{"id":"193d35a0-813b-41fc-9538-e72db6751560","arxiv_id":"1908.03065","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The author proves several 1997 Borwein-Bradley-Broadhurst conjectures and general reduction formulas for alternating multiple zeta values using iterated integrals.","lead":"This paper proves new identities for alternating multiple zeta values, including several conjectures from 1997, by manipulating iterated integrals of multiple polylogarithms. The results give a systematic way to express complicated alternating sums in terms of simpler unit-exponent alternating sums.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equations (1.13)–(1.14) contain first-exponent-1 multiple zeta values that diverge in the paper's own convergence domain; no regularization is defined, so the claimed proofs of these BBB conjectures are not identities among convergent zeta values.","rationale":"The reader's weakest assumption is the compressed proof of Theorem 2.2, but the more concrete and more damaging issue is that the central identities (1.13) and (1.14) are not even stated for convergent zeta values. The paper defines MZVs by convergent limits and never introduces a regularization for series with first exponent 1. Since (1.13) is one of the six BBB conjectures the paper claims to prove, and its proof via (4.9) appears to mix Li evaluated at -1 (which would give a leading bar) with the non-alternating ζ(m+1,{1}^n), the argument for this conjecture is not sound as written. This is not a disagreement with the expected consensus on the BBB identities; it is a correctness risk internal to the manuscript's own definitions. The issue could be resolved either by restoring missing bars or by explicitly working in a regularized MZV algebra and proving the cancellation of divergent parts. Because the mathematical framework is standard and the remaining identities (1.9)–(1.12) are plausible and independently checkable, the appropriate verdict remains conditional rather than outright rejection.","tokens_in":29856,"tokens_out":35532,"duration_ms":316171,"concrete_test":"Take m=1,n=1 in (1.13). Evaluate the RHS partial sums up to N: ζ_N(1,1,1)=Σ_{N≥n1>n2>n3≥1}1/(n1 n2 n3) and ζ_N(1,1,\\bar1)=Σ_{N≥n1>n2>n3≥1}(-1)^{n3}/(n1 n2 n3). These grow like (log N)^2/2 and do not converge, so the RHS is not a well-defined MZV; the LHS is ζ(2,1)=ζ(3)=1.2020569. If the intended statement had a bar, instead evaluate the two convergent alternating sums ζ(\\bar1,1,1) and ζ(\\bar1,1,\\bar1) to high precision and check whether -ε_+ζ(\\bar1,1,1)-ε_-ζ(\\bar1,1,\\bar1) equals ζ(3).","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In the paper's own definition (1.5)–(1.7), a multiple zeta value is a limit of finite sums and convergence is tied to the condition ℜ(s1+...+sj)>j. In (1.13), each summand ζ(1,{1}^n,S_k) has first exponent 1 and all later exponents ±1. For m≥1 the depth is at least 2, and the series diverges: after summing the inner variables with a positive first index, the remaining tail behaves like a nonzero constant divided by n1, so Σ 1/n1 diverges; alternating signs in S_k do not rescue a positive first sign. Thus the right-hand side is not a convergent multiple zeta value in the sense defined by the paper. Similarly, the left-hand side of (1.14) is ζ(1,m+1,{1}^n), also with first exponent 1, hence divergent for all m≥0 and n≥0. The derivation quoted for (1.13), Eq. (4.9), compounds the problem: its right-hand side is Li(-1,{1}^n,σ1,σ2/σ1,...), which is ζ(\\bar1,{1}^n,...), not ζ(1,{1}^n,...). So either a bar has been lost in transcription, or the identities are asserted for a regularized value that is never defined. Since Theorem 3.4 and Theorem 4.1 are supposed to prove exactly these formulas, this gap is load-bearing for the paper's central claim.