{"id":"367b8137-d72b-43b1-9b0f-60861fd48fc3","arxiv_id":"2411.17638","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The density of primes for which the first coefficient of U_l(eta^r) is odd is proven to exist, is classified, bounded, and computed for dihedral CM families.","lead":"This paper studies the parity of a special subsequence of coefficients of powers of the Dedekind eta function, analogous to the indices on which partition congruences live. It proves that the associated density of odd coefficients exists, classifies when it vanishes, bounds it, and computes it for infinite families of eta powers.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The level-9 nonvanishing theorem (Thm 6.2(ii)) is asserted by a bare citation to [B.Im, Theorem I], whose level-1 version is Thm 4.3; Theorem B's converse and Theorem C's strictness rest on this unproved level-9 extension.","rationale":"The reader's conditional verdict is well-founded. I agree that Proposition 6.1's five omitted cases are a gap: they feed Corollary 6.3, Theorem 6.2(iii), Theorem C(iii), and Remark 1.2. However, the more upstream and more load-bearing issue is Theorem 6.2(ii). The proof is a one-line citation to [B.Im, Theorem I], while the only version of that theorem quoted in the paper (Theorem 4.3) is level 1. The converse of Theorem B requires knowing no C^n with gcd(n,6)=1, n>1 has zero density; Theorem 6.2(iii)'s strictness requires δ(g)>0 for the complementary form g. If the cited theorem is not level-9, both fail. This is not a mathematical contradiction I can exhibit from the manuscript alone—it is a missing support/reference, exactly the kind of gap the review rules ask to flag. The proposed check (verify the scope of [B.Im, Theorem I], and if necessary prove the K(9) analogue) would settle it. I would not move the verdict to reject: both the Prop 6.1 checks and Theorem 6.2(ii) are likely fillable and the numerics in the paper are consistent. Hence CONDITIONAL remains the right disposition.","tokens_in":36208,"tokens_out":31134,"duration_ms":257956,"concrete_test":"Obtain [B.Im] and check the exact hypotheses of Theorem I: if it is stated for mod-2 forms of arbitrary level N with p∤N (or with the analogous exceptional form), the proof of Thm 6.2(ii) is a valid citation; if it is level 1 only, require the authors to supply a proof for K(9) (or a precise reference to a published level-9 version) before the manuscript can be accepted. As a secondary computational sanity check, for n in {5,7,11,13,17,19} compute a_l(C^n) mod 2 for all primes l≤10^6 and compare the observed densities with Corollary 6.3/Prop 6.1; any mismatch would indicate the level-9 density claims need revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 6.3 states Theorem 6.2(ii): for f in K(9), δ(f)=0 iff f=C, with proof 'See Bellaïche [B.Im, Theorem I]'. But the only theorem from [B.Im] actually stated in this paper is Theorem 4.3, which is explicitly for the level-1 space K and has exceptional form Δ. Since K(9) is a new space constructed in Section 6.1, the level-9 statement is not an immediate corollary of Theorem 4.3 unless [B.Im, Theorem I] is itself a level-N theorem; the paper does not quote its hypotheses or location. This matters because the converse half of Theorem B uses Corollary 6.3 to conclude δ(C^{b_r})=0 only when b_r=1 or even, and Theorem 6.2(iii) uses δ(g)>0 to rule out all but six density-1/8 forms; these are exactly the steps that exclude additional r with D(r)=0 and that make Theorem C(iii) strict. If [B.Im, Theorem I] does not cover Γ0(9), the central classification is unsupported at its most load-bearing point.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a notion D(r) measuring, for each normalized eta-power eta^r, the natural density of primes ell for which the leading coefficient of U_ell(eta^r) modulo 2 is nonzero (equivalently, for which the order at infinity of U_ell(eta^r) modulo 2 is minimal). It proves four main results: Theorem A states that D(r) always exists and is a dyadic rational; Theorem B classifies exactly when D(r)=0, namely when r divides or is a multiple of 32 or 48; Theorem C gives unconditional upper bounds D(n)<1, D(2n)<1/2, D(4n)<1/4 with four explicit exceptions; Theorem D computes D(r) exactly for several infinite families of eta-powers whose reductions are dihedral mod-2 forms, giving densities such as 