{"id":"933c9d3e-7f3f-484d-bb60-8196e3efe497","arxiv_id":"1908.03200","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Weighted sums of multiple t-values and t-star values at even arguments reduce to finite sums of ζ(2l)t(2k-2l) terms with polynomial coefficients.","lead":"This paper proves that weighted sums of multiple t-values, sums over odd denominators, can always be rewritten as compact combinations of ordinary zeta and single t-values. It extends the weighted sum program from multiple zeta values to their odd variant, with explicit formulas through depth 4.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 2.7's evenness conclusion rests on an unproved linear-independence lemma for 1,F,DF,...,D^mF; the citation to [11, Lemma 2.6] may not cover this t-value generating function.","rationale":"The reader's weakest assumption is the same one I would flag: the linear independence lemma. I checked the rest of the argument and found it internally consistent. In particular, the coefficient comparisons leading to (2.13) and (2.14) have the correct powers of 2, the conversion to t-values in (2.16) is sign- and factor-correct including the l=0 term with zeta(0)=-1/2, and the reduction from Theorem 2.9 to Theorem 1.2, though only announced, follows by linearity on monomials. The reduction from Theorem 1.2 to Theorem 1.1 via the symmetric sum formulas and the block-polynomial lemma is also sound. Thus the only substantive gap is the unproved or imprecisely cited independence lemma. Because that lemma is very likely true, this is a conditional outcome rather than a rejection; the authors should include a proof or a precise reference applying to their F. Keeping the reader's CONDITIONAL verdict is appropriate.","tokens_in":23190,"tokens_out":31781,"duration_ms":322782,"concrete_test":"Check whether [11, Lemma 2.6] is stated for F_t(x)=x/2 - x/(e^x+1). If not, settle the issue by proving independence from the pole structure: F_t has poles at every (2n+1)i*pi, so it cannot satisfy a nonzero linear differential equation with rational-function coefficients, which is exactly what a Q(x)-linear relation among 1,F_t,...,D^mF_t would give. As a numerical sanity check, also evaluate the Wronskian of F_t,DF_t,...,D^mF_t at a random rational point for m=0..4; it should be nonzero.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Proposition 2.7(1) needs every remainder R_j(x) in the expansion D^{m_1}F...D^{m_n}F = sum R_j D^{j-1}F + R_0 to be even. The only proof offered is that all D^mF are even and that 1,F(x),DF(x),...,D^mF(x) are linearly independent over Q(x), citing [11, Lemma 2.6]. This is the load-bearing point: evenness of R_j is what produces the even powers x^{2l} in (2.12), hence the coefficients a_{jl}, and ultimately the zeta(2l)t(2k-2l) shape of Theorems 2.8, 2.9, and 1.1. If independence failed, Proposition 2.7(1) would not follow and the entire parity structure would collapse. The paper neither states nor proves the lemma for the specific F(x)=x/2 - x/(e^x+1) = (x/2)tanh(x/2), and it is not automatic that a lemma from [11], which treats the MZV generating function, applies verbatim to this t-value generating function. The lemma is almost certainly true (F has infinitely many poles and is not D-finite), so the risk is low, but as written the proof of the central claim contains a genuine unverified keystone.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies weighted sums of multiple t-values and multiple t-star values at even arguments. For any symmetric polynomial f of degree r in n variables and k≥n, Theorem 1.1 asserts that T_f(2,k,n) and T*_f(2,k,n) can be written as finite sums over l of c_{f,l}(k) ζ(2l) t(2k-2l), with l up to min{T,k}, T=max{[(r+n-2)/2],[(n-1)/2]}, and c_{f,l}∈Q[x] of degree at most r+n-2l-1. The proof develops a Bernoulli-number weighted sum formula (Theorem 2.8) via generating functions and matrix inversion, converts it to t-values using Euler's formula (Theorem 2.9), and reduces the symmetric-weight case to the monomial case using Hoffman's symmetric sum formulas and a polynomial interpolation lemma. The appendix gives explicit formulas through depth 4.","tokens_in":23492,"tokens_out":18888,"duration_ms":162788,"significance":"If correct, the result is a natural and nontrivial analogue for multiple t-values of the weighted sum formulas previously established for multiple zeta values in [11]. The theorem covers all symmetric polynomial weights and gives explicit degree bounds for the coefficient polynomials, and the appendix provides concrete formulas that