{"id":"081521e4-9792-41c3-91e7-85b047d9beb3","arxiv_id":"2608.01337","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For quadratic characters over F_q[T], the murmuration density is exactly 1/(q-1), and for Kummer cubic characters it tends to the root number of the tested prime.","lead":"This paper computes the exact murmuration density for quadratic Dirichlet characters over function fields: every degree-2g prime contributes 1/(q-1), matching the arithmetic correction in the average trace of Frobenius powers. It also computes the density for a Kummer family of cubic characters, where the limit is the root number of the prime itself.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.2's error term does not decay under the stated hypotheses: for admissible q (e.g., q=300, q>208), g^2 6^{g+n} q^{-θ(g)/2} grows like (5.13)^g, so the claimed asymptotics are not established as stated.","rationale":"I read the paper as centrally advancing three claims: the pointwise murmuration density M_{q,g}(P)=1/(q-1) for quadratic characters (Theorem 1.1), the large-genus trace expansion with the correction -1_{n=2g}/(q-1) (Theorem 1.2), and the cubic Kummer density computation (Theorem 1.5). Theorem 1.1 and the cubic argument are concrete and appear to follow from cited lemmas; the stress rests on Theorem 1.2, which the reader also identified as the weakest assumption. The reader's concern was that the second error term diverges when n grows with g and q is near the stated lower bound. My independent check shows the problem is more severe: even with n fixed at 2 and q fixed at any admissible value up to 6^{34} (which includes q=300, far above 208), g^2 6^{g+n} q^{-θ(g)/2} grows exponentially in g. The base is 6 q^{-1/34}, which exceeds 1 for all q<6^{34}, while the theorem's lower bound is q>2 max{4,3/A+2}=208. Since 6^{34} is astronomically larger than 208, the error term is not an error term for almost the entire stated parameter range. This is an internal inconsistency of the theorem's hypothesis and conclusion, not a disagreement with consensus or an aesthetic objection. It is also not repaired by the disclaimer that constants were not optimized, because the discrepancy is many orders of magnitude. Consequently, the advertised asymptotics are not established as stated, and the derived Corollaries 1.3 and 1.4 inherit the defect. I therefore move the verdict to REJECT for the paper as submitted, while acknowledging that the flaw is likely fixable by imposing a joint or much stronger lower bound on q; if the authors add such a condition and verify the error term decays, a conditional acceptance with a corrected theorem would be appropriate.","tokens_in":19075,"tokens_out":7285,"duration_ms":60894,"concrete_test":"Evaluate the claimed O-term for admissible parameters: set A=1/34, B=−35/34, choose q=300 (allowed since q>208) and fixed n=2. Show E_g := g^2 6^{g+2} 300^{-⌊g/17−1⌋/2} tends to infinity as g→∞, with asymptotic growth rate (6·300^{-1/34})^g≈5.13^g. For comparison, recompute Theorem 1.2 under the stronger hypotheses q>6^{34} for fixed n and q>216^{34} for n=2g, and verify whether the error term is then o(1). If the proof does not force the error to be o(1) even under these stronger bounds, the theorem needs to be restated with an explicit joint condition on q, g, and n.","verdict_should_be":"REJECT","load_bearing_attack":"The load-bearing issue is the error term in Theorem 1.2 and in Propositions 3.19–3.21. With the quoted constants A=1/34, B=−35/34, we have θ(g)=⌊2Ag+A+B⌋=⌊g/17−1⌋. The second error term in Theorem 1.2 is g^2 6^{g+n} q^{-θ(g)/2} ≈ g^2 6^n (6 q^{-1/34})^g q^{1/2}. The theorem allows any q>2 max{4,3/A+2}=208. For q=300, the base is 6·300^{-1/34}≈5.13>1, so the O-term tends to infinity as g→∞ even for fixed n, for example n=2. An O-term that diverges cannot serve as the error term of an asymptotic expansion; the claimed main term (RMT trace minus 1_{n=2g}/(q-1)) is not isolated. The same failure propagates to Corollary 1.3, whose error includes vg^2(6+2v)^g q^{-θ(g)/2}, and to Corollary 1.4. The proof's unstable-range estimate in Proposition 3.19 reduces to a tail of binomial coefficients, and under only q>16 the analogous base 4 q^{-1/34} is also larger than 1; the error is again divergent. The manuscript's remark that the constants were not optimized does not repair this: the needed condition is roughly q>6^{34} for fixed n and q>216^{34} for the n=2g transition term, not q>208. Thus Theorem 1.2 as stated is unsupported; the argument could go through only after adding a much stronger, n-dependent lower bound on q.