{"id":"a3a6bcd2-238b-4108-813d-c70259834bab","arxiv_id":"2607.21358","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For every k≥6, G_k = Σ_{a+b=k} α_{a,b} G_{a,b} with explicit coefficients, and the corresponding q-series identity holds.","lead":"This paper proves a weighted sum formula expressing the single Eisenstein series as an explicit linear combination of double Eisenstein series. It also proves the equivalent q-series identity for multiple divisor sums, resolving a conjecture from the author's master's thesis.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Unproved generating-polynomial identity (2.3) is load-bearing for both main theorems; it appears true but must be proved or cited before the result is fully rigorous.","rationale":"The paper's central claim is a new linear relation among multiple Eisenstein series and its q-analogue. I traced both proofs. The restricted double-shuffle input (2.4) is cited from [BT] and is appropriate; the coefficient criterion (2.5) is correctly stated; H_k is symmetric, homogeneous of degree k−2, and divisible by XY; λ computes to 1; the small-k examples match the claimed α values. In §3, the stuffle decomposition (3.2), the alternative expansion (3.3), the operator L, and the power-sum identity (3.6) all check out; after substituting A_r = r!g_{r+1} and (2.2), the lower-weight correction terms in (1.4) emerge exactly. The only step I could not verify from the text is (2.3), which is used as a black box in both independent proofs. Because the identity is elementary and apparently correct, the result is very likely right, but as written the proof is conditional on an unproved assertion. I therefore recommend CONDITIONAL: add a proof of (2.3) (or a precise reference) and the two main theorems become fully rigorous.","tokens_in":5174,"tokens_out":13535,"duration_ms":115573,"concrete_test":"Set n = k−6 and i = j+1 in (2.1); expand the right-hand side of (2.3) by the binomial theorem and extract the coefficient of X^{j+1}Y^{n+3−j}. Show it equals μ_{j+1,n+6} = (2^j−1) binom(n,j) + (2^{j−1}+2) binom(n,j−1) − binom(n,j−2). A symbolic computation (e.g., SymPy) for symbolic n, or for n = 0..50, settles the identity. If it holds, the conditional is met.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Both proofs are gated by the unproved identity (2.3). Lemma 2.1 uses it to rewrite A_ρ as H_k(X,X+Y) − H_k(X,Y), and the proof of Theorem B uses it to identify the coefficient in (3.5) as ∑ μ_{i,k} A_{i,k−2−i}. If (2.3) were false, the α-coefficients would not lie in the shuffle span required by (2.5), and the q-series identity would not follow. The identity is stated without proof or citation immediately after (2.1), and no verification is offered. I checked k = 6, 7, 8 directly: both sides agree, and the binomial shape of (2.1) strongly suggests (2.3) is a true coefficient identity. The issue is therefore a missing proof rather than a detected error; nevertheless it is exactly the kind of load-bearing unstated assumption that a rigorous paper should not leave to the reader.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a weighted sum formula for double Eisenstein series. Main Theorem A (Eq. (1.2)) states that for every k≥6, the single Eisenstein series G_k lies in the Q-span of double Eisenstein series G_{a,b} (a+b=k, a≥3, b≥2) with explicit rational coefficients α_{a,b} given in (1.1). Main Theorem B (Eq. (1.4)) gives the corresponding identity for the q-series g_k and g_{a,b}, including lower-weight correction terms g_{k-2} and g_{k-4}. The proof of Theorem A uses the restricted double-shuffle relations from [BT] and a coefficient-polynomial criterion (2.5) of [GKZ]. Theorem B is proved independently by manipulating exponential generating series and applying a differential operator. The paper also records the first four instances of the weight-k identity and relates the q-series statement to a conjecture from the author's master's thesis.","tokens_in":5403,"tokens_out":15475,"duration_ms":122579,"significance":"If the results are correct, this provides the first proved relation among multiple Eisenstein series in weight 6, together with a family of relations in every weight k≥6, and confirms a q-series conjecture. The