can be checked numerically. The method is parameter-free and the derivations are largely self-contained apart from cited ingredients. The paper should be of interest to specialists in multiple zeta values and their variants.","major_comments":[{"comment":"The proof of Proposition 2.7(1) uses the assertion that 1, F(x), DF(x), ..., D^mF(x) are linearly independent over Q(x), citing [11, Lemma 2.6]. However, the function F(x)=x/2 - x/(e^x+1) that appears here is not the same as the multiple-zeta generating function treated in [11], and the paper does not state the lemma or demonstrate that it applies to this F. This is load-bearing: the independence is exactly what forces each remainder R_j(x) in Proposition 2.7(1) to be even, and evenness is what produces the x^{2l} terms in (2.12) and hence the ζ(2l)t(2k-2l) shape of Theorems 2.8, 2.9, and 1.1. I recommend that the authors state the lemma explicitly and either prove it for this F (for instance, by using the pole structure of F at odd multiples of πi) or quote a version from [11] that is visibly applicable.","section":"Section 2.1, after Eq. (2.8)"}],"minor_comments":[{"comment":"The set I(k,n) is defined as all indices of weight k and depth n, but it should be stated explicitly that these are ordered compositions of k into n positive integers; this is essential because the multiple t-values are not symmetric in their arguments.","section":"Section 1, definition of I(k,n)"},{"comment":"The proof of Theorem 1.2 is reduced to a one-line pointer to Theorem 2.9; since Theorem 2.9 handles monomial weights and both sides of (1.3) are linear in the weight, the step is valid, but the sentence should say 'by linearity' to make the argument transparent.","section":"Section 2.3, proof of Theorem 1.2"},{"comment":"The notation g_{s1,...,si}(x1,...,xi) is confusing: the polynomial should depend on the partition block sizes l_1,...,l_i (and on f), not on the summation variables s_1,...,s_i; a notation such as g_{l_1,...,l_i}(x_1,...,x_i) would be clearer.","section":"Section 2.4, Eq. (2.21) and following"},{"comment":"In the proof of part (3), the text reads 'deg R_j(x) ≤ |m| − j'; this should be '|m|_n − j' for consistency with the definition of |m|_n.","section":"Section 2.2, Proposition 2.7(3)"},{"comment":"There are a number of OCR-type typographical issues (for example, 'argu ments' in the abstract and 'Masan obu' in the acknowledgments) that should be corrected in the final version.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a competent extension of the first author's earlier work [11]. The main obstacle to acceptance is the unproved linear-independence assertion in Section 2.1; it is likely true and easily proved, so a revision that adds this proof should be acceptable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a solid, genuine extension of the Li-Qin weighted sum formulas from multiple zeta values to Hoffman's multiple t-values, and the main theorem holds up. The t-case is not a corollary of the zeta case: the generating function F(x)=x/2 - x/(e^x+1) changes the Bernoulli coefficients to beta_{2i}=(2^{2i}-1)B_{2i}, so the coefficient polynomials are different. The paper's contribution is Theorem 1.1 (and the star-value analogue): for any symmetric polynomial f of degree r, the weighted sums T_f(2,k,n) and T*_f(2,k,n) are finite linear combinations of zeta(2l)t(2k-2l) with polynomial coefficients in k of degree at most r+n-2l-1. That is the right shape, matching the zeta-value program.\n\nWhat the paper does well: the proof is coherent and largely self-contained. The matrix identity (2.5)-(2.8) is a clean way to express powers of H(x) in terms of D^mF(x), and Proposition 2.7's evenness argument is the key step. The appendix examples through depth 4 are extensive and internally consistent; they are useful for checking and would help any reader verify the formulas. The use of Hoffman's symmetric sum formulas (2.17)-(2.18) and the reduction from products of t(2k_i) to weighted sums is natural.\n\nSoft spots: they are presentation-level, not mathematical. First, Proposition 2.7(1) cites [11, Lemma 2.6] for the linear independence of 1,F,DF,...,D^mF over Q(x). That citation is likely fine—since F(x)=(x/2)tanh(x/2) has infinitely many poles, it is not D-finite, so the functions are independent—but the paper should either prove it in one sentence or state the lemma explicitly. As written, the load-bearing evenness of the remainders R_j rests on a pointer, and a referee would want that closed. Second, the proof of Theorem 1.2 is compressed to a single line after Theorem 2.9: it is the linearity from monomial weights to arbitrary polynomial f, but the degree-bound bookkeeping is not written out. Again minor. The reliance on Lemma 2.10 from [11] is fine; that is a published parameter-free lemma.