{"id":"eabbce77-1fbe-45c6-9f4f-85bf1717de82","arxiv_id":"2411.15431","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors generalize Takeyama's Ohno relation to regularized refined symmetric multiple zeta values, yielding a generating-function identity with gamma factors.","lead":"This paper proves a new Ohno relation for regularized refined symmetric multiple zeta values, extending a known theorem of Takeyama. The identity involves gamma factors and an exponential factor, and is stated for a wide class of non-admissible iterated integrals from 0 to 0.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Final-step mismatch: Lemma 3.5 gives factor (e^{2πiT}−1)/(2πiT), while Theorem 1.14 requires (1−e^{−2πiT})/(2πiT); the two differ by e^{2πiT}, so the proof does not derive the stated theorem.","rationale":"The reader's weakest_assumption, the identity ρ(w(1−xT))=~ρ(w)(1−zT), is actually correct. Recomputing with the correct order in ρ(x)=(1−yT)^{-1}x shows both sides of the identity agree, and the explicit w=y expansion gives −y^2xT^2 on both sides, not the claimed −yxy versus −y^2x. So the primary objection raised by the reader does not land. However, a different load-bearing flaw is present: Lemma 3.5 produces a factor (e^{2πiT}−1)/(2πiT), while Theorem 1.14 states (1−e^{−2πiT})/(2πiT). These differ by a factor e^{2πiT}, and the A=B=0 specialization shows the theorem's factor is the one consistent with Takeyama's relation. This is an internal inconsistency in the proof of the central claim, not a disagreement with consensus. Because the main theorem is proved by combining Lemma 3.4 and Lemma 3.5, the stated result is not established by the paper's derivation. Thus the verdict should remain REJECT, despite the specific counterexample proposed by the reader being incorrect.","tokens_in":10702,"tokens_out":35297,"duration_ms":271331,"concrete_test":"Set A=B=0 in Theorem 1.14 and compare the resulting factor with Theorem 1.8. Theorem 1.14 gives (1−e^{−2πiT})/(2πiT), while Lemma 3.5 gives (e^{2πiT}−1)/(2πiT). Compute the coefficient of T in each: 1−πiT+O(T^2) versus 1+πiT+O(T^2). If these disagree, then Lemmas 3.4 and 3.5 do not imply Theorem 1.14, and the proof must be corrected before the main theorem can be accepted.","verdict_should_be":"REJECT","load_bearing_attack":"The proof's final step is internally inconsistent. Lemma 3.4 says ZRS(~ρ(...)) = ZRS(1/(1−xA) · (−y)/(1+yT) · 1/(1−xB)) · ZRS(~σ(...)). Lemma 3.5 then computes the first factor as (e^{2πiT}−1)/(2πiT) times the displayed Γ-factors. But Theorem 1.14 asserts the factor is (1−e^{−2πiT})/(2πiT) times the same Γ-factors. These are not equal: (1−e^{−2πiT})/(2πiT) = 1−πiT+O(T^2), while (e^{2πiT}−1)/(2πiT) = 1+πiT+O(T^2); the ratio is e^{2πiT}. The theorem's factor is the one needed to recover Takeyama's Theorem 1.8 when A=B=0, so the discrepancy is in Lemma 3.4 or Lemma 3.5. Since Theorem 1.14 is obtained verbatim from Lemmas 3.4 and 3.5, the stated identity is not established as written. Note that the reader's cited defect in Lemma 3.3 is not valid: with ρ=τ∘σ∘τ one has ρ(x)=(1−yT)^{-1}x, so ρ(w(1−xT))=ρ(w)(1−yT)^{-1}(1−zT)=~ρ(w)(1−zT); for w=y both sides have T^2-coefficient −y^2x, not −yxy. The genuine obstruction is the final factor mismatch, not the Lemma 3.3 identity.