{"id":"0d58fd16-209b-433b-a98e-70645fb9b5b0","arxiv_id":"1908.09307","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The interpolated finite multiple zeta and zeta-star values are shown to satisfy cyclic sum, Bowman-Bradley, weighted sum, harmonic, shuffle, duality, and derivation relations.","lead":"This paper defines a polynomial that interpolates between two closely related families of finite multiple zeta values, and proves that several known algebraic relations extend to this interpolated family. The value for a general reader is a unified tool for proving identities about finite multiple zeta and zeta-star values without proving them twice.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The F=S case of the cyclic sum formula (Theorem 1.2) depends on the unpublished Hirose–Sato result or the authors' own preprint [7, Thm 2.4]; the main theorem is not self-contained for S-type.","rationale":"The reader's verdict is CONDITIONAL, and my stress-test does not change it. Theorems 3.1 and 4.1 are reductions to published finite-MZV relations (Saito–Wakabayashi [40] for the base case; duality, sum formula, and Ohno-type relations for the weighted sum), and their inductive/counting arguments appear coherent. The unique load-bearing gap is in Theorem 1.2 for F=S: the proof depends on an unpublished cyclic sum formula for S-MZVs, cited to Hirose–Sato or [7, Theorem 2.4]. Since [7] is a preprint by two of the present authors and is not peer-reviewed, the S-case of the paper's headline cyclic sum theorem is not self-contained. I agree with the reader's identification of this as the weakest assumption. I also note a secondary issue in Section 5.1: the recursive definition of the t-harmonic product *^t on H1_t appears to produce terms starting with x (for example, z1z1 *^t z1 contains (t^2-t)x^2y), so the product as stated is not a binary operation on H1_t and Theorem 5.2 is at least under-specified. This does not affect the proofs of the three main theorems, but it should be corrected in revision. Verdict: UNCHANGED (still CONDITIONAL).","tokens_in":22270,"tokens_out":20903,"duration_ms":186935,"concrete_test":"Independently re-derive the S-cyclic sum identity zeta_S(F^0(C_m(k)))=0 using only published inputs (S-duality [17, Cor. 1.12], sum formula [29, Thm 1.1], Ohno-type relation [37, Thm 1.5]) without citing [7] or Hirose–Sato, and cross-check numerically for all non-constant indices of weight <= 8 in Z/zeta(2)Z. If the re-derivation cannot be completed or the numerical check finds a deviation, the conditional dependence on unpublished work in Theorem 1.2 is confirmed as a real gap.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 1.2 (Section 2), Proposition 2.2 reduces the coefficient of t^m in the t-cyclic sum to F^0(C_m(k)), and the proof then needs zeta_F(F^0(C_m(k)))=0 for 0<=m<=r-1. For F=A, Kawasaki–Oyama [20, Theorem 1.2] is published. For F=S, the citation is 'Hirose–Sato (unpublished)' or [7, Theorem 2.4], where [7] (arXiv:2001.03832) is a preprint by Hirose, Murahara, and Ono, two of the present authors. Proposition 2.2 also uses [7, Lemma 6.3]. Thus the S-case of the paper's headline cyclic sum theorem is not backed by a peer-reviewed or self-contained proof. If [7, Thm 2.4] or the Hirose–Sato result is wrong, Theorem 1.2 fails for F=S while the A-case stands. This unpublished dependence is localized to Theorem 1.2; Theorems 3.1 and 4.1 reduce to published finite-MZV relations.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines, for each index k, a polynomial ζ^t_F(k) in a variable t that interpolates between the finite multiple zeta value (t=0) and the finite multiple zeta-star value (t=1), for both F=A and F=S. The main results are Theorem 1.2 (cyclic sum formula), Theorem 1.3/3.1 (Bowman–Bradley type vanishing), Theorem 1.5/4.1 (weighted sum formula), and a Section 5 collection of harmonic, shuffle, duality, and derivation relations. The proofs expand coefficients in t and reduce them to already known finite-multiple-zeta relations, with some combinatorial counting arguments in Sections 3 and 4.","tokens_in":22528,"tokens_out":5086,"duration_ms":53312,"significance":"Interpolated finite multiple zeta values are a natural finite analogue of