{"id":"4e718a4b-259d-438f-9dbb-2793dd1fe8b0","arxiv_id":"2607.05795","paper_version":1,"verdict":"CONDITIONAL","confidence":"UNKNOWN","novelty_score":4.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"A simpler proof and explicit algorithm express multiple zeta values in terms of symmetric multiple zeta values and their products.","lead":"The paper gives a simpler proof that symmetric multiple zeta values (SMZVs) generate all multiple zeta values, and provides an algorithm to compute such expressions. This matters because it makes a key structural result about MZVs more accessible and computationally explicit.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"Proof structure is sound; the load-bearing risk is the elided generating-function calculation connecting A₂ to ζ₂ coefficients, not the duality dependence the reader flags.","rationale":"The paper's central claim (Theorem 1.3) is a reproof of Yasuda's theorem using a different technique. The proof strategy — reducing to Theorem 2.5, then using generating functions, Theorem 2.3, the shuffle-antipode relation, and Lemma 2.6 — is coherent and the algebraic steps I checked are correct. The duality dependence the reader flags is a genuine scope limitation (acknowledged by the author) but not a correctness risk for the stated theorem.\n\nThe actual correctness risk is concentrated in the 'straightforward calculation' that connects the generating function A₂ to the multitangent coefficients ζ₂ via formula (2.1). This is the linchpin: without it, Theorem 2.3 cannot be invoked, and equation (2.7) — which drives the entire proof of Theorem 2.5 — has no justification. The calculation involves expanding a sum of products of shuffle-regularized MZV generating functions and matching coefficients against an explicit combinatorial formula with binomial coefficients and signs. While such calculations are standard in the MZV literature, they are error-prone, and the paper provides no intermediate steps.\n\nThe explicit examples (ζ(2,3) and ζ(1,2,2) formulas) serve as partial sanity checks, but they are outputs of the algorithm in Section 3 rather than direct verifications of the A₂–ζ₂ identity. A numerical check of these examples would test the entire pipeline end-to-end.\n\nThe new results in Sections 4–5 (Theorem 4.3, Theorem 5.7, Propositions 5.3–5.9, Theorem A.1) are genuine contributions that build on the same framework. Theorem A.1 in particular is a substantial computation involving Bernoulli number congruences.\n\nGiven that the proof structure is sound, the elided calculation is standard but unverified, and there is no formal verification or shipped code, the CONDITIONAL verdict is appropriate. The confidence should perhaps be higher than UNKNOWN — the algebraic steps I checked are correct, and the main risk is a mechanical calculation error, not a conceptual flaw.","tokens_in":25104,"tokens_out":12427,"duration_ms":670408,"concrete_test":"Numerically verify the explicit example given in Section 3: ζ(2,3) = 1/5 ζ(2)ζ(3) + 6/5 ζˣ_S(2,3) + 4/5 ζˣ_S(3,2) − 4/5 ζˣ_S(1,2,2) − 2/5 ζˣ_S(2,1,2) − ζˣ_S(2,2,1). This identity is a direct output of the algorithm in Section 3, which is derived from the same generating-function identities (particularly the A₂–ζ₂ connection) used in the proof of Theorem 2.5. Compute both sides to high precision (e.g., 50 digits) using known MZV values and the definition (1.2) of ζˣ_S. If they disagree, the elided calculation contains an error and the proof of Theorem 1.3 fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader identifies the dependence on duality relations (via Theorem 2.3) as the weakest assumption. This is a real limitation on the proof's *scope* — it prevents generalization to formal SMZVs — but it does not threaten the *correctness* of Theorem 1.3 for actual MZVs, since duality is well-established there. The author honestly acknowledges this in the Remark after Lemma 2.6.\n\nThe more load-bearing concern for correctness is the identity stated as following from 'a straightforward calculation together with (1.2) and (2.1)':\n\n∂²A₂/∂X₁∂Xᵣ = Σ k₁kᵣ ζ₂(k₁+1, k₂, …, kᵣ₋₁, kᵣ+1) X₁^{k₁-1}…Xᵣ^{kᵣ-1}.\n\nThis identity is the sole bridge between the generating-function framework and Theorem 2.3 (Hirose's result connecting SMZVs to ζ₂ coefficients of multitangent functions). If the binomial-coefficient manipulations or sign conventions in this calculation contain an error, the connection to Theorem 2.3 fails, and the proof of Theorem 2.5 — and hence Theorem 1.3 — collapses. The calculation is mechanical but nontrivial: it involves expanding A₂ (defined via a sum of products of Z-functions), matching against the explicit formula (2.1) for ζ₂, and tracking signs through the stuffle regularization.