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops iterated-integral identities for multiple polylogarithm functions and applies them to alternating multiple zeta values. Section 2 proves a transformation theorem (Theorem 2.2) and an integration-by-parts identity (Theorem 2.4). Section 3 derives a general reduction (Theorem 3.4) expressing alternating MZVs of the form ζ(\\bar1,{1}^{m1-1},p1+1,{1}^{m2-1},...,pk+1,{1}^{mk+1-1}) in terms of ordinary MZVs and infinite sums involving multiple harmonic star sums. Section 4 applies these results to prove the six Borwein-Bradley-Broadhurst conjectures numbered (1.9)-(1.14). Section 5 extends the method to Kaneko-Yamamoto multiple zeta values, and Section 6 gives general linear relations via (p+2)-posets.","tokens_in":30177,"tokens_out":13116,"duration_ms":118669,"significance":"If the statements are corrected as indicated below, the paper offers explicit, checkable identities and a uniform method that proves several long-standing conjectures. The approach is not circular: the identities are derived from the iterated integral representation, and the intermediate known identities such as (4.2) and (2.4) are cited from published sources. The paper is systematic and gives many concrete examples and corollaries. However, the displayed BBB identities (1.13)-(1.14) are not well-defined as written, and this directly affects the central claim that those conjectures are proved.","major_comments":[{"comment":"As printed, equations (1.13) and (1.14) are not identities among convergent multiple zeta values under the paper's own convergence criterion ℜ(s1+...+sj)>j quoted for (1.3). In (1.13), every summand ζ(1,{1}^n,S_k) has first exponent +1, so its defining series over n1 is a sum of a term that does not tend to zero divided by n1, which diverges; the alternating signs in S_k do not affect the first summation variable. Similarly, the left-hand side of (1.14) is ζ(1,m+1,{1}^n), which also has first exponent +1 and diverges. The derivation in Section 4 indicates that barred exponents are intended: Eq. (4.9) is obtained from Li_{m+1,{1}^n}(-1), which is ζ(\\bar{m+1},{1}^n), and its right-hand side is a sum of Li(-1,{1}^n,...)=ζ(\\bar1,{1}^n,...), not ζ(1,{1}^n,...). Thus either a bar (and in (4.10) also a sign) has been lost in transcription, or the identities are asserted for a regularized value that is never defined. Since Theorem 3.4 and Theorem 4.1 are advertised as proofs of (1.13)-(1.14), this must be corrected and the corrected statements verified before the central claim can be accepted.","section":"Eqs. (1.13)-(1.14) and Section 4 (Eqs. (4.9), (4.14))"},{"comment":"The proof of Theorem 2.2, which is the foundation for Theorem 4.1 and for equations (4.9) and (4.14), is compressed into the sentence \"by a direct calculation.\" The summation over σ_j∈{1,a} with η(1)=1 and η(a)=-a, the product ∏ η(σ_j)/σ_j, and the bookkeeping of the Cat blocks are exactly the steps that produce the exponent sequences in (2.2). Please provide the full expansion, including the degenerate cases m1=0 and p_i=0, because the one-to-one correspondence in (2.4) and the later reductions depend on the precise placement of the arguments a/σ_i and σ_i/σ_{i-1}.","section":"Theorem 2.2, Eq. (2.2)"},{"comment":"The proof of Theorem 3.4 says \"Continuing this process k times,\" and the notation E_i, F_i with the □ operation is intricate. The step in which products of harmonic sums are expanded via the stuffle product is asserted in a paragraph rather than shown. Since Theorem 3.4 is the main reduction result underlying Theorem 4.1 and the BBB identities, please supply the induction in detail and state exactly how the ζ(E_i,{1}^{p_i-j}) factors and the infinite harmonic-star sums in (3.9) are obtained, especially in degenerate cases where some p_i=0 or m_i=1.","section":"Theorem 3.4, Eq. (3.9)"}],"minor_comments":[{"comment":"The convention for m1=0 is explained only after the statement of Proposition 2.1, but the symbol {1}^{-1} appears inside the displayed formula before that convention is introduced. Please state the convention before the proposition.","section":"Proposition 2.1"},{"comment":"Reference [18] is incomplete: it gives authors and a title but no journal, year, arXiv number, or DOI. Please supply the full publication data.","section":"References"},{"comment":"The proofs of Theorems 5.1 and 5.2 are described as \"similar\" to earlier proofs. Because these theorems are used for Corollary 5.6, please include enough detail (or a supplement) to make the derivations verifiable.","section":"Theorems 5.1 and 5.2"},{"comment":"The manuscript contains numerous typographical and OCR artifacts, such as \"pol ylogarithm\" in the abstract, malformed diagrams in Section 6, and inconsistent spacing in several displayed equations. A careful proofread is needed.