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a generalization of Takeyama's Ohno relation for refined symmetric multiple zeta values to the regularized setting. The main theorem (Theorem 1.14) asserts an identity for the regularized refined symmetric multiple zeta value map ZRS evaluated at a generating series involving the maps σ~ and ρ~. The proof is algebraic: it introduces an anti-automorphism φ, a symmetric harmonic product, and reduces the main identity to two lemmas about the evaluation of ZRS on certain noncommutative generating series.","tokens_in":11124,"tokens_out":39865,"duration_ms":296053,"significance":"If the main theorem were established, it would be a natural and interesting extension of both the authors' earlier regularized Ohno relation and Takeyama's refined symmetric Ohno relation, with a clean generating-function formulation. The paper is careful about the underlying noncommutative algebra and connects to prior published work. However, the proof as written contains false identities in load-bearing lemmas, so the central claim is not proven by this manuscript.","major_comments":[{"comment":"Lemma 3.3 is false. Taking w=y, we have σ~(y)=y, φ(σ~(y))=φ(y)=-y, and hence the right-hand side of the lemma equals φ((-y/(1+yT)) *~ (-y)) = φ(y/(1+yT)) = -y(1-yT)^{-1}. But by definition ρ~(y)=ρ(y)(1-yT)^{-1}=y(1-yT)^{-1}. The two sides differ by the sign of y(1-yT)^{-1}. The error enters in the proof when expanding φ(( -y/(1+yT)+y) *~ A): the contribution φ(y *~ A) equals φ(A)=σ~(w), not -φ(φ(σ~(w)))=-σ~(w). This sign error invalidates the proof of Lemma 3.3 and therefore also the proof of Lemma 3.4, which is derived directly from Lemma 3.3.","section":"Lemma 3.3"},{"comment":"The factor stated in Lemma 3.5 is (e^{2πiT}-1)/(2πiT), whereas Theorem 1.14 requires (1-e^{-2πiT})/(2πiT). These differ by the multiplicative factor e^{2πiT}. The theorem's factor is the one needed to recover Takeyama's Theorem 1.8 at A=B=0. Moreover, the proof of Lemma 3.5 itself contains a sign inconsistency: the third displayed equation changes -1/(2πi) to 1/(2πi) without explanation. Direct evaluation using Definition 1.13 gives ZRS(-y/(1+yT)) = (1-e^{2πiT})/(2πiT), which is not the lemma's stated value and also not the value required by Lemma 3.4 and Theorem 1.14 for w=y. Thus the final step of the proof does not establish the stated theorem.","section":"Lemma 3.5 vs Theorem 1.14"},{"comment":"Because Lemma 3.3 is false and Lemma 3.5 does not match the theorem, the derivation of Theorem 1.14 as stated fails at two independent points. These are not typographical slips: the sign error changes the content of Lemma 3.3, and the exponential factor in Lemma 3.5 changes the generating series. The main theorem might be true, but it is not proved by the arguments in this manuscript.","section":"Overall proof of Theorem 1.14"}],"minor_comments":[{"comment":"The proof of Proposition 2.2 uses paths β and β' but the notation is compressed; labeling the path composition in a displayed equation would improve readability.","section":"Section 2.2"},{"comment":"The manuscript uses both H0 and h0 for the same space; the proof of Lemma 3.4 introduces H0 without defining it, though the intended identification with h0 is clear from context.","section":"Section 3"}],"recommendation":"reject","confidential_remarks":"The manuscript is clearly written and the intended generalization is natural, but the proof contains serious errors in Lemmas 3.3 and 3.5 that cannot be repaired locally. The authors would need to rework the proof substantially, so I recommend rejection in the current form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this paper with the expectation that the main theorem is plausible but the proof is not yet trustworthy. The good news: Theorem 1.14 is a natural extension of Takeyama's Ohno relation to regularized refined symmetric MZVs, and the paper's setup with ZRS in Section 2 is genuinely useful. The duality (Prop 2.2) and the symmetric harmonic product (Thm 2.4) are coherent and likely correct.\n\nThe problems start in Section 3. Lemma 3.3 is false as stated. Take w=y. The left side ~ρ(y) = y(1-yT)^{-1}. The right side, φ(-y/(1+yT) ~∗ φ(~σ(y))) = φ(-y/(1+yT) ~∗ (-y)) = φ(y/(1+yT)) = -y(1-yT)^{-1}. Sign error. The stress-test note correctly says that the identity ρ(w(1-xT)) = ~ρ(w)(1-zT) is valid, so the reviewer's specific complaint about that identity is misdirected. But the lemma itself still breaks.