\n\nI independently traced the algebraic steps (2.2)→(2.5)→(2.6)→(2.7)→(2.8)→(2.9)→(2.10)→(2.11)→(2.12) and verified that the variable substitutions, additions, and the application of Lemma 2.6 (with the implicit change to the Y-variable system Yᵢ = Xᵢ₊₁ − X₁) are all correct. The shuffle-antipode relation giving A₁ = 0 is cited from [1, Prop. 3.3]. So the proof structure is sound; the risk is concentrated in the elided coefficient-matching calculation.","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","summary":"This paper provides a simpler proof of Yasuda's theorem (Theorem 1.3) that the space of multiple zeta values (MZVs) of weight k is generated by *-symmetric MZVs (SMZVs). The proof replaces Yasuda's auxiliary real numbers with the coefficients of multitangent functions, leveraging Hirose's result (Theorem 2.3) connecting SMZVs to these coefficients via Hoffman duality. The core argument (Theorem 2.4 implies Theorem 1.3 via weight induction) proceeds through generating function manipulations in Section 2.2. Based on this proof, Section 3 provides an algorithm for expressing MZVs in terms of SMZVs, explicitly carried out for depths 1 and 2. Sections 4 and 5 give results on depth-2 and depth-3 SMZVs and finite MZVs (FMZVs), including a formula for Z_S(k1, k2) and a proof that the space of triple SMZVs equals the space of depth-2 SMZVs for even weight. Appendix A partially addresses the equivalence of two definitions of Z_A(r,s).","tokens_in":25457,"tokens_out":1392,"duration_ms":260526,"significance":"The paper provides a genuine simplification of Yasuda's proof, replacing the technically involved construction of real numbers with the more structured framework of multitangent functions. The explicit algorithm in Section 3, along with the worked examples for depths 1 and 2, adds practical value. The results in Sections 4 and 5 on the structure of triple SMZV/FMZV spaces, particularly Theorem 5.7 and the depth-2 formula in Theorem 4.3, are substantive contributions. The partial result in Appendix A on the equivalence of definitions of Z_A(r,s) is a useful incremental step. The author honestly acknowledges the limitation regarding duality dependence in the Remark after Lemma 2.6.","major_comments":[{"comment":"§2.2, the identity for ∂²A₂/∂X₁∂Xᵣ: This identity is stated as following from 'a straightforward calculation together with (1.2) and (2.1),' but it is the sole bridge between the generating-function framework and Theorem 2.3 (Hirose's result). The calculation involves expanding A₂ (defined via a sum of products of Z-functions), matching against the explicit formula (2.1) for ζ₂, and tracking signs through stuffle regularization. Given its load-bearing role, this calculation should be expanded or verified in an appendix. The skeptic's note correctly identifies this as the primary correctness risk.","section":null},{"comment":"§2.2, the identity A₁(X₁,…,Xᵣ) = 0: This is stated to follow from the shuffle-antipode relation and 'a straightforward calculation.' Since A₁ = 0 is used to derive equation (2.3), which is essential for the proof of Theorem 2.5, the elision here is also load-bearing. Providing the intermediate steps would strengthen the verification.","section":null},{"comment":"§5, Proposition 5.4: In the proof, equation (5.6) is derived from Lemma 4.1, but the sign convention and the factor (1 - δ_{r,a}) need careful verification. The transition from (5.6) to (5.7) involves a sum over r from b-1 to k-2, and the subsequent manipulation using Bernoulli number identities is intricate. A minor error in the binomial coefficient tracking could propagate into the formulas (5.4) and (5.5), which are used in Theorem 5.7.","section":null}],"minor_comments":[{"comment":"§1, p.1: The notation ζ^X_S(k) in (1.2) uses a superscript X, but later the text refers to 'X-symmetric multiple zeta value ζ^X_S(k)'. Consistency in superscript placement would improve readability.","section":null},{"comment":"§2.2, p.4: The generating function Z_S(X₁,…,Xᵣ) is defined with a sum from i=0 to r, but the terms Z(X₁,…,Xᵢ) and Z(-Xᵣ,…,-Xᵢ₊₁) need clearer specification of what happens at the boundary cases i=0 and i=r (e.g., Z() = 1?).","section":null},{"comment":"§3.1, p.7: In the derivation of (3.1), the step from the generating function expansion to the coefficient comparison could benefit from one intermediate line