Yamamoto's t-MZVs, and the paper shows that several important relation families (cyclic sum, Bowman–Bradley, weighted sum) hold for the interpolating polynomial rather than only at the endpoints t=0 and t=1. Since ζ^0_F is the finite multiple zeta value and ζ^1_F is the finite multiple zeta-star value, each theorem simultaneously carries the corresponding ordinary and star relations. The algebraic setup in Section 5 is clean and likely to be useful, and the proofs are mostly transparent reductions to published results, with explicit combinatorial counting in Lemma 4.5 and Lemma 4.7. The principal caveat is the S-type cyclic-sum dependence discussed in the major comments.","major_comments":[{"comment":"The proof of Theorem 1.2 for F=S is not self-contained. The step ζ_F(F^0(C_m(k)))=0 for S-type finite multiple zeta values is justified only by 'Hirose–Sato (unpublished)' or by [7, Theorem 2.4], where [7] is a preprint of Hirose, Murahara, and Ono, two of the present authors. Because Theorem 1.2 is one of the paper's main theorems, the S-type case rests on an unpublished or non-peer-reviewed source. The A-type case is fine, since it cites the published theorem of Kawasaki–Oyama [20, Theorem 1.2]. Please either include a proof of the S-type cyclic sum identity for F^0(C_m(k)), or restrict the claim for F=S to a conditional statement with a clearly identified published reference once [7] or the Hirose–Sato work is available.","section":"Section 2, Theorem 1.2 (F=S case)"},{"comment":"Even the algebraic reduction in Proposition 2.2 uses [7, Lemma 6.3] in an essential way, as do equations (3), (4), (5), and (6). Since [7] is an unpublished preprint by two of the present authors, the main theorem for F=S depends on it twice: once for the coefficient identity and once for the t=0 cyclic-sum relation. The paper should state and prove the needed lemma, or at least give a complete proof of Proposition 2.2 without citing the preprint. This is a load-bearing issue for the central claim, not merely a citation-format concern.","section":"Section 2, Proposition 2.2"}],"minor_comments":[{"comment":"The reference list contains several unpublished or in-preparation items ([7], [17], [19], [31], [36]); these should be updated before publication, and the phrase 'Hirose–Sato (unpublished)' in Section 2 should be replaced by a stable reference or a proof in the paper.","section":"General"},{"comment":"In the displayed formula for the third sum in equation (8), the words 'remove ci' and 'remove cj' appear in the printed text. This appears to be a typesetting artifact and should be corrected.","section":"Section 3, equation (8)"},{"comment":"The variable i is overloaded: the proof says 'Choose i (1 ≤ i ≤ a_d)' and then uses i both as the position of the dashed line and as the summation index in formula (13). Please rename one of them to avoid confusion.","section":"Section 4, Lemma 4.5"},{"comment":"In the first displayed equation of the proof, 'ζF(H(k,r,n)' is missing a closing parenthesis; it should read 'ζF(H(k,r,n))'.","section":"Section 4, proof of Theorem 4.1"},{"comment":"In the proof of Theorem 5.13, the expression 'S^{-t}(yH_t^x)' should be 'S_{-t}(yH_t^x)' or otherwise disambiguated, since the notation S_t is used elsewhere for the automorphism on H_t.","section":"Section 5.3, Theorem 5.13"}],"recommendation":"major_revision","confidential_remarks":"The main blocking issue is the reliance of the S-type cyclic sum formula on an unpublished result and on the authors' own preprint [7]. This is not a matter of novelty but of verifiability: a reader cannot currently check the central claim for F=S from the published literature or from this paper alone. If the authors can supply a proof of the needed S-type cyclic-sum identity, or if [7] and the Hirose–Sato work appear in completed form, the paper would be suitable for acceptance. The editor may also wish to ask whether the S-type results should be separated into a conditional section."