\n\nLemma 3.5 also has a mismatch with Theorem 1.14. The theorem needs (1-e^{-2πiT})/(2πiT); Lemma 3.5 produces (e^{2πiT}-1)/(2πiT). These differ by e^{2πiT}. At A=B=0, Lemma 3.5 claims ZRS(-y/(1+yT)) = 1 + πiT + ..., while direct computation gives -1 - πiT + .... So the sign in the lemma is off.\n\nThe authors know the subject; the errors look like sign slips, not conceptual confusion. The main theorem passed a simple sanity check (w=y). So the likely truth of the result does not rescue the current proof. Both Lemma 3.3 and Lemma 3.5 need fixing before the paper can be trusted.\n\nWho is this for? Specialists in multiple zeta values, particularly those interested in refined symmetric MZVs and Ohno-type relations. It deserves a serious referee: the result is meaningful, and the corrections are probably local. But as it stands, the proof is not coherent. I would send it out, but with a clear request for a corrected revision rather than accepting anything on faith.","headline":"Natural extension of Takeyama's Ohno relation, but the proof has load-bearing sign errors in Lemmas 3.3 and 3.5 that must be fixed.","tokens_in":11650,"tokens_out":31480,"would_cite":false,"duration_ms":226811,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M32"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves an Ohno-type generating-function relation for regularized refined symmetric multiple zeta values, with explicit gamma and exponential factors, extending the classical Ohno and refined symmetric relations to non-admissible…","keywords":["multiple zeta values","Ohno relation","refined symmetric multiple zeta values","regularized multiple zeta values","generating functions","iterated integrals","gamma factors"],"falsifier":"Expand both sides of the identity $\\rho(y(1-xT))=\\tilde{\\rho}(y)(1-(x+y)T)$ and compare the coefficients of $T^2$: the left-hand side contains the word $-yxy$ and the right-hand side contains $-y^2x$. Whether these coefficients agree or differ decides whether the key lemma's premise holds; in parallel, evaluating the full Theorem 1.14 numerically for a small word at $A=B=0$ would directly test the final relation.","tokens_in":10484,"feed_emoji":"🧮","tokens_out":13305,"duration_ms":121015,"temperature":0.7,"pith_summary":"The paper proves an Ohno-type relation for regularized refined symmetric multiple zeta values, the iterated-integrals-from-0-to-0 analogue of the classical multiple zeta values. The main theorem is a generating-function identity in formal parameters $A$ and $B$ and a series parameter $T$, comparing a symmetric rho transform with a symmetric $\\sigma$ transform on every noncommutative word in $x,y$ that begins and ends with $y$. If the theorem is correct, it provides one source for sum-formula and duality relations at all regularized weights, and specializing $A=B=0$ recovers the earlier Ohno relation for refined symmetric multiple zeta values. The proof works through a regularization map $Z_{\\mathrm{RS}}$, a duality involution, a symmetric harmonic product, and a gamma-function evaluation of a universal factor.","feed_headline":"Ohno relation extended to regularized refined zeta values","feed_subtitle":"Gamma-factor identity covers non-admissible iterated integrals and recovers the refined symmetric case at A=B=0.","key_machinery":"The load-bearing object is the $\\mathbb{Q}$-linear map $Z_{\\mathrm{RS}}\\colon h\\to\\mathbb{C}$ on noncommutative polynomials in $x,y$, which extends refined symmetric multiple zeta values to all words by the explicit formula\n$$Z_{\\mathrm{RS}}(u_1\\cdots u_k)=\\sum_{\\substack{0\\le p\\le q\\le k\\\\ u_{p+1}=\\cdots=u_q=y}}\\frac{(-2\\pi i)^{q-p-1}}{(q-p)!}(-1)^{k-q}Z_x(u_1\\cdots u_p)Z_x(u_k\\cdots u_{q+1}).