showing the binomial expansion explicitly.","section":null},{"comment":"§4, p.10: In Lemma 4.1, the sum over i from 1 to k₂-1 uses binomial coefficients, but the case k₂=1 (where the sum is empty) should be explicitly noted to confirm the 'in particular' statement.","section":null},{"comment":"§5, p.13: In Proposition 5.4, the Bernoulli number B_{r-b+1} appears with r ranging from b-1, so B_0 = 1 is used. This should be stated for clarity.","section":null},{"comment":"Appendix A, p.17: The definition of ėZ_A(a,b) uses Z'_A(k-1), which involves B_{2p-1-k}. The range of k for which this is well-defined (given the Bernoulli number vanishing properties) should be specified.","section":null},{"comment":"References: Reference [6] (Hirose) is listed as a 2024 arXiv preprint. If published by the time of acceptance, the reference should be updated.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The paper is a solid contribution with a clean proof strategy. The main concern is the level of detail in the 'straightforward calculation' steps, which are load-bearing but mechanical. The author should be asked to provide these details, perhaps in an appendix, to allow full verification. The results in Sections 4-5 and Appendix A appear correct but are dense; the author should ensure all steps are traceable. The duality dependence flagged by the reader is a real scope limitation but not a correctness issue for actual MZVs, and the author acknowledges it."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for the careful reading and constructive suggestions. All three major comments request expanded verification of calculations that were elided as 'straightforward.' We agree with all three requests and will add the missing details in the revised manuscript.","responses":[{"response":"The referee is correct that this identity is load-bearing and that the elision is not appropriate given its role in the proof. We will add a detailed verification as an appendix (or as a subsection of Section 2) in the revised manuscript. The calculation proceeds by expanding the definition of $A_2$ as a sum of products of $Z$-functions, substituting the explicit formula (2.1) for $ζ_2(k)$, and matching coefficients using the definition (1.2) of $ζ_S^X$. The key steps are: (i) expanding each $Z$-factor in $A_2$ as a power series, (ii) collecting terms of bidegree $(k_1-1, k_r-1)$ after taking $∂^2/∂X_1∂X_r$, and (iii) identifying the resulting coefficient with $k_1 k_r ζ_2(k_1+1, k_2, …, k_{r-1}, k_r+1)$ via (2.1), tracking signs through the stuffle regularization. We agree that presenting these steps explicitly will strengthen the paper and allow independent verification.","revision_made":"yes","referee_comment":"§2.2, the identity for ∂²A₂/∂X₁∂Xᵣ: This identity is stated as following from 'a straightforward calculation together with (1.2) and (2.1),' but it is the sole bridge between the generating-function framework and Theorem 2.3 (Hirose's result). The calculation should be expanded or verified in an appendix."},{"response":"We agree. The vanishing $A_1 = 0$ follows from the shuffle-antipode relation (Proposition 3.3 in [1]) by expanding $A_1$ and matching the resulting sum against the antipode identity. Specifically, expanding $A_1(X_1, …, X_r)$ and collecting the coefficient of $X_1^{k_1-1} ⋯ X_r^{k_r-1}$ yields exactly the left-hand side of the shuffle-antipode relation as displayed in the manuscript. We will include these intermediate steps in the revised version, making the connection to the antipode relation explicit rather than leaving it as a 'straightforward calculation.'","revision_made":"yes","referee_comment":"§2.2, the identity A₁(X₁,…,Xᵣ) = 0: This is stated to follow from the shuffle-antipode relation and 'a straightforward calculation.' Since A₁ = 0 is used to derive equation (2.3), providing the intermediate steps would strengthen the verification."