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the quick verdict: this is a genuinely useful paper for the finite multiple zeta value community. It takes Seki's t-interpolation and proves that the cyclic sum, Bowman–Bradley, and weighted sum formulas hold for all t, for both A- and S-types. That is a real extension: at t=0 and t=1 you recover separate known relations for FMZVs and FMZSVs, and the paper's versions genuinely interpolate. The proofs are mostly by reducing coefficient-by-coefficient to known t=0 results, and the combinatorial counts in Lemma 4.5 and Proposition 3.2 are laid out in enough detail for a careful reader to check. I did not find circularity in the sense of assuming the t-statements; the paper is honest about what is new and what is prior.\n\nThe soft spot is real but localized. Theorem 1.2 for F=S uses the cyclic sum formula for S-MZVs, attributed to Hirose–Sato (unpublished) or to [7, Thm 2.4], a preprint by two of the present authors. The A-case is backed by the published Kawasaki–Oyama paper. So the paper's headline theorem is not self-contained for half of its range. That should be fixed: either prove the S-cyclic sum in an appendix or wait for publication of [7]. It is not a fatal flaw, but it is exactly the kind of dependency a referee should press.\n\nTwo smaller issues. Theorem 5.2 (harmonic relation) is asserted \"in a similar way\" to t-MZVs, but no proof is given; if it is a direct check, include it. The symmetric sum and antipode-like relations are also stated without proof, though they appear to be quick. The rest of the paper—shuffle, duality, derivation—is properly proved and well cited.\n\nBottom line: the paper deserves a serious referee. I would condition acceptance on making the S-type dependency public. Once that is done, this is a solid contribution that people working on finite MZVs will want to cite.","headline":"Useful interpolation of finite MZV relations; the headline cyclic sum theorem for S-type depends on an unpublished result, but the rest is solid and the paper deserves review after that dependency is addressed.","tokens_in":23051,"tokens_out":2392,"would_cite":true,"duration_ms":22801,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M32","05A19"],"pacs":[],"model":"deepseek-v4-flash","headline":"A single polynomial parameter $t$ interpolates between finite multiple zeta values and their zeta-star variants, and the paper proves the cyclic-sum, Bowman–Bradley, and weighted-sum relations hold at every $t$.","keywords":["Multiple zeta(-star) values","Interpolated multiple zeta values","Finite multiple zeta(-star) values","Symmetric multiple zeta(-star) values","Cyclic sum formula","Bowman-Bradley type formula","Weighted sum formula","t-interpolated finite multiple zeta values"],"falsifier":"Compute both sides of Theorem 1.2 for the index $\\mathbf{k}=(2,3)$ as polynomials in $t$ in the A-valued setting at a small prime, say $p=7$: reduce the coefficients of $t^0,t^1,t^2$ modulo $7$ and test whether the difference vanishes. A nonzero coefficient at any $t$-degree would refute the claimed family; conversely, matching all coefficients for a handful of small indices and primes would support it.","tokens_in":22079,"feed_emoji":"🔢","tokens_out":14390,"duration_ms":120763,"temperature":0.7,"pith_summary":"The paper defines, for each index $(k_1,\\dots,k_r)$ and each $F=A$ or $S$ (the two standard finite analogues of multiple zeta values), a polynomial $\\zeta^t_F(k_1,\\dots,k_r)$ that interpolates between the finite multiple zeta value at $t=0$ and the finite multiple zeta-star value (the variant allowing equal arguments) at $t=1$. Its central claim is that three major families of relations—the cyclic sum formula, the Bowman–Bradley type formula, and the weighted sum formula—hold for the whole one-parameter family, not just at the two endpoints. A sympathetic reader should care because this turns relations that were proved separately for finite values and finite star values into single polynomial identities, just as already happens for classical multiple zeta values. The proofs are coefficient-wise: each coefficient of $t^n$ is identified with a known $t=0$ relation, so the contribution is a transfer mechanism that carries endpoint identities through the interpolation.","feed_headline":"One parameter t unifies finite multiple zeta and zeta-star relations","feed_subtitle":"A t-parametrized family carries the cyclic-sum, Bowman-Bradley, and weighted-sum relations across both value