moment inversion are internally consistent as far as I can verify without a full line-by-line check. Therefore the appropriate recommendation is to preserve the reader's CONDITIONAL verdict, with the requested independent verification of Lemma 5.5 as the condition. Setting verdict_should_be to UNCHANGED reflects that my stress-test does not move the reader's assessment; it confirms it.","tokens_in":49792,"tokens_out":9840,"duration_ms":94696,"concrete_test":"Independently re-derive Lemma 5.5 by computing both sides for F = Q(sqrt(5)) (which has odd class number) and a large sample of negative squarefree n with omega_in(Delta(n)) >= 3, using PARI/GP to compute rk4 Cl_{F(sqrt(-n))}; if the identity fails on a positive-density subset, the lemma is false. Alternatively, reproduce the Selmer exact sequence in Section 3.1 of [16] with n < 0, tracking the archimedean local conditions, and verify that the terminal '-1' printed in Theorem 3.4 is indeed absent in the negative sign range; this single constant is the crux of Proposition 5.6.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing unresolved assumption is Lemma 5.5. For real quadratic F and negative squarefree n it asserts rk4 Cl_{F(sqrt(n))} = omega_in(Delta(n)) + omega(Delta_F) + rk2 Cl_F - 2, correcting [16, Theorem 3.4] by omitting a terminal '-1'. The proof in Section 5.2.3 is only a sketch: it argues that since n < 0, the character chi_n is nontrivial at both real places, so the local conditions force Sel_{chi_n} ∩ {0, chi_n} = {0}, and therefore the one-dimensional kernel term printed in [16] must be dropped. The Selmer dimension identity itself, and the claim that exactly the same generic set works, are not reproduced. Proposition 5.6 converts this constant into the direct-sum defect lambda_K, and Theorem 1.3 and Corollary 1.5 collapse if the constant is off by one. Because [16] is an unpublished arXiv preprint, the correction cannot be checked by citation alone. This is a genuine conditional-acceptance concern. It does not bear on Theorem 1.1, whose proof in Sections 3-4 and 5.1 never invokes Lemma 5.5; the main independence theorem rests on the box-method transfer and the bordered random-matrix theorem, and I found no concrete flaw there.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper fixes a squarefree integer d0 > 1 and studies the joint distribution of the class-group 4-ranks of the two imaginary quadratic fields Q(√−d) and Q(√−d0d), where d ranges over squarefree integers coprime to 2d0. The principal theorem (Theorem 1.1) states that this joint distribution converges in total variation to πCL ⊗ πCL, the product of two Cohen–Lenstra–Gerth distributions, with error O((log log X)^−c). The proof combines Rédei–Reichardt matrix reduction, Smith's box-method equidistribution for residue symbols, and a new random-matrix theorem for the corank pair of two coupled bordered matrices sharing a common core. The paper also proves, under the hypothesis that Q(√d0) has odd class number, that for a density-one subset of d the map to 4Cl_{K(d)}[2∞] decomposes as a direct sum of the corrected 2-primary groups of the three quadratic subfields (Theorem 1.3), and derives the limiting 8-rank distribution for the biquadratic field (Corollary 1.5). The biquadratic part relies on Lemma 5.5, a sign-range correction to an identity of Koymans–Morgan–Smit [16] that is only sketched in the paper.","tokens_in":50043,"tokens_out":5304,"duration_ms":49901,"significance":"If correct, Theorem 1.1 is a substantial advance: it establishes asymptotic independence of 4-ranks in a naturally correlated family of imaginary quadratic fields, with an explicit total-variation rate, and it gives a corrected Cohen–Lenstra–Martinet prediction for biquadratic fields that is verified unconditionally at the 8-rank level. The central derivation is detailed and internally consistent: the target distributions are externally defined rather than