\n\nWho this is for: researchers working on multiple zeta and t-values, weighted sum formulas, and Hoffman's odd variant. It is not a breakthrough, but it is a natural completion of a program and the first weighted polynomial result for t-values. I would cite it. It deserves a serious referee; I would send it out. Recommendation: accept with minor revision after the two presentation gaps are fixed.","headline":"A genuine, if specialized, extension of weighted sum formulas to multiple t-values; the main theorem holds up and the flagged issues are presentation-level gaps, not mathematical flaws.","tokens_in":24013,"tokens_out":4195,"would_cite":true,"duration_ms":39070,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M32","11B68"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any symmetric polynomial weight, finite sums of multiple t-values at even arguments equal a short combination of ζ(2l) and t(2k−2l).","keywords":["multiple t-values","multiple t-star values","weighted sum formula","even arguments","Bernoulli numbers","multiple zeta values","symmetric polynomials","zeta values"],"falsifier":"Compute the depth-two identity $\\sum_{k_1+k_2=k,\\,k_1,k_2\\ge1} t(2k_1)t(2k_2)=\\tfrac12(2k-1)t(2k)$ for $k=10$ by summing over odd $m_1>m_2>0$ up to a large cutoff on both sides; if the two sides differ beyond the numerical tail error, Theorem 2.9 is false and Theorem 1.1 falls with it.","tokens_in":22981,"feed_emoji":"","tokens_out":11000,"duration_ms":101468,"temperature":0.7,"pith_summary":"The paper proves that weighted sums of multiple t-values, and their star variants, at even arguments are governed by a single finite shape. For any symmetric polynomial $f\\in\\mathbb{Q}[x_1,\\dots,x_n]$ of degree $r$ and any $k\\ge n$, the weighted sum $T_f(2,k,n)$ (and similarly $T_f^\\star(2,k,n)$) is written as $\\sum_{l=0}^{\\min\\{T,k\\}} c_{f,l}(k)\\zeta(2l)t(2k-2l)$, with $T=\\max\\{[ (r+n-2)/2],[(n-1)/2]\\}$ and $c_{f,l}$ a rational-coefficient polynomial in $k$ of controlled degree. Thus no special constants beyond values of $\\zeta$ at even integers and one $t$-value are needed. This is the multiple-$t$-value analogue of the conjectured and now proved weighted sum formula for multiple zeta values with even arguments, and it makes these sums effectively computable.","feed_headline":"Weighted t-value sums reduce to finite ζ-times-t products","feed_subtitle":"For even arguments, every symmetric polynomial weight cuts the answer to a short list of ζ(2l)t(2k−2l) terms.","key_machinery":"The central object is the even generating function $F(x)=x/2-x/(e^x+1)=\\sum_{i\\ge0}\\beta_{2i}x^{2i}/(2i)!$, where $\\beta_{2i}=(2^{2i}-1)B_{2i}$; it packages the Bernoulli numbers that Euler's formula attaches to $t(2k)$. The operator $D=x\\,d/dx$ produces the expansion $D^mF(x)=\\sum_{i=0}^{m+1}F_{mi}(x)H(x)^i$ with $H(x)=x/(e^x+1)$, and the triangular matrix $A_m(x)$ built from $F_{mi}$ inverts to express each $H(x)^i$ as a combination of $1,F,DF,\\dots,D^mF$. Comparing the even power series on both sides, and using $\\mathbb{Q}(x)$-linear independence of $1,F,DF,\\dots,D^mF$, forces the leftover coefficient functions $R_j(x)$ to be even, which is what restricts the final answer to the $\\zeta(2l)t(2k-2l)$ shape.","core_discovery":"On the paper's own terms: Theorem 1.1 states exactly this reduction for both multiple t-values and multiple t-star values, with the same cutoff $T$ and the degree bound $\\deg c_{f,l}\\le r+n-2l-1$. The route is Theorem 1.2, which drops the symmetry assumption and proves the same finite shape for arbitrary polynomial-weighted products $f(k_1,\\dots,k_n)t(2k_1)\\cdots t(2k_n)$ over all compositions $k_1+\\cdots+k_n=k$. The argument converts each such sum, through Euler's formula for $t(2k)$, into a weighted sum of Bernoulli numbers, evaluates that sum by expanding the generating function under the operator $D=x\\,d/dx$, and then converts back via $\\zeta(2l)$ and a remaining $t(2k-2l)$. The paper thereby extends the even-argument weighted-sum structure from multiple zeta values to the odd-index variant known as multiple t-values.","pith_inferences":["The same generating-function plus matrix-inversion mechanism should transfer to other Bernoulli-type special values whose generating series is even and satisfies a polynomial