$$\nIt is also an iterated integral along a loop from $0$ to $0$ encircling $1$ once. The argument is carried by the deformed maps $\\tilde{\\sigma}(w)=\\sigma(w)(1-xT)$ and $\\tilde{\\rho}(w)=\\rho(w)(1-yT)^{-1}$, the duality involution $\\phi(x)=x+y$, $\\phi(y)=-y$, and the symmetric harmonic product $\\tilde{*}$; Lemma 3.4 uses these to rewrite $\\tilde{\\rho}$-evaluation as $\\tilde{\\sigma}$-evaluation times one universal word, and Lemma 3.5 evaluates that word through gamma-function identities.","core_discovery":"The central claim is Theorem 1.14: for every $w\\in h_0$, the regularized refined symmetric evaluation satisfies\n$$Z_{\\mathrm{RS}}\\left(\\tilde{\\rho}\\left(\\frac{1}{1-xA}w\\frac{1}{1-xB}\\right)\\right)\n=\\frac{1-$e^{{-2\\pi i T}}$}{2\\pi i T}\n\\left(2-\\frac{\\Gamma(1-T)\\Gamma(1+A)}{\\Gamma(1-T+A)}\\right)\n\\left(2-\\frac{\\Gamma(1+T)\\Gamma(1-B)}{\\Gamma(1+T-B)}\\right)\nZ_{\\mathrm{RS}}\\left(\\tilde{\\$\\sigma$}\\left(\\frac{1}{1-xA}w\\frac{1}{1-xB}\\right)\\right).$$\nThis is offered as the refined-symmetric counterpart of the authors' earlier regularized Ohno relation for ordinary multiple zeta values, and as a regularization of the known refined-symmetric Ohno relation: when $A=B=0$, the factor in front reproduces the coefficient matching that earlier theorem. The proof reduces the $\\tilde{\\rho}$ side to the $\\tilde{\\sigma}$ side times a single universal word, then evaluates that word by known gamma-function identities for regularized shuffle products.","pith_inferences":["A direct check of the identity $\\rho(w(1-xT))=\\tilde{\\rho}(w)(1-(x+y)T)$ at order $T^2$ for $w=y$ is not included in the paper; this coefficient-level test would settle whether the proof of Lemma 3.3, and hence the main theorem, survives as written.","One could test the theorem numerically for small words such as $w=yxy$ by comparing both sides as power series in $A,B,T$ against known values of refined symmetric multiple zeta values; the paper contains no numerical sample.","The loop-integral description of $Z_{\\mathrm{RS}}$ suggests the same relation may transport to cyclotomic or $t$-adic refinements, since the proof uses only the symmetric harmonic product and duality structure.","If the gamma-factor shape is not an artifact of the regularization, then specializations such as $A=B=1/2$ would yield identities among regularized refined symmetric zeta values at shifted integer weights, a consequence the authors do not spell out."],"forward_implications":["Setting $A=B=0$ recovers the Ohno relation for refined symmetric multiple zeta values, including the factorial denominator $(j+1)!$ that the earlier statement omitted.","The theorem applies to non-admissible words, so it gives regularized, divergent-from-0-to-0 analogues of the sum and duality formulas, not just convergent cases.","Because the identity is a generating function in $T$ with two extra parameters, one computation packages infinitely many Ohno-type relations of all weights.","The explicit gamma factors suggest the relation admits analytic continuation in $A$ and $B$, so it can be specialized outside the formal-power-series setting."],"supporting_citations":[{"why":"The original Ohno relation, the classical sum-duality statement that the regularized and refined settings generalize.","marker":"[8]"},{"why":"The authors' earlier regularized Ohno relation; Proposition 3.1 and the overall proof pattern are imported from it.","marker":"[4]"},{"why":"The preceding Ohno relation for refined symmetric multiple zeta values, which the main theorem extends and recovers at $A=B=0$.","marker":"[10]"},{"why":"Gives the definition of refined symmetric multiple zeta values, their iterated-integral description, and the duality relation used in Proposition 2.2.","marker":"[2]"},{"why":"Supplies