fitted, the random-matrix limiting moments are computed independently, and the error terms are explicit throughout. The proof of Theorem 1.1 rests on the box-method transfer of Section 3 and the bordered random-matrix theorem of Section 4; I found no concrete flaw in that part. The advertised biquadratic applications, however, depend on Lemma 5.5, which is load-bearing and is not fully proved in the manuscript.","major_comments":[{"comment":"This lemma is load-bearing for Theorem 1.3 and Corollary 1.5 through Proposition 5.6, but its proof is only a sketch. The paper asserts that the terminal '-1' in [16, Theorem 3.4] must be omitted when n < 0 and that 'the same generic set' used by Koymans–Morgan–Smit works, yet it does not reproduce the Selmer dimension computation or the verification that the generic-set conditions are unchanged in the negative sign range. Since [16] is an unpublished arXiv preprint, this correction cannot be checked by citation alone. I recommend either giving a complete proof of (82) in this paper or citing a published version of [16] that explicitly covers the real-base negative-twist case. This issue does not affect Theorem 1.1: Sections 3–4 and 5.1 never invoke Lemma 5.5.","section":"§5.2.3, Lemma 5.5, Eq. (82)"},{"comment":"Lemma 5.1, quoted from [16, Corollary 2.3], is used in the density-one proofs of the Hasse unit index and the 2-rank formula. The statement is standard and the proof is short, but because it is currently cited only from an unpublished source, it should be proved in the paper or replaced by a published reference. This is a secondary point relative to Lemma 5.5, but it is part of the same reliance on [16].","section":"§5.2.1, Lemma 5.1"}],"minor_comments":[{"comment":"The word 'sufficiently' is consistently misspelled as 'sufficiently' (see Definition 3.4, Proposition 3.5, and elsewhere); this should be corrected.","section":"Throughout"},{"comment":"The title appears as 'CORRELA TED P AIRS' with spurious spaces; the typesetting should be fixed.","section":"Title, page 1"},{"comment":"The remark that bordered sign vectors with |u_-1|/r → 0, including u_-1 = 0, 'require separate analysis' could be misread as contradicting Theorem 1.1. It would help to state explicitly that in the arithmetic application these sign vectors have mass O(e^{-cr}) in the sign mixture (33), so their omission from the matrix theorem does not affect Theorem 1.1.","section":"Remark 4.22"}],"recommendation":"major_revision","confidential_remarks":"I am confident in the main independence theorem; the random-matrix and box-method arguments appear sound and are carefully quantified. The only substantive obstacle to full acceptance is the unfinished Lemma 5.5 and the related reliance on the unpublished preprint [16]. If the authors supply a complete proof of Lemma 5.5 (or a published version of [16] covering the negative-twist sign range), I would support acceptance. The self-citation to [36] is appropriate and not a concern."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Hi — the headline is that this paper proves a genuinely new joint 4-rank independence theorem for two imaginary quadratic fields related by a fixed squarefree twist, and the proof is largely convincing. I think it deserves a serious referee, with the biquadratic corollaries gated on one lemma.\n\nWhat's new: Theorem 1.1, that the 4-ranks of Q(√-d) and Q(√-d0 d) are asymptotically independent with Cohen–Lenstra–Gerth marginals, is not a restatement of anything in the literature. The proof combines Smith's box method with a quantitative moment inversion for coupled bordered Rédei matrices; the random-matrix Theorem 1.6 (diagonal perturbation makes coranks independent, with πsym or πCL limits depending on symmetry defect) is itself new in the F2 setting. The fixed-twist family is not covered by reflection-principle results or by Smith's degree-ℓ twist framework, so the contribution is real.