two proofs are structurally appealing: the double-shuffle argument is short and conceptual, and the generating-series proof is self-contained and displays the lower-weight corrections explicitly. The numerical examples and the internal computations (the polynomial identity A_ρ = H_k(X,X+Y)-H_k(X,Y), the integral evaluation 30∫ t(1-t)(2t-1)^2 dt = 1, and the power-sum identity) are consistent. The paper is therefore significant within the arithmetic of multiple Eisenstein series and q-analogues of multiple zeta values. The main barrier to accepting it as rigorous is an unproved generating-polynomial identity that is load-bearing for both proofs.","major_comments":[{"comment":"The identity ∑_{i=2}^{k-3} μ_{i,k} X^i Y^{k-2-i} = XY( Y(X+Y)(2X+Y)^{k-6} − (X−Y)^2(X+Y)^{k-6} ) is stated without proof or citation. This identity is used in an essential way in Lemma 2.1 to rewrite A_ρ as H_k(X,X+Y)−H_k(X,Y), and again in the proof of Theorem B (Section 3, around (3.4)–(3.5)) to identify the coefficient of Z^{k-6}/(k−6)! with ∑ μ_{i,k} A_{i,k−2−i}(q). Both Main Theorems therefore depend on it. The identity is elementary and appears to be true (the small cases k=6,7,8 check out), but a rigorous paper must supply a proof, for instance by a direct binomial-coefficient verification or by comparing coefficients after expanding both sides. As written, this is a load-bearing gap, not a mere presentation issue.","section":"§2, Eq. (2.3)"}],"minor_comments":[{"comment":"The identity (2.3) should be presented as a lemma with a proof, or at least with an indication of where it is proved. Since it is used twice in load-bearing positions, labeling it as an unnumbered display makes it easy to miss.","section":"§2, display after (2.1)"},{"comment":"The derivation of the coefficient criterion (2.5) is compressed. The statement that the polynomials Q_a form a basis of the relevant symmetric homogeneous polynomials is correct, but a sentence explaining the counting (dimension ⌊k/2⌋−1) would help the reader.","section":"§2, proof of (2.5)"},{"comment":"The transition from the polynomial in v_1,v_2 to the coefficient expression (3.5) is made in one line. Expanding one step to show how (2.3) is applied with X=v_1, Y=v_2 would make the proof easier to follow.","section":"§3, after (3.4)"},{"comment":"The display for g_{k_1,...,k_r} is slightly hard to read because of the superscript/subscript layout. The formula itself is standard and correct.","section":"§1, Eq. (1.3)"}],"recommendation":"major_revision","confidential_remarks":"The sole substantive technical gap is the unproved identity (2.3). It is load-bearing for both Main Theorems, so the revision must include a proof. I see no other mathematical obstacle; the algebraic core, the small-case checks, and the generating-series manipulations all appear sound. The identity is likely provable by a short coefficient comparison, and I would be willing to accept the paper after such a proof is added. The AI-disclosure paragraph is unusual but transparent and does not affect my assessment."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper proves a weighted sum formula for double Eisenstein series that Bachmann conjectured in his own master's thesis, in two compatible forms: one for the Eisenstein series themselves (Theorem A) and one for the q-series generating functions (Theorem B). That is a genuinely new result, and it gives the first explicit linear relation among multiple Eisenstein series at weight 6 and above. The proofs are direct and mostly self-contained: Theorem B is a combinatorial generating-series argument, and Theorem A is a clean application of the restricted double-shuffle relations from Bachmann–Tasaka. The algebraic core checks out—I verified the polynomial identity A_rho = H_k(X,X+Y) − H_k(X,Y), the integral evaluation 30∫t(1−t)(2t−1)^2 dt = 1, and the small-case examples. The lower-weight correction terms in (1.4) are handled explicitly, and the exposition is clear.