2^{-n} and 3*2^{-(n+2)}. The proofs proceed by reducing eta-powers modulo 2 to powers of Delta or of the level-9 form C (Proposition 2.4), expressing D(r) as a finite sum of Bellaiche densities delta(f) (Proposition 7.1 and Corollary 7.2), and then computing delta(f) for dihedral and abelian forms. The paper also supplies proofs of two unpublished Bellaiche results, including a combinatorial formula for the density of dihedral basis elements m(a,0), m(0,a) in terms of the base-2 digits of a (Theorem 4.6).","tokens_in":36514,"tokens_out":8598,"duration_ms":79380,"significance":"If the central claims are correct, the paper gives a complete structural classification of the vanishing of D(r) and near-optimal upper bounds, together with the first exact computations for infinite families of eta-powers; the reduction to Bellaiche density and the combinatorial density formula are elegant and parameter-free. The paper is also valuable for communicating, with proof, two of Bellaiche's unpublished results and for extending some level-1 techniques to level 9. The main theorems are falsifiable and are checked against extensive numerics, and no fitted parameters appear anywhere in the argument. The significance is somewhat reduced, but not destroyed, by the fact that two load-bearing points are not fully proved inside the manuscript: the level-9 nonvanishing theorem is imported by a citation whose exact statement and hypotheses are not given, and five of the six cases of Proposition 6.1 are dismissed as straightforward without proof.","major_comments":[{"comment":"The proof of Theorem 6.2(ii) is only the sentence 'See Bellaiche [B.Im, Theorem I]'. The only nonvanishing theorem for mod-2 forms actually stated in this paper is Theorem 4.3, which is explicitly for the level-1 space K. The level-9 space K(9) is constructed later in Section 6.1, so Theorem 6.2(ii) is not an immediate corollary of Theorem 4.3 unless [B.Im, Theorem I] is itself a level-N theorem. This is load-bearing: the converse half of Theorem B uses Corollary 6.3 to conclude that delta(C^{b_r})=0 only for the stated b_r, and Theorem 6.2(iii) uses delta(g)>0 to exclude all but six forms; without a valid level-9 nonvanishing statement, the claimed classification of r with D(r)=0 and the strictness in Theorem C(iii) are unsupported. Please quote the precise statement of [B.Im, Theorem I], verify that its hypotheses cover the space K(9), or supply a direct proof for this level-9 case.","section":"Section 6.3, Theorem 6.2(ii)"},{"comment":"Proposition 6.1 asserts that the six forms alpha_5, alpha_7, alpha_11, alpha_13, alpha_17, alpha_19 are abelian with field of determination Q(mu_24) and satisfy the coefficient criterion a_ell(alpha_i)=1 if and only if ell ≡ i mod 24. The proof gives detailed arguments only for i=11; the other five cases are disposed of with the sentence 'The other cases are straightforward.' These omitted checks are load-bearing because Corollary 6.3, Theorem 6.2(iii), the strictness statement in Theorem C(iii), and the level-9 entries in Remark 1.2 all depend on the coefficient criterion and on the exceptional status of exactly these six forms. An error in any omitted case would change the exceptional set in Theorem C(iii) and the claimed densities. Please provide complete verifications for all five remaining cases, including the representation/uniqueness arguments for the quadratic forms listed in the table.","section":"Section 6.2, Proposition 6.1"},{"comment":"In the subcase r=2s with s prime to 6, the text says that T_5 C^s = 0 'immediately gives D(r) ≤ 3/8 < 1/4', but the target inequality in Theorem C(ii) is D(2n)<1/2, and 3/8 is not less than 1/4. This appears to be a typographical slip rather than a mathematical error, since the displayed computation gives D(r) ≤ 3/8 < 1/2. Please correct the inequality.","section":"Section 7.3, proof of Theorem C(ii)"}],"minor_comments":[{"comment":"The names 'Nicholas-Serre' and 'Nicolas-Serre' are used inconsistently; the correct spelling in the references is Nicolas-Serre.","section":"Abstract and throughout"},{"comment":"The notation 'C^i ∈ K(9)_i for every i relatively prime to 24' would be clearer if the condition that i is taken modulo 24 were repeated, since C^i is supported on exponents congruent to i modulo 24.","section":"Section 6.1(viii)"},{"comment":"In the proof of Theorem B, the phrase 'r = 16n with n > 1 odd' should say 'with n odd and prime to 3', because the argument uses the level-9 reduction and Corollary 7.2 for r prime to 6.","section":"Section 7.2"},{"comment":"In the proof of Lemma 4.12, the sentence 'we establish that c, cg, or g^{2^d} are in the kernel of t_f' should read 'are not in the kernel'; otherwise the displayed Cases contradict the conclusion.","section":"Section 4.2, Lemma 4.12"},{"comment":"The table lists the field K for alpha_7 and alpha_13 as Q(sqrt(-3)), yet the proposition states that the field of determination is Q(mu_24); please clarify the relation between the quadratic field used for representation by the quadratic form and the full abelian field of determination.","section":"Section 6.2, table in Proposition 6.1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's central classification is plausible and the main framework is well motivated, but two load-bearing gaps need to be resolved before publication: the exact status of [B.Im, Theorem I] for level 9, and the five unproved cases of Proposition 6.1. The authors should also make available, or at least precisely locate, the unpublished Bellaiche material on which several steps depend. If the missing level-9 nonvanishing theorem turns out to be unavailable, the converse half of Theorem B and the strict inequalities in Theorem C would need to be substantially reformulated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First: the paper is real. It defines a new statistic D(r) for eta powers, shows it exists and is dyadic, classifies exactly when it vanishes, gives sharp-ish upper bounds, and computes explicit dyadic values for infinite dihedral/CM families. It also supplies proofs of two of Bellaiche's unpublished theorems, which is a service to the field. The central line of argument—relating D(r) to Bellaiche's delta via Proposition 7.1—is sound and clearly presented.\n\nThe soft spots are, in order:\n\n1. Theorem 6.2(ii) is asserted with \"See Bellaiche [B.Im, Theorem I]\" but the only theorem the paper states from that reference is level-1 (Theorem 4.3). The level-9 extension is not automatic from the displayed results; it is load-bearing for the converse of Theorem B and for the strictness in Theorem C(iii). The authors need to either quote the hypotheses of [B.Im, Theorem I], show it applies to Gamma_0(9), or give a proof. This is likely a fixable exposition gap, but as written it is a black box.\n\n2. Proposition 6.1 leaves five of six verification cases to the reader. The i=11 case is written out; the others are \"straightforward\". Since Corollary 6.3, Theorem C(iii), and the level-9 entries in Remark 1.2 feed off those five cases, a referee should ask for the details or a precise reference. This is a completeness issue, not a fatal one.\n\n3. Minor: the proof of Theorem C(ii) contains \"3/8 < 1/4\" where it clearly means 3/8 < 1/2; there are a few other typos. Nothing substantive.\n\nThere is no fitted parameter and no circularity I can see: D(r) is not used to define delta, and the combinatorial lemma (Theorem 4.15) is proved in full with a nice base-2 argument.\n\nWho is this for: anyone working on mod-2 modular forms, partition parity, or eta-quotients. A serious referee should get this; I would accept it for review and ask for revision. I would likely cite it for the D(r) classification and the Bellaiche proofs.","headline":"A genuinely useful paper on parity densities of eta powers, with a real but fixable gap in the level-9 citation and some omitted verifications.","tokens_in":36992,"tokens_out":3545,"would_cite":true,"duration_ms":30780,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F33","11F03","11F20","11F80","11P83","11R45"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper establishes a complete structural picture of the parity density of eta powers: D(r) exists, is dyadic rational, vanishes exactly for r dividing or divisible by 32 or 48, and obeys the upper bounds D(n)<1, D(2n)<1/2, D(4n)<1/4…","keywords":["Dedekind eta function","modular forms modulo 2","Galois representations","density of Fourier coefficients","dihedral forms","Chebotarev density theorem","partition parity","level 9"],"falsifier":"Compute, for each i∈{5,7,13,17,19}, the coefficients a_ℓ(α_i) for primes ℓ up to a few thousand; a single prime with a_ℓ(α_i)=1 while ℓ≢i (mod 24), or a_ℓ(α_i)=0 while ℓ≡i (mod 24), would disprove Proposition 