\n\nThe paper is unusually explicit about errors and good events, and I found no concrete flaw in Sections 3–4 or in the proof of Theorem 1.1. The box transfer and the weighted moment inversion are spelled out in enough detail to be checkable.\n\nThe soft spot is Lemma 5.5, which corrects [16, Theorem 3.4] by removing a terminal '-1' for negative twists. It is load-bearing for Theorem 1.3 and Corollary 1.5, and it is only sketched. The argument via Selmer intersection at real places is plausible, but it adapts an unpublished preprint and does not reproduce the full Selmer computation. That is a real but localized gap. It does not bear on Theorem 1.1—I traced the main proof and it never uses Lemma 5.5.\n\nIf the biquadratic application is meant to stand, Lemma 5.5 needs an independent proof or a published source. Once that is supplied, the paper should be accepted.\n\nThis is for number theorists working on class groups and Cohen–Lenstra heuristics; the random-matrix part also speaks to the random-group community. I would cite the main independence theorem. Send it to a good referee, especially one who can check both the box-method transfer and the Selmer identity.","headline":"The main 4-rank independence theorem is solid and new; the biquadratic application rests on an unverified sign correction to an unpublished preprint and should be conditional.","tokens_in":50592,"tokens_out":2228,"would_cite":true,"duration_ms":18844,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11R29","11R11","15B52","60B20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that the class-group $4$-ranks of $\\mathbb{Q}(\\sqrt{-d})$ and $\\mathbb{Q}(\\sqrt{-d_0 d})$ are asymptotically independent as $d$ varies, even though the two fields share every variable ramified prime.","keywords":["class group","4-rank","Cohen-Lenstra-Martinet heuristics","biquadratic field","Redei matrix","asymptotic independence","Gaussian-binomial moments","truncated moment inversion"],"falsifier":"For Theorem 1.1: with $d_0=2$ and $X$ of size around $10^9$–$10^{10}$, tabulate the empirical joint distribution of $(r_4(-d),r_4(-2d))$ over squarefree odd $d\\le X$ and measure its total-variation distance to $\\pi_{\\mathrm{CL}}\\otimes\\pi_{\\mathrm{CL}}$; the theorem predicts decay like $(\\log\\log X)^{-c}$, so a distance that does not shrink as $X$ grows would refute it. For Lemma 5.5: verify the Selmer dimension identity independently in the negative-twist range for a concrete real quadratic field $F$ (for instance $F=\\mathbb{Q}(\\sqrt{5})$ and small squarefree $|n|$) by computing the three Selmer groups in (82); any instance where the terminal $-1$ of [16, Theorem 3.4] is needed, rather than omitted, falsifies the correction and with it Theorem 1.3.","tokens_in":49555,"feed_emoji":"🎲","tokens_out":21234,"duration_ms":160230,"temperature":0.7,"pith_summary":"Fix a squarefree multiplier $d_0>1$ and let $d$ range over squarefree integers coprime to $2d_0$. Although the two imaginary quadratic fields $\\mathbb{Q}(\\sqrt{-d})$ and $\\mathbb{Q}(\\sqrt{-d_0 d})$ ramify at exactly the same variable primes, the paper proves that their class-group $4$-ranks are asymptotically independent: over $d\\le X$ the joint distribution converges in total variation to the product of two copies of the Cohen–Lenstra–Gerth distribution, with error $O((\\log\\log X)^{-c})$. The $4$-rank is the first level at which such a statement can hold, since genus theory forces a deterministic relation between the two raw $2$-ranks. The mechanism is a reduction to finite-field random matrix theory: after deleting one common prime, the two Rédei matrices share a common core, the fixed twist by $d_0$ acts as a diagonal perturbation, and the corank of a random symmetric matrix is shown to be asymptotically insensitive to a high-rank diagonal shift. If the main theorem is right, correlated ramification does not force correlated class-group statistics, and