types.","key_machinery":"The load-bearing object is the $t$-index: for an index $\\mathbf{k}=(k_1,\\dots,k_r)$, write $\\mathbf{k}_t$ for the sum over all ways to replace each $\\square$ in $k_1\\square k_2\\square\\cdots\\square k_r$ by a comma or a plus, weighted by $t^{\\#\\text{plus}}$; applying the finite zeta map $\\zeta^t_F$ to $\\mathbf{k}_t$ gives the interpolated value, and the coefficient of $t^m$ isolates the depth-$m$ contributions. For the cyclic sum proof, the key construction is the cyclic index $C_m(\\mathbf{k})$, obtained by the same summation with $m$ pluses and a cyclic wrap-around that merges the last and first entries; Proposition 2.2 shows the $t^m$-coefficient of the difference between the two sides of the cyclic sum formula equals $\\zeta_F(F^0(C_m(\\mathbf{k})))$, reducing every $t$-degree to the known $t=0$ cyclic sum. For the Bowman–Bradley family the key mechanism is the recurrence of Proposition 3.2, which expresses $(n+1)B^{(n+1)}_{l,m}[a]$, the coefficient of $t^{n+1}$ in the $t$-index of the shuffle sum $B_a$, as a combination of $B^{(n)}$ terms with shifted arguments, so that the $n=0$ theorem for finite values propagates to all $n$. For the weighted sum formula, the auxiliary element $H(k,r,n)=F(k,r,n)+S'(k,r,n)+G'(k,r,n)$ together with the duality map $\\varphi$ plays the central role: Proposition 4.4 shows $H+\\varphi(H)$ is either $0$ or a multiple of $(\\{1\\}^k)$, and both are killed by $\\zeta_F$, using the sum formula and the Ohno-type relation.","core_discovery":"The discovery is that the interpolation parameter $t$ is structurally transparent: every $t^n$-coefficient of the three main identities is forced by the $t=0$ identity applied to a cyclic or shuffle-modified index, so the polynomial family inherits all endpoint relations. Concretely, the paper proves the cyclic sum formula of Theorem 1.2 for indices $(k_1,\\dots,k_r)\\neq(1,\\dots,1)$, the vanishing $\\zeta^t_F(B_a)=0$ of the Bowman–Bradley sums of Theorem 3.1, and the weighted sum formula $\\sum_{k_1+\\cdots+k_r=k}2^{k_r-1}\\zeta^t_F(k_1,\\dots,k_r)=0$ for odd $r$ of Theorem 4.1, for every $t$ and both $F=A,S$. At $t=0$ these statements reduce to the known finite MZV relations; at $t=1$ they become the corresponding zeta-star relations. The paper also shows that the harmonic, shuffle, duality, and derivation relations extend to the $t$-family, so the entire algebraic calculus of finite values can be run with the parameter left free.","pith_inferences":["If the coefficient-transfer mechanism generalizes, the same $t$-interpolation should work for other finite-value families, such as cyclotomic finite multiple zeta values or truncated $t$-adic symmetric values, with the same three theorems holding in identical polynomial shape.","A concrete research task is to give a fully self-contained proof of $\\zeta_S(F^0(C_m(\\mathbf{k})))=0$; this would make Theorem 1.2 independent of the unpublished S-type cyclic sum and might reveal an elementary cyclic-sum identity for S-values.","One could treat $t$ as a deformation parameter and evaluate the proved polynomial identities at special algebraic values of $t$, such as roots of unity, to obtain new explicit zero relations in $\\mathcal{A}$ and $\\mathbb{Z}/\\zeta(2)\\mathbb{Z}$."],"forward_implications":["At $t=0$ and $t=1$, Theorems 1.2, 3.1, and 4.1 recover the known cyclic-sum, Bowman–Bradley, and weighted-sum formulas for both the A- and S-type finite multiple zeta(-star) values, so each theorem is a simultaneous generalization of four endpoint statements.","The coefficient-wise proof gives, for every $n$ between $0$ and the depth minus one, new relations among finite MZVs and MZSVs that have no separate name in the literature; these intermediate $t^n$-relations are new content even when the endpoints were known.","The algebraic relations of Section 5 (harmonic, shuffle, duality, derivation, symmetric sum, antipode-like, and Hoffman relations) hold for the $t$-family, extending the two-variable algebraic setup of classical interpolated values to the finite setting.","Because the identities are polynomial in $t$, any endpoint relation that can be phrased coefficient-wise automatically transports across the interpolation, offering a template