},{"response":"We appreciate the referee's careful attention to the sign conventions and binomial coefficient tracking in this proof. We have re-examined the derivation and confirm that the sign $(-1)^r$ and the factor $(1 - δ_{r,a})$ in (5.6) are correct: the sign comes from the $(-1)^{k_1+k_3}$ factor in Lemma 4.1 with $k_1 = r$, $k_3 = 1$, and the Kronecker delta excludes the trivial term $a = r$ where the binomial coefficient $¥binom{a}{r}$ would give a degenerate contribution. The transition from (5.6) to (5.7) involves multiplying both sides of (5.6) by $¥frac{1}{k-b}¥binom{a}{b}(k-a)B_{r-b+1}$ and summing over $r$ from $b-1$ to $a-1$, then swapping the order of summation. We will add these intermediate steps explicitly in the revised manuscript, including the justification for the summation range change and the application of the Bernoulli number identity (5.9), so that the binomial coefficient tracking can be verified step by step.","revision_made":"yes","referee_comment":"§5, Proposition 5.4: In the proof, equation (5.6) is derived from Lemma 4.1, but the sign convention and the factor (1 - δ_{r,a}) need careful verification. The transition from (5.6) to (5.7) involves a sum over r from b-1 to k-2, and the subsequent manipulation using Bernoulli number identities is intricate. A minor error in the binomial coefficient tracking could propagate into the formulas (5.4) and (5.5), which are used in Theorem 5.7."}],"tokens_in":24883,"tokens_out":1358,"duration_ms":36664,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"The headline: Kina gives a simpler proof of Yasuda's result that symmetric MZVs generate all MZVs (Theorem 1.3), using multitangent functions and generating-function manipulations instead of Yasuda's ζ^{♮,F}(k). Beyond the re-proof, the paper ships an explicit algorithm for expressing MZVs in terms of SMZVs (Section 3, worked out for depths 1 and 2), a depth-2 formula (Theorem 4.3), structural results for depth-3 SMZVs and FMZVs (Theorem 5.7, Propositions 5.3–5.5), and a partial equivalence of two definitions of Z_A(r,s) (Theorem A.1). These are genuine new contributions within the Kaneko-Zagier SMZV program. The generating-function framework is cleanly laid out, and the worked examples (the ζ(2,3) formula, the depth-2 explicit formula) are concrete and checkable. The author is also honest about limitations — the Remark after Lemma 2.6 openly states that the proof needs duality relations, which Yasuda's did not, and that this matters for formal SMZVs. That candor is appreciated. On the soft spots: the reader flags the duality dependence as the weakest assumption, but I agree with the stress-test that this is a scope limitation, not a correctness risk. Duality is well-established for actual MZVs. The real load-bearing concern is the identity for ∂²A₂/∂X₁∂Xᵣ that bridges the generating-function framework to Theorem 2.3 (Hirose). It is labeled 'straightforward calculation' but involves expanding A₂, matching against the explicit formula (2.1) for ζ₂, and tracking signs through stuffle regularization. If there is an error in the binomial-coefficient manipulations or sign conventions there, the connection to Theorem 2.3 fails and the proof of Theorem 2.5 collapses. The stress-test independently traced the chain (2.2)→(2.12) and found the substitutions and Lemma 2.6 application correct, which is reassuring. But the elided coefficient-matching step itself was not verified. Several other transitions are also called 'straightforward' without full detail. No machine-checked proofs or shipped code, though the algorithm is described concretely enough to implement. This is a paper for specialists in multiple zeta values, particularly those working on the Kaneko-Zagier SMZV/FMZV program. The depth-3 results and Theorem A.1 will be of independent interest. It deserves a serious referee who can verify the elided calculations in Section 2.2 — particularly the A₂ identity — and check the Bernoulli number manipulations in Appendix A.","headline":"Simpler proof of Yasuda's theorem that SMZVs generate MZVs, plus new depth-3 structural results and an explicit algorithm. The proof structure is sound; the main risk is in elided generating-function calculations, not the duality dependence.","tokens_in":26290,"tokens_out":673,"would_cite":false,"duration_ms":76894,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Symmetric zeta values generate all multiple zeta values, new proof shows","keywords":[],"falsifier":"If a weight k MZV were found that cannot be expressed as a Q-linear combination of ∗-SMZVs of weight k, the main theorem would fail. Concretely, the algorithm in Section 3, when run on such an MZV, would produce an inconsistent system.","tokens_in":25193,"feed_emoji":"ζ","tokens_out":878,"duration_ms":97997,"temperature":0.7,"pith_summary":"The paper proves that every multiple zeta value — a number built from nested infinite sums of fractions — can be written as a linear combination of symmetric multiple zeta values (SMZVs) and products of lower-weight MZVs. SMZVs are a special class of zeta values introduced by Kaneko and Zagier, defined by a symmetrization procedure that makes them well-behaved modulo ζ(2). The author gives a simpler proof