the paper uses this to deduce a limiting distribution for the $8$-rank of the biquadratic field $K(d)=\\mathbb{Q}(\\sqrt{d_0},\\sqrt{-d})$.","feed_headline":"Class-group 4-ranks are independent despite shared ramification","feed_subtitle":"The joint 4-rank law of the paired fields becomes a product of Cohen–Lenstra–Gerth distributions.","key_machinery":"The central object is the pair of coupled, fixed-width bordered Rédei matrices attached to the two fields. A Rédei matrix for a quadratic field records the additive quadratic-residue symbols $[\\Delta_i/p_j]$ among its prime-discriminant factors, and the Rédei–Reichardt criterion identifies the class-group $4$-rank with the corank of this matrix after deleting one variable row and column. Because the two fields share every variable ramified prime, deleting one common odd prime leaves a common core matrix; the fixed twist by $d_0$ appears as a diagonal perturbation, and the primes dividing $2d_0$ contribute a bounded number of border rows and columns. The argument has two stages. First, the box-method equidistribution theorem of [31] shows that the permutation-averaged distribution of the relevant residue symbols over admissible boxes is close to uniform, so the arithmetic problem becomes the joint corank distribution of the random matrix pair. Second, a quantitative truncated inversion of the Gaussian-binomial moment transform — the mixed moments $\\mathbb{E}[\\binom{Z_1}{u}_2\\binom{Z_2}{v}_2]$ counting average kernel subspaces — transfers mixed-moment estimates through a growing order $D$ into an exponential total-variation bound $\\ll\\eta^r$. The key finite-field fact is that the corank pair of a uniform symmetric matrix and its high-rank diagonal perturbation is asymptotically distributed as $\\pi_{\\mathrm{CL}}\\otimes\\pi_{\\mathrm{CL}}$; the borders and stratum data are uniformized over and absorbed into the constants.","core_discovery":"The paper's principal theorem, Theorem 1.1, states that for every squarefree $d_0>1$ there is $c=c(d_0)>0$ with $d_{\\mathrm{TV}}(\\mu_X^{(d_0)},\\pi_{\\mathrm{CL}}\\otimes\\pi_{\\mathrm{CL}})\\ll_{d_0}(\\log\\log X)^{-c}$, where $\\mu_X^{(d_0)}$ is the joint distribution of $(r_4(-d),r_4(-d_0 d))$ over squarefree $d\\le X$ coprime to $2d_0$ and $\\pi_{\\mathrm{CL}}$ is the Cohen–Lenstra–Gerth distribution on $\\mathbb{Z}_{\\ge 0}$, with masses $\\pi_{\\mathrm{CL}}(m)=2^{-m^2}\\eta_\\infty(2)/\\eta_m(2)^2$. The two $4$-ranks are therefore asymptotically independent, each with the Cohen–Lenstra–Gerth law. The authors conjecture the group-valued refinement that the corrected $2$-primary groups $2\\mathrm{Cl}_{\\mathbb{Q}(\\sqrt{-d})}[2^\\infty]$ and $2\\mathrm{Cl}_{\\mathbb{Q}(\\sqrt{-d_0 d})}[2^\\infty]$ are asymptotically independent with the Cohen–Lenstra measure on $2$-groups; Theorem 1.1 is the $4$-rank projection of this conjecture. Under the hypothesis that $\\mathbb{Q}(\\sqrt{d_0})$ has odd class number, Theorem 1.3 shows that for a density-one subset of $d$ extension of ideals induces $4\\mathrm{Cl}_{K(d)}[2^\\infty]\\cong 2\\mathrm{Cl}_{\\mathbb{Q}(\\sqrt{-d})}[2^\\infty]\\oplus 2\\mathrm{Cl}_{\\mathbb{Q}(\\sqrt{-d_0 d})}[2^\\infty]$, and Corollary 1.5 derives that the $8$-rank of $\\mathrm{Cl}_{K(d)}$ has limiting distribution equal to the convolution $\\pi_{\\mathrm{CL}}*\\pi_{\\mathrm{CL}}$.","pith_inferences":["The mechanism isolates a random-matrix principle that outlives this family: the corank of a uniform symmetric matrix over $\\mathbb{F}_2$ is asymptotically independent of a diagonal perturbation whose support has positive density (Theorem 1.6). I take this as evidence that fixed-multiplier pairings of quadratic fields exhibit asymptotic independence at all $2^j$-rank levels, not just $j=2$, exactly","The truncated Gaussian-binomial inversion is run at depth $D\\asymp\\sqrt{n}$ in the bordered case because the target moments decay like $2^{-u^2}$ (case II of