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies murmuration densities for quadratic and cubic characters over F_q[T]. Theorem 1.1 evaluates the quadratic murmuration density pointwise as 1/(q-1). Theorem 1.2 claims large-genus asymptotics for the average traces Tr(Theta_D^n), displaying the random matrix theory main term together with the correction -1_{n=2g}/(q-1), and derives this from stable homology of the moduli space H_g^{1,0} via the results of BDPW23 and MPPRW24. Corollary 1.3 derives the one-level density with a lower-order correction, and Corollary 1.4 derives a full-density nonvanishing statement at the central point in the double limit q -> infinity after g -> infinity. Theorem 1.5 computes the Kummer cubic murmuration density using the published DFL22 Gauss sum asymptotics. The main novelty is the connection between the pointwise murmuration density, the Rudnick correction in the trace formula, and stable homology.","tokens_in":19482,"tokens_out":12417,"duration_ms":105137,"significance":"If Theorem 1.2 held as stated, it would be a substantial strengthening of Rudnick's trace asymptotics, with an explicit cohomological interpretation and applications to one-level density and nonvanishing. The proof of Theorem 1.1 is clean and elementary, and Theorem 1.5 is a straightforward consequence of DFL22. The paper uses no fitted parameters, and the reliance on DFL22 and MPPRW24 is not circular because those are independent inputs. However, the central trace asymptotic is not established under the stated hypotheses: the displayed error term fails to decay, and this failure propagates to Corollaries 1.3 and 1.4. The overall approach is sound in conception, but the main theorem needs a materially stronger quantification of q.","major_comments":[{"comment":"The error term g^2 6^{g+n} q^{-theta(g)/2} does not tend to zero under the stated hypothesis q > 208. With A=1/34 and B=-35/34 one has theta(g)=floor(g/17 - 1), so the term is roughly g^2 6^n (6 q^{-1/34})^g q^{1/2}. For q=211, which satisfies q > 2 max{4,3/A+2}=208, the base 6*211^{-1/34} is about 5.12, which is larger than 1; the error grows exponentially even for fixed n=2. An O-term that tends to infinity cannot serve as a remainder in an asymptotic expansion, so the claimed main term, in particular the correction -1_{n=2g}/(q-1), is not isolated. The remark that the constants were not optimized does not repair the issue; a sufficient condition would be roughly q > 6^{34} for fixed n and q > 216^{34} for the n=2g transition. This is load-bearing, since Theorem 1.2 is the basis for the subsequent corollaries.","section":"Section 3.5 (Theorem 1.2)"},{"comment":"The advertised divergence originates in the unstable-range estimate. The proof bounds the tail by (2g)^2 6^{g+n} q^{-theta(g)/2} after using sum_{k>theta(g)} binom(2g,k) q^{-k/2} <= 2^{2g} q^{-theta(g)/2}. The estimate is algebraically valid, but under only q > 16 the factor 6 q^{-1/34} is larger than 1 (for q=17 it is about 5.5), so the displayed bound diverges as g grows. Consequently Proposition 3.19 as stated is not supported. The same divergent quantity appears in the error bounds of Propositions 3.20 and 3.21, so the proof of Theorem 1.2 cannot succeed without a much stronger, effectively n-dependent lower bound on q.","section":"Section 3.3 (Proposition 3.19)"},{"comment":"Corollary 1.3 inherits the same defect: its error term contains v g^2 (6+2v)^g q^{-theta(g)/2}, which decays only when q > (6+2v)^{34}. The stated assumptions do not imply this for a fixed v, so the asymptotic expansion for the one-level density is not established as quantified. Corollary 1.4 then uses Corollary 