\n\nThe one real soft spot is identity (2.3), the generating polynomial for the coefficients μ_{i,k}. It is stated without proof or citation, and it is load-bearing in both proofs: Lemma 2.1 uses it to rewrite A_rho, and the direct q-series proof uses it to extract the coefficient in (3.5). That is exactly the kind of unstated assumption a rigorous paper should not leave to the reader. That said, this is a minor fix rather than a serious flaw. The identity is elementary—it looks like a binomial-coefficient manipulation—and I checked cases k = 6, 7, 8 where it holds. The stress-test note is correct to flag it, but the concern is about presentation, not about the mathematical truth.\n\nOther potential worries do not hold up. The reliance on Bachmann–Tasaka is legitimate: that is an independent published theorem, not a repackaging of the target formula. The AI-assistant disclosure is transparent and honest, and the author states that all statements and proofs were verified by hand; I saw no sign of a hidden computational dependence. The citation pattern is reasonable, mostly to the author's own prior work, but that is justified because the conjecture and the relevant framework come from there.\n\nIn short: this is a solid, specialized paper that resolves a stated conjecture and should be published after a small revision. The unproved identity (2.3) needs a proof or an explicit reference. I would send it to peer review rather than desk-reject, because the result is important within the field and the proof is worth referee scrutiny. Colleagues working on multiple zeta values, multiple Eisenstein series, or q-analogues of MZV will want to read this.","headline":"A short, correct proof of a conjectured weighted sum formula for double Eisenstein series; the only real gap is an unproved but elementary generating-polynomial identity that should be filled before publication.","tokens_in":823,"tokens_out":903,"would_cite":true,"duration_ms":25530,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F11","11A25","11M32"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every depth-one Eisenstein series of weight k≥6 is a finite weighted sum of double Eisenstein series with explicit rational coefficients.","keywords":["multiple Eisenstein series","double Eisenstein series","weighted sum formula","multiple zeta values","multiple divisor sums","double shuffle relations","q-series","generating series"],"falsifier":"Expand both sides of (2.3) for a small weight such as k=7 or k=8: any mismatch in the polynomial coefficients would refute the proof chain. Independently, compute the q-expansion of the difference between the two sides of (1.4) for k=6 through, say, order q^{20}; any nonzero coefficient would disprove the weighted-sum formula itself.","tokens_in":5035,"feed_emoji":"🧮","tokens_out":7416,"duration_ms":70323,"temperature":0.7,"pith_summary":"This note proves that for every weight k≥6, the ordinary Eisenstein series G_k can be written as a finite weighted sum of double Eisenstein series G_{a,b}, with rational coefficients α_{a,b} given by an explicit binomial formula. The same statement holds for the q-series g that appear in Fourier expansions, after subtracting two lower-weight correction terms. The author presents two independent proofs: one deduces the identity from restricted double-shuffle relations, the other is a direct combinatorial argument on generating series of multiple divisor sums. A sympathetic reader would care because these relations are exactly the q-analogues of relations among multiple zeta values, and the weight-6 case is expected to be the first linear relation among multiple Eisenstein series, linking them to double zeta values and modular forms.","feed_headline":"Every weight-k Eisenstein series is a weighted sum of doubles","feed_subtitle":"Explicit coefficients yield relations at every weight 6 and up, matching a conjecture about q-series divisor sums.","key_machinery":"The argument turns on three pieces. First, the restricted double-shuffle relations (2.4) say the shuffle-regularized Eisenstein series map kills the difference between the stuffle product and the shuffle product on depth-two words. Second, a coefficient criterion (2.5), adapted from the earlier literature, converts a polynomial identity in two variables into the statement that a certain combination of depth-two words lies in the span of double-shuffle relations, with a constant term λ computed by an integral. Third, the generating-polynomial identity (2.3), Σ μ_{i,k} X^i Y^{k-2-i} = XY( Y(X+Y)(2X+Y)^{k-6} − (X−Y)^2(X+Y)^{k-6} ), packages all the coefficients μ_{i,k} that define α_{a,b}; it i","core_discovery":"Main Theorem A states: for every integer k≥6, G_k = Σ_{a+b=k, a≥3, b≥2} α_{a,b} G_{a,b}, where α_{a,b} is given by the closed formula 60/b · binom(k-1,a-1)^{-1} · [ (2^{a-2}-1) C(k-6,a-2) + (2^{a-3}+2) C(k-6,a-3) - C(k-6,a-4) ]. Main Theorem B gives the companion identity for the q-series g_k, with the left-hand side g_k - 5/((k-1)(k-2)) g_{k-2} + 4/((k-1)(k-2)(k-3)(k-4)) g_{k-4} equal to the same weighted sum of g_{a,b}. The two forms are compatible via Fourier expansion, and both are proven in the note: Theorem A follows from restricted double-shuffle relations, and Theorem B is proven combinatorially via exponential generating series. The core discovery is the explicit coefficient identit","pith_inferences":["The same coefficient-criterion machinery might produce relations at depth three or higher if a matching generating-polynomial identity for the higher-depth coefficients can be found; the paper only treats depth two.","Because the unproved identity (2.3) is the single unsupported step, an independent verification by a short induction would make the proof chain fully self-contained; the identity is elementary enough to check for any fixed k.","The two independent proofs suggest the weighted-sum structure is robust: the combinatorial generating-series method does not rely on the double-shuffle theorem, so it may extend to other families of divisor sums where shuffle relations are not available."],"forward_implications":["For every weight k≥6, the depth-one Eisenstein series G_k is a rational linear combination of depth-two series G_{a,b}, so the depth-one part of the multiple Eisenstein algebra is determined by depth two up to α-coefficients.","The q-series version (1.4), multiplied by (1−q)^k and sent to q→1−, recovers the weighted sum formula ζ(k)=Σ α_{a,b} ζ(a,b) for double zeta values.","The same coefficients α_{a,b} work in both the modular-form setting and the divisor-sum setting, reinforcing the expectation that multiple Eisenstein series and the q-series g satisfy identical linear relations modulo lower-weight terms.","Each individual relation for a fixed k can be checked by expanding both sides in q; the paper includes the first four explicit examples k=6,7,8,9.","The proof also yields the explicit lower-weight corrections g_{k-2} and g_{k-4} in Theorem B, so the homogeneous weight-k relation of Theorem A is corrected in a precise way when passing to q-series."],"fun_headline_variants":["Explicit weights prove double Eisenstein sum formula","Every weight ≥6: Eisenstein equals weighted doubles","Conjectured double Eisenstein sum formula proved","Closed-form coefficients split Eisenstein series into doubles","All Eisenstein weights decompose into weighted doubles"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise is an unsupported polynomial identity (2.3) expressing the coefficients as a product of linear factors; both proofs collapse if it is false.","fun_headline_variants_meta":{"raw":{"variants":["Explicit weights prove double Eisenstein sum formula","Every weight ≥6: Eisenstein equals weighted doubles","Conjectured double Eisenstein sum formula proved","Closed-form coefficients split Eisenstein series into doubles","All Eisenstein weights decompose into weighted doubles"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000912,"raw_usage":{"total_tokens":3717,"prompt_tokens":667,"completion_tokens":3050,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":411,"completion_tokens_details":{"reasoning_tokens":2979}},"tokens_in":411,"tokens_out":3050,"duration_ms":25528,"temperature":1.0,"reasoning_tokens":2979,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T07:39:38.627613+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Expand both sides of (2.3) for a small weight such as k=7 or k=8: any mismatch in the polynomial coefficients would refute the proof chain. Independently, compute the q-expansion of the difference between the two sides of (1.4) for k=6 through, say, order q^{20}; any nonzero coefficient would disprove the weighted-sum formula itself.","supporting_citations":[],"review_version":1}