than the original one by Yasuda, replacing Yasuda's auxiliary real numbers with the coefficients ζ₂(k) that arise naturally from multitangent functions — higher-depth analogues of the cotangent function. The key link is a theorem of Hirose connecting these multitangent coefficients to SMZVs via Hoffman duality. The proof proceeds by showing that the generating function for regularized MZVs is cyclically invariant modulo SMZVs and products, then applying an algebraic lemma to extract vanishing derivatives, which forces every MZV into the span of SMZVs. Beyond the proof, the paper provides an explicit algorithm for computing such decompositions (worked out for depths one and two), and studies the structure of the space spanned by depth-three SMZVs and their finite analogues, proving that this space equals the space of depth-two values for even weights.","feed_headline":"Symmetric zeta values generate all multiple zeta values","feed_subtitle":"New proof via multitangent functions replaces Yasuda's auxiliary numbers, plus an algorithm for explicit decomposition.","key_machinery":"Multitangent functions Ψ_k(z) and their ζ₂(k) coefficients; Hoffman duality connecting multitangent coefficients to SMZVs (Hirose's theorem); generating functions for regularized MZVs and their cyclic invariance modulo SMZVs and products; a differential-algebraic lemma (Lemma 2.6) extracting vanishing partial derivatives from homogeneous polynomial identities","core_discovery":"The space of all multiple zeta values of any given weight k is exactly the space spanned by the ∗-symmetric multiple zeta values of that weight. The proof replaces Yasuda's ζ^{♮,F}(k) numbers with the ζ₂(k) coefficients of multitangent functions, connected to SMZVs via Hirose's duality theorem. The argument works through generating function identities: cyclic invariance of the MZV generating function modulo SMZVs and products, combined with a differential lemma, forces all MZVs into the SMZV span. The paper also gives an algorithmic procedure for explicit decomposition and proves that for even weights, the space of triple SMZVs (and their finite analogues) coincides with the space of depth-2","pith_inferences":[],"forward_implications":["Any multiple zeta value can be algorithmically decomposed into symmetric MZVs and products, making the SMZV basis computationally accessible for low-depth cases.","The identification of the triple-SMZV space with the depth-2 space for even weights gives a concrete structural result that can be checked against dimension conjectures involving cusp forms.","The algorithm produces explicit formulas with controlled coefficients (in 1/k·Z) and depth structure, which could serve as a tool for numerical experimentation and relation-hunting among MZVs.","The partial equivalence of two definitions of Z_A(r,s) (Theorem A.1) advances the Kaneko–Zagier program of establishing a full isomorphism between finite MZVs and SMZVs."],"fun_headline_variants":["Symmetric MZVs span all multiple zeta values","Multitangent proof: symmetric MZVs generate all MZVs","Algorithm decomposes MZVs into symmetric MZVs and products","Simpler proof that symmetric MZVs span the full MZV space","Cyclic symmetry forces all MZVs into symmetric MZV span"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The proof relies on duality relations among MZVs (via Hirose's theorem connecting multitangent coefficients to SMZVs), whereas Yasuda's original proof used only extended double shuffle relations. If duality does not follow from double shuffle in a given formal setting, the proof does not go through there.","fun_headline_variants_meta":{"raw":{"variants":["Symmetric MZVs span all multiple zeta values","Multitangent proof: symmetric MZVs generate all MZVs","Algorithm decomposes MZVs into symmetric MZVs and products","Simpler proof that symmetric MZVs span the full MZV space","Cyclic symmetry forces all MZVs into symmetric MZV span"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":530,"prompt_tokens":437,"completion_tokens":93,"prompt_tokens_details":null},"tokens_in":437,"tokens_out":93,"duration_ms":24649,"temperature":1.0,"reasoning_tokens":null,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-08T23:43:39.546335+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"If a weight k MZV were found that cannot be expressed as a Q-linear combination of ∗-SMZVs of weight k, the main theorem would fail. Concretely, the algorithm in Section 3, when run on such an MZV, would produce an inconsistent system.","supporting_citations":[],"review_version":1}