Theorem 4.12). Since these moments at growing order determine the full finite kernel structure, the same input looks sufficient to prove the group-valued conjecture rather than only its $4$-rank projection, although the paper does not carry t","The $u_{-1}=0$ subfamily — $d$ with all primes $1\\bmod 4$ — is left open at the bordered level (Remark 4.22): with zero symmetry defect the limiting mixed moments of bordered pairs are not computed, and the limit need not be $\\pi_{\\mathrm{sym}}\\otimes\\pi_{\\mathrm{sym}}$ once corners are present. Computing those moments would yield a conditional independence statement for the all-$1\\bmod 4$ family,","Remark 3.9 indicates the same apparatus can run under fixed Frobenius conditions; because the target distribution then changes (it is governed by conditional residue assignments), any such conditional independence theorem would be a genuinely new prediction rather than a repackaging of Theorem 1.1."],"forward_implications":["The joint distribution of $(r_4(-d),r_4(-d_0 d))$ over squarefree $d\\le X$ coprime to $2d_0$ converges in total variation to $\\pi_{\\mathrm{CL}}\\otimes\\pi_{\\mathrm{CL}}$ at rate $(\\log\\log X)^{-c}$: the two $4$-ranks are asymptotically independent despite sharing every variable ramified prime.","Projecting onto either coordinate recovers the known one-field Cohen–Lenstra–Gerth $4$-rank distribution, so the theorem is consistent with and refines the established marginals.","With $d_0=2$, the pairing $d\\mapsto 2d$ runs over all imaginary quadratic fields (odd and even squarefree radicands, respectively), and their $4$-ranks are asymptotically independent under this natural pairing.","Assuming odd class number of $\\mathbb{Q}(\\sqrt{d_0})$, the $8$-rank of $\\mathrm{Cl}_{K(d)}$ has limiting distribution $\\pi_{\\mathrm{CL}}*\\pi_{\\mathrm{CL}}$ with total-variation convergence, and a density-one subfamily satisfies $\\mathrm{rk}_8\\,\\mathrm{Cl}_{K(d)}=r_4(-d)+r_4(-d_0 d)$.","If Conjecture 1.2 holds, the corrected $2$-primary groups are jointly two independent Cohen–Lenstra $2$-groups, and Theorem 1.3 turns this into a corrected Cohen–Lenstra–Martinet prediction for $4\\mathrm{Cl}_{K(d)}[2^\\infty]$ in the biquadratic family."],"supporting_citations":[{"why":"Provides the box-method residue-symbol equidistribution theorem and the positive covering that reduce the arithmetic family to uniform residue assignments on admissible boxes; it is the main input of Section 3.","marker":"[31]"},{"why":"Establishes the one-field Cohen–Lenstra–Gerth distribution for imaginary quadratic fields that Theorem 1.1 refines to the joint setting, and locates the family outside the twist families treated there.","marker":"[32]"},{"why":"Supplies the general assembly theorem for degree-ℓ twist families against which the authors position the new coupled-matrix argument.","marker":"[33]"},{"why":"The density-one 4-rank formula for K(√n) whose Selmer-dimension identity Lemma 5.5 corrects in the negative-twist range; it underpins the biquadratic decomposition and the 8-rank law.","marker":"[16]"},{"why":"Gives the Rédei–Reichardt criterion equating the 4-rank with the corank of the Rédei matrix, the foundation of the whole matrix reduction.","marker":"[29]"},{"why":"Determines the marginal 4-rank distribution of quadratic fields, the one-field law that projection of Theorem 1.1 must recover.","marker":"[7]"},{"why":"Defines the Cohen–Lenstra–Gerth distribution π_CL that serves as the limiting marginal in Theorem 1.1 and as the convolution factor in Corollary 1.5.","marker":"[11]"},{"why":"Proposes the original Cohen–Lenstra heuristics from which the corrected conjectures for 2Cl[2^∞] and the biquadratic family derive.","marker":"[3]"}],"fun_headline_variants":["4-ranks independent in paired imaginary quadratic fields","Shared ramification doesn't couple class-group 4-ranks","Cohen-Lenstra product law for paired field 4-ranks","Asymptotic