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The missing-bar issue in (1.13)-(1.14) is likely a transcription typo rather than a substantive mathematical error, but because the paper's headline claim is the proof of those exact conjectures, the correction must be explicit and the corrected identities checked. I would not reject on this basis alone. The editor may also wish to check the degree of overlap with the author's prior papers [19,20], since those are used for several intermediate identities."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Xu proves a batch of Borwein–Bradley–Broadhurst identities (1.9)–(1.12) and gives general reduction theorems (2.2, 3.4, 4.1, 5.5) for alternating MZVs to unit-exponent ones. The method is standard iterated integrals, and the main body looks credible: Theorem 4.1 plus the corollary at (4.5) really does give (1.9) and (1.11), and the derivations of (1.10) and (1.12) from Corollary 3.6 are plausible. The paper also contains useful identities for Kaneko–Yamamoto zeta values and a linear relation in Section 6. Within the MZV subfield this is a meaningful result, and the novelty claim is honest: (1.10) is explicitly credited to Wang–Liu–Chen.\n\nThe soft spot is real and, as printed, load-bearing for two of the six conjectures. Equations (1.13)–(1.14) contain ζ(1,{1}^n,...) with first exponent 1. Under the paper's own convergence definition, a multiple zeta value with first exponent 1 diverges: after summing the inner variables, the tail behaves like a constant divided by n1, and alternating signs in later slots do not rescue a positive first sign. The derivation of (1.13) at (4.9) actually ends with Li{...}(-1,{1}^n,...), which is ζ(\\bar1,{1}^n,...), not ζ(1,{1}^n,...). So either a bar is missing in the statement or the identity uses an undefined regularized value. This needs fixing, and it matters because (1.13)–(1.14) are advertised as proven conjectures.\n\nOther soft spots are minor by comparison: Theorem 2.2's key step is 'by a direct calculation,' and Theorems 5.1 and 5.2 are described as 'similar' proofs. The extracted text is garbled enough that full line-by-line checking is impossible, but I found no contradiction in the convergent part. The reliance on the author's own earlier papers for some intermediate identities is not a red flag; those are published results.\n\nRecommendation: send it to a qualified referee if the author supplies a corrected version of (1.13)–(1.14) with the bars restored or a regularization defined. As it stands, the paper deserves referee time—the main theorems are likely right and the standard for this subfield is explicit identities—but it should not be accepted without addressing the divergence.","headline":"Standard-technique proofs of several Borwein–Bradley–Broadhurst conjectures, but (1.13)–(1.14) as printed use divergent first-exponent-1 zeta values and a missing bar appears to be the cause.","tokens_in":30780,"tokens_out":3725,"would_cite":false,"duration_ms":36005,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M06","11M40","40B05","33E20"],"pacs":[],"model":"deepseek-v4-flash","headline":"One integral method proves six zeta-value conjectures.","keywords":["multiple polylogarithm functions","iterated integrals","alternating multiple zeta values","unit-exponent alternating multiple zeta values","multiple harmonic star sums","six conjectured identities","circled-product multiple zeta values","linear relations of alternating multiple zeta values"],"falsifier":"Take identity (1.11) with $m=n=0$ and compute both sides as partial sums of their defining multiple harmonic series up to $N=10^6$ terms: the left side is $\\zeta(\\bar{1},2,2)$ and the right side is an explicit combination of unit-exponent alternating values, so a mismatch beyond truncation error would refute the central identity.","tokens_in":29641,"feed_emoji":"🧮","tokens_out":13008,"duration_ms":118307,"temperature":0.7,"pith_summary":"This paper proves that a broad family