6.1 and change the level-9 densities and the Theorem C(iii) exceptions.","tokens_in":2243,"feed_emoji":"🔢","tokens_out":2252,"duration_ms":129277,"temperature":0.7,"pith_summary":"The paper studies, for every positive integer r, the density D(r) of primes ℓ for which the leading coefficient of the Hecke operator U_ℓ applied to the normalized q-series η^r(m_r τ) is odd. It proves three structural theorems: D(r) always exists and is a dyadic rational; D(r)=0 exactly when r is a divisor or a multiple of 32, or a divisor or a multiple of 48; and D(r) obeys the strict bounds D(n)<1, D(2n)<1/2, and D(4n)<1/4 with exactly four exceptions (n=9,15,18,30). It also computes D(r) explicitly for several infinite families of r attached to dihedral and abelian mod-2 modular forms, obtaining values that are powers of 1/2. The motivation is the parity problem for the partition function: the tested subsequence is modelled on the sequence δ_ℓ=$24^{{-1}}$ mod ℓ that supports partition congruences.","feed_headline":"Zero parity density for eta powers comes only from 32 and 48","feed_subtitle":"Exact parity densities for eta powers: zero density sits at 32 and 48; dihedral families computed.","key_machinery":"The central object is the sequence δ_{ℓ,r}, defined as the least positive integer n satisfying n≡b_r (mod m_r) and ℓ|n, so that p_r(δ_{ℓ,r}) is the leading coefficient of U_ℓ(P_r); D(r) is the density of primes for which this coefficient is odd. The workhorse identity is Proposition 2.4: P_r(q)≡$Δ^{{b_r}}$ (mod 2) when 3|r, and P_r(q)≡$C^{{b_r}}$ (mod 2) otherwise, with C=η(3τ)^8. This reduces eta-power questions to the prime coefficients of the mod-2 modular forms Δ^s and C^s. Those coefficients are frobenian, meaning that the value at ℓ depends only on the Frobenius conjugacy class in a finite Galois group, so Chebotarev counting computes the relevant densities; for the level-1 basis forms m(a,0) and m(0,a), the density δ(m(a,0))=1/$2^{{u(a)+v(a)+1}}$ follows from counting residue classes modulo $2^{{d(a)+1}}$ in which the coefficient of x^a in a modified Chebyshev polynomial is 1, a count governed by the base-2 digit statistics of a.","core_discovery":"The paper's central claim is a complete description of the parity density D(r) of eta powers. For the normalized series P_r(q)=η^r(m_r τ), the quantity D(r) measures the proportion of primes ℓ for which the first formal coefficient of U_ℓ(P_r) is odd. The paper proves that D(r) always exists and is a dyadic rational (a fraction whose denominator is a power of 2), that D(r)=0 if and only if 32 or 48 divides r, or r divides 32 or 48, and that D(r) satisfies the strict upper bounds D(n)<1, D(2n)<1/2, D(4n)<1/4 with the four exceptions n=9,15,18,30. For the infinite families in Theorem D, where the relevant mod-2 forms are dihedral, the densities are explicit powers of 1/2 determined by the base-2 digits of the exponents. The proofs identify P_r modulo 2 with $Δ^{{b_r}}$ or $C^{{b_r}}$, where C=η(3τ)^8 is the unique normalized cusp form of weight 4 and level 9, then express D(r) as a sum of prime-coefficient densities of the resulting mod-2 forms, computed by counting Frobenius elements in finite Galois extensions of Q.","pith_inferences":["A structural reading of the proof of Theorem B is that every zero-density case is a pure congruence/support obstruction: P_r modulo 2 and the tested index δ_{ℓ,r} never align modulo a suitable power of 2 for almost all ℓ. If that reading is right, the zero locus of such densities for other moduli and other q-series would also be characterized by support conditions rather than by finer Galois struc","The formula δ(m(a,0))=1/2^{u(a)+v(a)+1} ties these densities to binary digit statistics, suggesting that other families of eta powers with dihedral reduction will have densities expressible as powers of 1/2 determined by the base-2 digits of their exponents.","Because the paper's Remark 7.3 shows the same frobenian argument applies to any integral modular form supported on an arithmetic progression, the framework should compute parity densities for other subsequences of coefficients of modular forms, not only powers of eta.","The paper does not settle the partition-function analogue D(-1); extending the method to η^{-1} would require a mod-2 model for that nonholomorphic case, and any such extension would directly inform the Partition Parity