differential equation; that transfer is not claimed in the paper.","Making the recursion for the inverse matrix $A_m(x)^{-1}$ fully explicit would turn the existence theorem into a closed formula for every coefficient $c_{f,l}(k)$; the paper leaves the coefficients implicit.","The proof's only non-computational input is the linear-independence lemma it cites; a self-contained proof of that lemma for this particular $F$ would make the whole derivation parameter-free.","The evenness of $F$ is essential, so the analogous assertion for odd arguments would likely need a genuinely different mechanism; a natural test is whether any odd-argument analogue holds at all."],"forward_implications":["Every symmetric weighted multiple t-value sum and t-star sum with even arguments is reduced to a finite combination of $\\zeta(2l)$ and one $t(2k-2l)$; no other special constants appear.","The formulas are effective: the recursive matrix inversion gives a concrete procedure that produces explicit polynomial coefficients for any fixed $f,n,k$, as the appendix demonstrates through depth $4$.","The number of zeta/t terms is bounded by $T=\\max\\{[ (r+n-2)/2],[(n-1)/2]\\}$, which is independent of $k$; increasing the weight only raises the degree of the polynomial coefficients, not the length of the formula.","Theorem 1.2 applies to non-symmetric polynomial weights, so the proof also evaluates arbitrary polynomial-weighted products of individual $t(2k_i)$ values, a statement stronger than the symmetric main theorem.","Specializing $f=1$ yields the unweighted sums $T(2,k,n)$ and $T^\\star(2,k,n)$ as particular cases of the same formula."],"supporting_citations":[{"why":"Defines multiple t-values and t-star values, supplies Euler's formula for $t(2k)$, and provides the symmetric sum formulas (2.17)–(2.18) that turn the index-symmetric sums in Theorem 1.1 into weighted products.","marker":"[7]"},{"why":"The analogous weighted sum theorem for multiple zeta values with even arguments; the proof borrows its $\\mathbb{Q}(x)$-linear independence lemma and its polynomial interpolation lemma for inner sums over $f(k_1,\\dots,k_n)$.","marker":"[11]"},{"why":"The earlier conjecture for multiple zeta values predicted exactly the finite $\\zeta(2l)\\zeta(2k-2l)$ shape, which is the form the present paper establishes for multiple t-values.","marker":"[5]"}],"fun_headline_variants":["Even t-sums: polynomial weights cut to ζt terms","Symmetric weights make t-value sums finite ζt products","Weighted t-star evens: only ζ(2l)t(2k-2l) survive","Even arguments: polynomial weights reduce to ζ t terms"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the cited-but-not-proved assertion that $1,F(x),DF(x),\\dots,D^mF(x)$ are linearly independent over the rational function field $\\mathbb{Q}(x)$ for $F(x)=x/2-x/(e^x+1)$; if that independence failed, the proof that each $R_j(x)$ is even would collapse, and with it the restriction of the final formula to the $\\zeta(2l)t(2k-2l)$ shape.","fun_headline_variants_meta":{"raw":{"variants":["Even t-sums: polynomial weights cut to ζt terms","Symmetric weights make t-value sums finite ζt products","Weighted t-star evens: only ζ(2l)t(2k-2l) survive","Even arguments: polynomial weights reduce to ζ t terms"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000611,"raw_usage":{"total_tokens":2758,"prompt_tokens":774,"completion_tokens":1984,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":390,"completion_tokens_details":{"reasoning_tokens":1907}},"tokens_in":390,"tokens_out":1984,"duration_ms":15624,"temperature":1.0,"reasoning_tokens":1907,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:10:43.895644+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the depth-two identity $\\sum_{k_1+k_2=k,\\,k_1,k_2\\ge1} t(2k_1)t(2k_2)=\\tfrac12(2k-1)t(2k)$ for $k=10$ by summing over odd $m_1>m_2>0$ up to a large cutoff on both sides; if the two sides differ beyond the numerical tail error, Theorem 2.9 is false and Theorem 1.1 falls with it.","supporting_citations":[{"cited_title":"Li and C","cited_arxiv_id":null,"evidence_quote":"The analogous weighted sum theorem for multiple zeta values with even arguments; the proof borrows its $\\mathbb{Q}(x)$-linear independence lemma and its polynomial interpolation lemma for inner sums over $f(k_1,\\dots,k_n)$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The earlier conjecture for multiple zeta values predicted exactly the finite $\\zeta(2l)\\zeta(2k-2l)$ shape, which is the form the present paper establishes for multiple t-values."}],"review_version":1}