the regularized symmetric harmonic relation (Theorem 2.4) used to split the $\\tilde{\\rho}$-side into a product in Lemma 3.4.","marker":"[3]"},{"why":"The regularization theorem for multiple zeta values; it underlies $\\rho$, shuffle regularization, and the gamma-function evaluation in Lemma 3.5.","marker":"[5]"},{"why":"The weak Ohno relation for symmetric multiple zeta values, the mod-$\\pi^2$ shadow of the refined result.","marker":"[9]"}],"fun_headline_variants":["Regularized refined Ohno relation proven via gamma-factor identity","Ohno identity extends to non-admissible 0-to-0 iterated integrals","Exact gamma factor yields Ohno relation for refined zeta values","Refined symmetric zeta: Ohno relation for non-admissible integrals","From admissible to regularized: Ohno relation for refined zeta"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof's key step assumes that a certain deformation of the word map can be pulled past multiplication by $1-xT$ in a simple way; the identity $\\rho(w(1-xT))=\\tilde{\\rho}(w)(1-(x+y)T)$ is exactly that assumption, and the main theorem collapses without it.","fun_headline_variants_meta":{"raw":{"variants":["Regularized refined Ohno relation proven via gamma-factor identity","Ohno identity extends to non-admissible 0-to-0 iterated integrals","Exact gamma factor yields Ohno relation for refined zeta values","Refined symmetric zeta: Ohno relation for non-admissible integrals","From admissible to regularized: Ohno relation for refined zeta"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000745,"raw_usage":{"total_tokens":3303,"prompt_tokens":910,"completion_tokens":2393,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":526,"completion_tokens_details":{"reasoning_tokens":2300}},"tokens_in":526,"tokens_out":2393,"duration_ms":16490,"temperature":1.0,"reasoning_tokens":2300,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:20:05.495153+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Expand both sides of the identity $\\rho(y(1-xT))=\\tilde{\\rho}(y)(1-(x+y)T)$ and compare the coefficients of $T^2$: the left-hand side contains the word $-yxy$ and the right-hand side contains $-y^2x$. Whether these coefficients agree or differ decides whether the key lemma's premise holds; in parallel, evaluating the full Theorem 1.14 numerically for a small word at $A=B=0$ would directly test the final relation.","supporting_citations":[{"cited_title":"Number Theory 74 (1999), no","cited_arxiv_id":null,"evidence_quote":"The original Ohno relation, the classical sum-duality statement that the regularized and refined settings generalize."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The authors' earlier regularized Ohno relation; Proposition 3.1 and the overall proof pattern are imported from it."},{"cited_title":"52 (2020), no","cited_arxiv_id":null,"evidence_quote":"The preceding Ohno relation for refined symmetric multiple zeta values, which the main theorem extends and recovers at $A=B=0$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the definition of refined symmetric multiple zeta values, their iterated-integral description, and the duality relation used in Proposition 2.2."},{"cited_title":"On a lifting of $t$-adic symmetric multiple zeta values","cited_arxiv_id":"2311.00473","evidence_quote":"Supplies the regularized symmetric harmonic relation (Theorem 2.4) used to split the $\\tilde{\\rho}$-side into a product in Lemma 3.4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The regularization theorem for multiple zeta values; it underlies $\\rho$, shuffle regularization, and the gamma-function evaluation in Lemma 3.5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The weak Ohno relation for symmetric multiple zeta values, the mod-$\\pi^2$ shadow of the refined result."}],"review_version":1}