Conjecture."],"forward_implications":["For every r≥1, the density D(r) is a dyadic rational, so the parity of eta-power coefficients in the tested subsequence has a well-defined, computable limiting frequency.","If the theorem holds, the vanishing classification is exact: the only powers of η whose tested coefficient is odd on a set of primes of density zero are those with r dividing or divisible by 32 or 48.","The upper bounds imply that for r of the form 2n or 4n, the order of infinity of U_ℓ(η^r) fails to be maximal for a positive proportion of primes—more than half of all primes for D(2n), and more than three-quarters for D(4n) except at the four listed n.","For the infinite families in Theorem D, the densities are explicit powers of 1/2, for example D(12·z_n)=2^{-(n+1)} and D(3·w_n)=3·2^{-(n+1)} for n≥2, making the values computable directly from the binary expansions of the exponents.","Conditional on the density expectation stated in Remark 7.4, most odd n should satisfy D(3n)=1/2 and D(6n)=1/4, so the explicitly computed small densities in Theorem D would be the atypical cases rather than the generic ones."],"supporting_citations":[{"why":"Supplies the density δ(f) of nonzero prime Fourier coefficients, its frobenian nature, and the positivity theorem used to classify vanishing.","marker":"[B.Im]"},{"why":"Provides the classification of abelian and dihedral mod-2 modular forms, the table used for the abelian computations in Remark 1.2, and the fields of determination Q(i) and Q(√-2).","marker":"[B.Sp]"},{"why":"Gives the structure of the level-1 mod-2 Hecke algebra A=F_2[[x,y]], the adapted basis m(a,b), and the Hecke action used in Theorem D.","marker":"[NS2]"},{"why":"Supplies the nilpotence order and Hecke recurrences for powers of ∆ that underlie the level-1 machinery and the identification of dihedral powers.","marker":"[NS1]"},{"why":"Proves the universal Galois pseudorepresentation t:G→A with t(Frob_ℓ)=T_ℓ, the level-1 Galois machinery extended here to level 9.","marker":"[B.Rep]"},{"why":"Establishes the theory of frobenian functions and the Chebotarev density consequences that make δ(f) and D(r) well-defined densities.","marker":"[S.Div]"},{"why":"Identifies the pure powers of ∆ that are dihedral, specifically the identities m(2^n−1,0)=∆^{z_n} and m(0,2^{n−1})=∆^{w_n} used in Theorem D.","marker":"[S.Let]"},{"why":"Supplies the class-field theory and quadratic-form representation results used in the proof of Proposition 6.1, on which the level-9 densities rest.","marker":"[Cox]"},{"why":"Provides the (Z/24Z)^×-grading and pseudorepresentation compatibility for level 9 used in Section 6.","marker":"[DM]"}],"fun_headline_variants":["Zero eta parity density only for divisors or multiples of 32 and 48","Parity density of eta powers zero iff 32|r or 48|r or r|32 or r|48","Explicit half-power parity densities for dihedral eta families","Strict upper bounds on eta parity density: D(n)<1, D(2n)<1/2"],"cache_read_input_tokens":39168,"weakest_assumption_plain":"The level-9 density values rest on Proposition 6.1, which asserts that six explicit forms α_i are abelian with field of determination Q(μ_24) and that a_ℓ(α_i)=1 if and only if ℓ≡i mod 24, but the proof is given in full only for i=11, the other five cases being dismissed as straightforward.","fun_headline_variants_meta":{"raw":{"variants":["Zero eta parity density only for divisors or multiples of 32 and 48","Parity density of eta powers zero iff 32|r or 48|r or r|32 or r|48","Explicit half-power parity densities for dihedral eta families","Strict upper bounds on eta parity density: D(n)<1, D(2n)<1/2"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.002174,"raw_usage":{"total_tokens":8469,"prompt_tokens":1035,"completion_tokens":7434,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":651,"completion_tokens_details":{"reasoning_tokens":7347}},"tokens_in":651,"tokens_out":7434,"duration_ms":47423,"temperature":1.0,"reasoning_tokens":7347,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:54:04.224175+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for each i∈{5,7,13,17,19}, the coefficients a_ℓ(α_i) for primes ℓ up to a few thousand; a single prime with a_ℓ(α_i)=1 while ℓ≢i (mod 24), or a_ℓ(α_i)=0 while ℓ≡i (mod 24), would disprove Proposition 6.1 and change the level-9 densities and the Theorem C(iii) exceptions.","supporting_citations":[],"review_version":1}