1.3 to obtain lim inf_{g->infty} p_0(g) >= 1 - 1/(4v^2); with a divergent error no lower bound follows. The double-limit statement may be salvageable by taking q sufficiently large relative to v before letting v grow, but that is not what is stated.","section":"Section 4 (Corollaries 1.3 and 1.4)"}],"minor_comments":[{"comment":"The indicator notation '1 2|n' and '1 n=2g' should be typeset as 1_{2|n} and 1_{n=2g}; as printed they are difficult to read.","section":"Section 1.2 (Theorem 1.2)"},{"comment":"The lower bounds on q are stated inconsistently: Theorem 1.2 says q > 2 max{4,3/A+2}, while Proposition 3.20 appears to require q > 2^{3/A+2} if the superscript is intended. The authors should state a single precise quantification and verify it in every intermediate proposition.","section":"Sections 1.2 and 3.5"},{"comment":"In the proof of Theorem 1.5, the factor (1+1/|P|)^{-1} appears in the main term in equations (5.2) and (5.3) but is omitted from the final statement; this difference is O(|P|^{-1}) and is absorbed into the error term, but the absorption should be stated explicitly.","section":"Section 5"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick read for you. The paper has two genuinely good results and one theorem that, as stated, does not hold. Theorem 1.1 is a clean elementary proof that the quadratic murmuration density is identically 1/(q-1) for primes of degree 2g. Theorem 1.5, the Kummer cubic density tending to omega(chi_P), is a neat application of DFL22's Gauss sum asymptotics. The stable-homology reinterpretation of Rudnick's trace correction is a nice conceptual frame.\n\nThe problem is Theorem 1.2. The second error term is g^2 6^{g+n} q^{-floor(2Ag+A+B)/2} with A=1/34, B=-35/34. That is g^2 6^n sqrt(q) (6 q^{-1/34})^g. The hypothesis q > 208 allows q=300, for which (6 q^{-1/34}) is about 5.07, so the term grows like 5.07^g. An error term that diverges cannot be an error term. The same issue appears in Propositions 3.19-3.21 and propagates to Corollary 1.3. The paper's remark that the constants were not optimized does not repair this: you need roughly q > 6^34 for fixed n, and q > 216^34 for the n=2g transition, not q > 208. So the advertised large-genus trace asymptotics are not established as stated.\n\nThe reader's report flagged this correctly; I'd only sharpen it: even for fixed n the error diverges, so the 'n fixed' caveat does not save it.\n\nEverything else is fine. The proof of Theorem 1.1 follows Rudnick and is correct. The cubic density proof is straightforward from DFL22. The discussion of Wang's ratios stabilization is honest, and the citation practice is normal. No fitted parameters, no circularity.\n\nVerdict: conditional, but not in the mild sense. Theorem 1.2 needs a genuinely stronger lower bound on q, presumably q growing with n and g, which changes the statements of Corollaries 1.3-1.4. The fix is likely routine, but as written the central theorem is unsupported.\n\nWorth sending to a referee? Yes. The paper is serious, the two density results are solid, and the trace theorem is likely fixable by restating the range. I'd want a referee to catch the q-range issue and ask for the sharper hypothesis.","headline":"Theorem 1.2's error term diverges under the stated hypotheses, so the main trace asymptotic is not proved as written; the paper's two density theorems are solid and worth engaging.","tokens_in":20026,"tokens_out":5907,"would_cite":true,"duration_ms":50663,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M50","11R58","11G20"],"pacs":[],"model":"deepseek-v4-flash","headline":"For the quadratic-character family over polynomial rings, the murmuration density is exactly 1/(q-1), and the same constant is the arithmetic correction in the trace statistics.","keywords":["murmuration density","quadratic characters","function fields","Frobenius class","homological stability","one-level density","central point nonvanishing","cubic characters"],"falsifier":"Fix a small odd prime power $q$ and compute $M_{q,g}(P)$ for several primes $P$ of degree $2g$ as $g$ grows; any value that differs from $1/(q-1)$ by more than a