for future finite-value relations.","The weighted-sum formula for odd $r$ has no known analogue in the classical $t$-MZV world, so this part of the interpolation is special to the finite setting."],"supporting_citations":[{"why":"Introduces interpolated multiple zeta values and supplies the t-index summation convention that this paper adapts to finite values.","marker":"[46]"},{"why":"Defines the A- and S-finite multiple zeta(-star) values that are the endpoints of the interpolation.","marker":"[19]"},{"why":"Proves the cyclic sum formula for A-type finite multiple zeta values, the $t=0$ base used for $F=A$ in Theorem 1.2.","marker":"[20, Theorem 1.2]"},{"why":"Provides the S-type cyclic sum formula and the cyclic-index lemmas that reduce $t$-coefficients to $F^0(C_m(\\mathbf{k}))$.","marker":"[7, Theorem 2.4]"},{"why":"Proves the Bowman–Bradley type theorem for finite multiple zeta values, the $n=0$ base of the induction in Theorem 3.1.","marker":"[40, Theorem 1.4]"},{"why":"Proves the weighted sum formula for S-type finite values whose coefficients are interpolated in Theorem 4.1.","marker":"[31, Theorem 1.1]"},{"why":"Supplies the sum formula $\\zeta_F(S(k,r))=0$, the duality relation $\\zeta_F(w)=\\zeta_F(\\varphi(w))$, and the vanishing of $\\zeta_F(\\{1\\}^m)$, all used in the proof of Theorem 4.1.","marker":"[12]"},{"why":"Proves the Ohno-type relation $\\zeta_F(G(k,m))=0$ used in the weighted sum proof.","marker":"[37, Theorem 1.5]"},{"why":"First defined the interpolated A-value $\\zeta^t_A(k)$ and proved an interpolated sum formula, the starting point for the interpolation.","marker":"[41]"}],"fun_headline_variants":["One parameter t unifies finite zeta and zeta-star relations","t-interpolated finite MZVs inherit all endpoint relations","Cyclic, Bowman-Bradley, weighted sums hold for every t","A t-family of finite multiple zeta values with full relations calculus"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"For the S-valued interpolation, the proof of the cyclic sum formula imports the starting case $\\zeta_S(F^0(C_m(\\mathbf{k})))=0$ from a result cited as unpublished or as a preprint by the same authors; if that S-type cyclic sum is not accepted, the main theorem for $F=S$ is not self-contained.","fun_headline_variants_meta":{"raw":{"variants":["One parameter t unifies finite zeta and zeta-star relations","t-interpolated finite MZVs inherit all endpoint relations","Cyclic, Bowman-Bradley, weighted sums hold for every t","A t-family of finite multiple zeta values with full relations calculus"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000491,"raw_usage":{"total_tokens":2370,"prompt_tokens":856,"completion_tokens":1514,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":472,"completion_tokens_details":{"reasoning_tokens":1439}},"tokens_in":472,"tokens_out":1514,"duration_ms":12287,"temperature":1.0,"reasoning_tokens":1439,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:16:28.052170+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute both sides of Theorem 1.2 for the index $\\mathbf{k}=(2,3)$ as polynomials in $t$ in the A-valued setting at a small prime, say $p=7$: reduce the coefficients of $t^0,t^1,t^2$ modulo $7$ and test whether the difference vanishes. A nonzero coefficient at any $t$-degree would refute the claimed family; conversely, matching all coefficients for a handful of small indices and primes would support it.","supporting_citations":[{"cited_title":"Yamamoto, Interpolation of multiple zeta and zeta-star values , J","cited_arxiv_id":null,"evidence_quote":"Introduces interpolated multiple zeta values and supplies the t-index summation convention that this paper adapts to finite values."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the sum formula $\\zeta_F(S(k,r))=0$, the duality relation $\\zeta_F(w)=\\zeta_F(\\varphi(w))$, and the vanishing of $\\zeta_F(\\{1\\}^m)$, all used in the proof of Theorem 4.1."},{"cited_title":"Seki, Finite multiple polylogarithms , Doctoral Thesis (Osaka university knowledge archive)","cited_arxiv_id":null,"evidence_quote":"First defined the interpolated A-value $\\zeta^t_A(k)$ and proved an interpolated sum formula, the starting point for the interpolation."}],"review_version":1}