of alternating multiple zeta values—zeta-type sums whose summands carry signs attached to the summation labels—can be rewritten using only ordinary multiple zeta values together with the simplest alternating ones, where every exponent after the first is 1. The proof passes through iterated integrals of multiple polylogarithm functions: a change of variables turns each integral into a finite sum over sign choices, and the special point $1/2$ links these integrals to the unit-exponent alternating values. This reduction yields explicit formulas for the six conjectured identities (1.9)--(1.14) stated in the introduction, and more generally expresses every value of the form ζ(¯1, {1}ᵀ₁⁻¹, p₁+1, {1}ᵀ₂⁻¹, …, p_k+1, {1}ᵀ_{k+1}⁻¹) in terms of unit-exponent alternating values. Section 5 extends the method to circled-product multiple zeta values, and Section 6 derives a general linear-relation identity for alternating multiple zeta values. If the identities are right, a family of special values previously approachable only case-by-case is captured by one uniform evaluation.","feed_headline":"One integral method proves six zeta-value conjectures","feed_subtitle":"Alternating multiple zeta values become ordinary ones, with explicit formulas in all six cases.","key_machinery":"The central machinery is the iterated-integral representation (2.1) of the multiple polylogarithm function—a nested series whose partial sums are multiple harmonic sums—combined with the change-of-variables identity (2.2). That identity rewrites the integral, after setting all parameters equal to $a$, as a sum over $2^{p_k}$ sign choices σ_j ∈ {1,a}, weighted by η(1)=1 and η(a)=−a. When $a=1/2$, equation (2.4) gives a one-to-one correspondence between these polylogarithm values and unit-exponent alternating multiple zeta values, and when $a=-1$, the same identity produces alternating-multiple-zeta-value relations directly. This single substitution rule is what carries every later evaluation in the paper.","core_discovery":"The paper's central discovery is that the alternating multiple zeta value ζ(¯1, {1}ᵀ₁⁻¹, p₁+1, {1}ᵀ₂⁻¹, …, p_k+1, {1}ᵀ_{k+1}⁻¹) is a finite 𝔺-linear combination of ordinary multiple zeta values and unit-exponent alternating multiple zeta values. This is Theorem 3.4 together with the correspondence at $1/2$; in the special case where all $p_i=1$ it gives Theorem 4.1, from which the six identities (1.9)--(1.14) follow as corollaries. The paper further proves that certain circled-product multiple zeta values, such as ζ(({2}ᵃ,3,{2}ᵇ) ⊛ (0,{2}ᵅ)⋆), lie in the ring 𝔺[ζ(2),ζ(3),ζ(4),…] generated by ordinary single zeta values, and it closes with a general linear-relation identity for alternating multiple zeta values obtained from an extended poset integral.","pith_inferences":["The reduction in Theorem 2.2 is algorithmic: the sign-sum expansion and the $1/2$ correspondence could be implemented as a normal-form procedure that rewrites any alternating multiple zeta value of the stated shape into a basis expression, giving a practical way to compute the 𝔺-span at fixed weight without case-by-case guessing.","Because the key change of variables works for a general real parameter $a$, the same proof scheme should produce analogous reduction identities for colored multiple zeta values at roots of unity, provided convergence conditions are handled; the paper does not itself make that extension.","The poset integral identity of Section 6 is strong enough to suggest a conjectural complete set of linear relations for alternating multiple zeta values, parallel to the conjecture for ordinary multiple zeta values that one integral-series identity generates all relations; the paper stops short of stating that conjecture."],"forward_implications":["Every alternating multiple zeta value of the shape ζ(¯1, {1}ᵀ₁⁻¹, p₁+1, {1}ᵀ₂⁻¹, …, p_k+1, {1}ᵀ_{k+1}⁻¹) can be evaluated in terms of ordinary multiple zeta values and unit-exponent alternating values, so no genuinely new class of constants is needed for these values.","The six conjectured identities (1.9)--(1.14) hold for all nonnegative integers $m,n$; for instance, ζ(¯1, {1}ᵀ, 2, {1}ⁿ) = ζ(¯1, {1}ⁿ, ¯1, ¯1, {1}ᵀ) − ζ(¯1, {1}ᵀ+ⁿ+²).","Several families of circled-product multiple zeta values, including ζ(({2}ᵃ,3,{2}ᵇ) ⊛ (0,{2}ᵅ)⋆), are shown to lie in 𝔺[ζ(2),ζ(3),ζ(4),…], with explicit rational-linear formulas such as the displayed evaluation of ζ((3,2) ⊛ (0,2,2)⋆).","The (p+2)-poset integral picture yields a family of linear relations among alternating multiple zeta values, with explicit small cases exhibited as equation (6.11).","The method proves the six conjectures uniformly, replacing a situation in which one of the identities had previously required a separate, later proof."],"supporting_citations":[{"why":"Supplies the six conjectured identities (1.9)--(1.14) that the paper sets out to prove; they are the paper's target.","marker":"[3]"},{"why":"Provides the one-to-one correspondence (Eq. (6.8)) between multiple polylogarithm values at 1/2 and unit-exponent alternating multiple zeta values, used to convert integral identities into zeta identities.","marker":"[4]"},{"why":"Independently states the same $1/2$ correspondence (Corollary 5), reinforcing the conversion step.","marker":"[29]"},{"why":"Gives the identity $\\operatorname{Li}_{\\{1\\}^j}(1/2)=-\\zeta(\\bar{j})$ used in the proof of identity (1.14).","marker":"[21]"},{"why":"Supplies the integral-series identity for circled-product multiple zeta values used in Section 5 to relate those values to ordinary multiple zeta values.","marker":"[13]"}],"fun_headline_variants":["Integral relations prove six alternating zeta conjectures","One integral identity tames alternating multiple zeta values","Proof: alternating zeta values reduce to ordinary ones","New integral method unifies alternating and standard zeta values","Borwein–Bradley–Broadhurst conjectures proven by integrals"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire reduction rests on the iterated-integral change-of-variables identity (2.2), whose proof is condensed into a 'direct calculation'; if that step is wrong, every later evaluation of an alternating multiple zeta value inherits the error.","fun_headline_variants_meta":{"raw":{"variants":["Integral relations prove six alternating zeta conjectures","One integral identity tames alternating multiple zeta values","Proof: alternating zeta values reduce to ordinary ones","New integral method unifies alternating and standard zeta values","Borwein–Bradley–Broadhurst conjectures proven by integrals"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000646,"raw_usage":{"total_tokens":2943,"prompt_tokens":898,"completion_tokens":2045,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":514,"completion_tokens_details":{"reasoning_tokens":1964}},"tokens_in":514,"tokens_out":2045,"duration_ms":13055,"temperature":1.0,"reasoning_tokens":1964,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:35:08.569884+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take identity (1.11) with $m=n=0$ and compute both sides as partial sums of their defining multiple harmonic series up to $N=10^6$ terms: the left side is $\\zeta(\\bar{1},2,2)$ and the right side is an explicit combination of unit-exponent alternating values, so a mismatch beyond truncation error would refute the central identity.","supporting_citations":[{"cited_title":"Borwein, D.M","cited_arxiv_id":null,"evidence_quote":"Supplies the six conjectured identities (1.9)--(1.14) that the paper sets out to prove; they are the paper's target."},{"cited_title":"Borwein, D.M","cited_arxiv_id":null,"evidence_quote":"Provides the one-to-one correspondence (Eq. (6.8)) between multiple polylogarithm values at 1/2 and unit-exponent alternating multiple zeta values, used to convert integral identities into zeta identities."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Independently states the same $1/2$ correspondence (Corollary 5), reinforcing the conversion step."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the identity $\\operatorname{Li}_{\\{1\\}^j}(1/2)=-\\zeta(\\bar{j})$ used in the proof of identity (1.14)."},{"cited_title":"Kaneko, S","cited_arxiv_id":null,"evidence_quote":"Supplies the integral-series identity for circled-product multiple zeta values used in Section 5 to relate those values to ordinary multiple zeta values."}],"review_version":1}