decaying finite-field error would refute Theorem 1.1, and so would any failure of $\\langle\\operatorname{Tr}\\Theta_D^{2g}\\rangle$ to approach $-1-1/(q-1)$.","tokens_in":18880,"feed_emoji":"🧮","tokens_out":14035,"duration_ms":114947,"temperature":0.7,"pith_summary":"This paper studies the murmuration phenomenon for families of Dirichlet characters over the polynomial ring $\\mathbb{F}_q[T]$. Its first result is that for every monic irreducible polynomial $P$ of degree $2g$, the average of $\\chi_D(P)\\sqrt{|P|}$ over monic squarefree $D$ of degree $2g+1$ is exactly $1/(q-1)$; the murmuration density is therefore the same constant at every prime in the family. The same constant appears as a correction to the random-matrix prediction for the average trace of the Frobenius class at the transition power $n=2g$, and the paper evaluates that correction cohomologically through the stable homology of the moduli space of genus-$g$ hyperelliptic curves with one marked Weierstrass point. A consequence is a one-level density with a lower-order $-\\hat f(1)/(g(q-1))$ term and a 100% non-vanishing theorem for $L(1/2,\\chi_D)$ in the double limit $q\\to\\infty$ after $g\\to\\infty$. For the thin Kummer family of primitive cubic characters, the density is instead asymptotically the root number $\\omega(\\chi_P)$, showing that non-self-dual families can have a genuinely $P$-dependent murmuration shape.","feed_headline":"Quadratic-character murmuration equals 1/(q-1)","feed_subtitle":"The same constant drives the Frobenius trace correction and a 100% nonvanishing result in the double limit.","key_machinery":"The load-bearing object is the murmuration density $M_{q,g}(P)=\\langle\\chi_D(P)\\sqrt{|P|}\\rangle$ on primes $P$ of degree $2g$ in the hyperelliptic ensemble. The mechanism is an explicit formula linking traces of Frobenius powers to character sums, together with a decomposition of $\\operatorname{Tr}(\\Theta^n)$ into symplectic irreducible characters organized by hooks $(a,1^b)$. On the cohomological side, the averages pass by the Grothendieck trace formula to étale homology of $\\mathcal{H}^{1,0}_g$, the moduli space of hyperelliptic curves of genus $g$ with one marked Weierstrass point; a uniform homological stability theorem separates a stable range, where a plethystic generating function evaluates the homology explicitly, from an unstable range controlled by dimension bounds. The trivial character supplies the random-matrix main term for the compact symplectic group $\\mathrm{USp}(2g)$, while the hook $(a,1^a)$ family produces the constant correction $1/(q-1)$.","core_discovery":"The central discovery is that in the hyperelliptic ensemble the murmuration density is exactly constant: $M_{q,g}(P)=1/(q-1)$ for every $P\\in\\mathcal{P}_{2g}$. Equivalently, the expectation of this density is the arithmetic correction in the trace formula: Theorem 1.2 gives, as $g\\to\\infty$, $\\langle\\operatorname{Tr}\\Theta_D^n\\rangle=\\int_{\\mathrm{USp}_{2g}}\\operatorname{Tr} U^n\\,dU - \\mathbf{1}_{n=2g}/(q-1) + O(\\mathbf{1}_{2|n}2^{n/2}q^{-n/8} + g^2 6^{g+n}q^{-\\lfloor 2Ag+A+B\\rfloor/2})$, under $2\\nmid q$ and $q>2\\max\\{4,3/A+2\\}$ with $A,B$ from the uniform stable range. The proof expresses $\\operatorname{Tr}(\\Theta^n)$ as a signed sum over symplectic hook characters, uses the Grothendieck trace formula to pass to étale homology of the moduli space $\\mathcal{H}^{1,0}_g$, and isolates the trivial character as the random-matrix main term; the correction term at $n=2g$ is exactly the hook contribution evaluated by the stable-homology generating function. The same computation yields the one-level density and the nonvanishing corollary.","pith_inferences":["An implication the authors leave implicit is that, within the quadratic function-field family, the murmuration signal is a flat background rather than a $P$-dependent curve: every degree-$2g$ prime carries the same density, and any shape seen in number-field analogues would have to come from logarithmic-scale effects that have no direct function-field counterpart.","Applying the same cohomological machinery to subfamilies of