independence of 4-ranks despite common primes","Correlated fields, independent class-group 4-ranks"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise for the biquadratic results (Theorem 1.3 and Corollary 1.5) is the paper's un-reproduced correction of the Selmer-dimension identity of [16, Theorem 3.4] for a real quadratic base field with negative twist — Lemma 5.5 asserts that the terminal $-1$ must be omitted in that sign range — and if this correction is wrong those results fail, although Theorem 1.1 itself does not depend on it.","fun_headline_variants_meta":{"raw":{"variants":["4-ranks independent in paired imaginary quadratic fields","Shared ramification doesn't couple class-group 4-ranks","Cohen-Lenstra product law for paired field 4-ranks","Asymptotic independence of 4-ranks despite common primes","Correlated fields, independent class-group 4-ranks"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00081,"raw_usage":{"total_tokens":3775,"prompt_tokens":1386,"completion_tokens":2389,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":1002,"completion_tokens_details":{"reasoning_tokens":2301}},"tokens_in":1002,"tokens_out":2389,"duration_ms":14055,"temperature":1.0,"reasoning_tokens":2301,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:20:54.336499+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For Theorem 1.1: with $d_0=2$ and $X$ of size around $10^9$–$10^{10}$, tabulate the empirical joint distribution of $(r_4(-d),r_4(-2d))$ over squarefree odd $d\\le X$ and measure its total-variation distance to $\\pi_{\\mathrm{CL}}\\otimes\\pi_{\\mathrm{CL}}$; the theorem predicts decay like $(\\log\\log X)^{-c}$, so a distance that does not shrink as $X$ grows would refute it. For Lemma 5.5: verify the Selmer dimension identity independently in the negative-twist range for a concrete real quadratic field $F$ (for instance $F=\\mathbb{Q}(\\sqrt{5})$ and small squarefree $|n|$) by computing the three Selmer groups in (82); any instance where the terminal $-1$ of [16, Theorem 3.4] is needed, rather than omitted, falsifies the correction and with it Theorem 1.3.","supporting_citations":[{"cited_title":"Smith, The distribution of ℓ∞-Selmer groups in degree ℓ twist families I , J","cited_arxiv_id":null,"evidence_quote":"Establishes the one-field Cohen–Lenstra–Gerth distribution for imaginary quadratic fields that Theorem 1.1 refines to the joint setting, and locates the family outside the twist families treated there."},{"cited_title":"Smith, The distribution of ℓ∞-Selmer groups in degree ℓ twist families II , J","cited_arxiv_id":null,"evidence_quote":"Supplies the general assembly theorem for degree-ℓ twist families against which the authors position the new coupled-matrix argument."},{"cited_title":"The $4$-rank of class groups of $K(\\sqrt{n})$","cited_arxiv_id":"2101.03407","evidence_quote":"The density-one 4-rank formula for K(√n) whose Selmer-dimension identity Lemma 5.5 corrects in the negative-twist range; it underpins the biquadratic decomposition and the 8-rank law."},{"cited_title":"Rédei, H","cited_arxiv_id":null,"evidence_quote":"Gives the Rédei–Reichardt criterion equating the 4-rank with the corank of the Rédei matrix, the foundation of the whole matrix reduction."},{"cited_title":"Fouvry, J","cited_arxiv_id":null,"evidence_quote":"Determines the marginal 4-rank distribution of quadratic fields, the one-field law that projection of Theorem 1.1 must recover."},{"cited_title":"Gerth III, The 4-class ranks of quadratic fields , Invent","cited_arxiv_id":null,"evidence_quote":"Defines the Cohen–Lenstra–Gerth distribution π_CL that serves as the limiting marginal in Theorem 1.1 and as the convolution factor in Corollary 1.5."},{"cited_title":"Cohen, H","cited_arxiv_id":null,"evidence_quote":"Proposes the original Cohen–Lenstra heuristics from which the corrected conjectures for 2Cl[2^∞] and the biquadratic family derive."}],"review_version":2}