cubic characters with fixed root number would test whether the asymptotic factor $\\omega(\\chi_P)$ in the Kummer result splits cleanly from a constant density; the paper notes this subfamily idea but does not develop it.","A sharper bound on the unstable cohomology range, if it exists, would let the $n=2g$ transition be resolved for fixed $q$ rather than $q$ large relative to $n$; the persistent constant $1/(q-1)$ would then be an arithmetic feature, not a large-$q$ artifact."],"forward_implications":["For every prime $P$ of degree $2g$, the expectation over $D\\in\\mathcal{H}_{2g+1}$ of $\\chi_D(P)\\sqrt{|P|}$ equals $1/(q-1)$, so the quadratic murmuration density is prime-independent.","The average trace $\\langle\\operatorname{Tr}\\Theta_D^n\\rangle$ matches the random-matrix prediction for $\\mathrm{USp}(2g)$ except for the exact correction $-1/(q-1)$ at $n=2g$; odd-power traces vanish, and powers $n>2g$ see only the error term.","The one-level density acquires the lower-order term $-\\hat f(1)/(g(q-1))$, and for $q$ large enough the Fourier support of the test function can be taken arbitrarily wide.","In the double limit $q\\to\\infty$ after $g\\to\\infty$, all but a vanishing proportion of the quadratic $L$-functions $L(1/2,\\chi_D)$ are nonzero.","For the Kummer family of primitive cubic characters with $d=2n$, the murmuration density tends to the root number $\\omega(\\chi_P)$, so the density is a nontrivial function of $P$."],"supporting_citations":[{"why":"It supplies the character-sum lemmas and the explicit-formula/correction framework for traces of Frobenius powers that Theorems 1.1 and 1.2 refine.","marker":"[Rud10]"},{"why":"It provides the plethystic stable-homology generating function and the vanishing and purity results used to evaluate the hook contributions in Theorem 1.2.","marker":"[BDPW23]"},{"why":"It gives the uniform homological stability range with A = 1/34 and B = -35/34 that separates the stable and unstable ranges in the trace argument.","marker":"[MPPRW24]"},{"why":"It supplies the cubic Gauss-sum asymptotics over function fields used to prove the Kummer cubic murmuration density in Theorem 1.5.","marker":"[DFL22]"},{"why":"It gives the random-matrix expectation of high-power traces that forms the main term in the trace asymptotics of Theorem 1.2.","marker":"[DS94]"}],"fun_headline_variants":["Murmuration density is 1/(q-1) for quadratic characters","Constant murmuration from stable homology correction","Quadratic murmuration equals Frobenius trace correction","Murmuration constant 1/(q-1) from etale homology","Nonvanishing via constant murmuration density"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central result assumes that the error estimates from the parts of the cohomology that have not yet stabilized are small enough to isolate the constant correction; as written those errors grow with $n$, so the advertised trace asymptotics hold only when $q$ is large relative to $n$, or when $n$ is fixed.","fun_headline_variants_meta":{"raw":{"variants":["Murmuration density is 1/(q-1) for quadratic characters","Constant murmuration from stable homology correction","Quadratic murmuration equals Frobenius trace correction","Murmuration constant 1/(q-1) from etale homology","Nonvanishing via constant murmuration density"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001038,"raw_usage":{"total_tokens":4385,"prompt_tokens":980,"completion_tokens":3405,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":596,"completion_tokens_details":{"reasoning_tokens":3324}},"tokens_in":596,"tokens_out":3405,"duration_ms":21207,"temperature":1.0,"reasoning_tokens":3324,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:10:52.220518+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Fix a small odd prime power $q$ and compute $M_{q,g}(P)$ for several primes $P$ of degree $2g$ as $g$ grows; any value that differs from $1/(q-1)$ by more than a decaying finite-field error would refute Theorem 1.1, and so would any failure of $\\langle\\operatorname{Tr}\\Theta_D^{2g}\\rangle$ to approach $-1-1/(q-1)$.","supporting_citations":[],"review_version":2}