{"id":"961e90c9-a24a-46b6-a03c-545e1fcda31e","arxiv_id":"2505.07221","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every multiple zeta value is an integer linear combination of multiple zeta values whose index entries are all at least 2, via an explicit elementary algorithm.","lead":"This paper proves that every multiple zeta value, a type of infinite sum used in number theory, can be expressed as a whole-number combination of similar sums whose entries are all at least 2. It also gives an explicit step-by-step algorithm for these expansions, replacing heavy machinery and settling a question about integer coefficients.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The transfer from finite ζ^♢-identities to ordinary MZVs rests on an unstated vanishing estimate for the A≠∅ correction terms in Definition 2.3; this is the sole analytic bridge and is deferred to a citation.","rationale":"The reader's weakest_assumption isolates the same step: the passage lim_N ζ^♢_N(k)=ζ(k). I agree this is the most load-bearing point. Everything else in the proof of the main theorem—the difference equation, the D-recurrence, the integer coefficients—is finite and algebraic; if it is correct, it yields an exact identity of finite sums. The only place where those finite sums are connected to actual multiple zeta values is the limit argument in Section 2, and there the manuscript relies on 'the same mechanism' as a lemma in [37] without stating the lemma or verifying its hypotheses for all A. I do not think this is a fatal flaw: the cited mechanism is standard and the correction terms very likely do vanish, with decay at least O(N^{-1}) up to log powers. But because the central theorem is false if the estimate fails, and because the estimate is not proved in the text, the CONDITIONAL verdict is appropriate. I would not change the reader's verdict. The other issues noted by the reader (the compressed proof of Theorem 2.5 and the Theorem Counter macro errors) are real but secondary; they concern exposition and verification burden rather than a specific unproved necessary condition. The concrete test above would settle the limit concern directly.","tokens_in":28760,"tokens_out":21381,"duration_ms":215341,"concrete_test":"Prove or disprove the needed estimate directly: for an arbitrary admissible k=(k_1,...,k_r) and nonempty A⊂[r]^1_k, bound the correction sum C_{N,A}(k)=Σ_{(n_i)∈S_{r,N}(A)} ∏_{i∈A} 1/(N-n_i) ∏_{i∉A} 1/n_i^{k_i} by C_k (log N)^{r-1}/N (or any o(1) bound), using that k_r≥2 and that every i∈A is followed by a chain ending in a factor with exponent ≥2. If the bound holds, the transfer is justified; if a counterexample with C_{N,A}(k) not tending to 0 is found, Theorem 1.7 is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 2.5 proves an identity of finite sums: for every admissible k and every N, ζ^♢_N(k) = Σ_l c_{k;l} ζ_N(l) with l∈I_k^{≥2}. The final Theorem 1.7 is obtained by letting N→∞. The left side is not ζ_N(k); Definition 2.3 adds correction sums over nonempty A⊂[r]^1_k containing factors 1/(N-n_i). The paper asserts (Section 2, after Definition 2.3) that these corrections vanish in the limit 'by the same mechanism as in the proof of [37, Lemma 2.2 (ii)]', but gives no estimate and does not state the lemma or check that its hypotheses cover arbitrary A and the summation regions S_{r,N}(A). This is load-bearing because it is the only analytic step in the proof: all other arguments are finite algebraic manipulations. If for some admissible k the corrections had a nonzero limit, Theorem 2.5 would still hold as an identity of sequences, but its limit would be a nonzero defect ζ(k)-Σ_l c_{k;l}ζ(l), so Theorem 1.7 would fail. The citation makes the estimate plausible, but as written the central claim depends on an unverified black box.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims two main theorems. Theorem 1.7 states that for every weight k and every admissible index k, the multiple zeta value ζ(k) is a Z-linear combination of ζ(l) with l ranging over indices of weight k whose entries are all at least 2, with explicitly given integer coefficients c_{k;l}; equivalently, the Z-module spanned by weight-k MZVs equals the Z-module spanned by the 'no ones' MZVs. The proof introduces modified multiple harmonic sums ζ♢_N(k) (Definition 2.3), which satisfy a restricted harmonic product formula and a discrete iterated integral expression, and then uses a difference calculus (Section 4) to prove that every ζ♢_N(k) is an integer linear combination of ζ♢_N(l) with l∈I_k^{≥2}. Passing to the limit N→∞, after a cited vanishing estimate for the correction terms, yields Theorem 1.7. The paper also proves Theorem 2.6, identifying the kernel of Z♢ with the new relation space Drop1, and Theorem 1.17, showing LinKaw*⊂Drop1, and derives the elementary upper bound dim_Q Z_k ≤ F_{k−1}.","tokens_in":28960,"tokens_out":12035,"duration_ms":125699,"significance":"If the proof is completed, this is a substantial new result: it gives an elementary, explicit, integral expansion of individual multiple zeta values in the 'no ones' basis, without the theory of mixed Tate motives. The ζ♢ construction is an original device that makes a nontrivial part of the MZV relations visible at the level of finite sums, and the identification Ker(Z♢)=Drop1 together with the inclusion of Kawashima's relations is of independent structural interest. The algebraic difference calculus in Section 4 is developed in real detail, the coefficients are produced by a recurrence rather than by numerical fitting, and the examples are consistent with the stated expansion. The main caveat is that the only analytic step, the N→∞ transfer, is not proved in the manuscript and is deferred to a citation; with that estimate supplied, the paper would meet the standards of a strong number theory journal.","major_comments":[{"comment":"The only analytic step in the paper, namely the claim that the nonempty-A correction terms in ζ♢_N(k)−ζ_N(k) tend to 0, is not proved. The sentence 'by the same mechanism as in the proof of [37, Lemma 2.2 (ii)]' does not state that lemma or verify that its hypotheses cover the summation regions S_{r,N}(A) for arbitrary A⊂[r]_k^1 and the exponents k_i. This is not a routine one-line estimate: when A has several elements, the multiple factors 1/(N−n_i) can contribute powers of log N, and the claimed decay must come from the ordering constraints together with the final exponent k_r≥2. Since Theorem 1.7 is obtained by taking N→∞ in Theorem 2.5, any nonzero limiting defect in these corrections would invalidate the main theorem. Please include a self-contained estimate, or state the cited lemma and verify its hypotheses explicitly.","section":"Section 2, after Definition 2.3"},{"comment":"The proof of Theorem 2.5 is compressed into the single sentence 'by induction on the weight of c, it suffices to take the sum from n=1 to N−1 of the difference relations given in Theorem 4.5.' The displayed summation identities are for ordinary multiple harmonic sums ζ_n(k), whereas the terms in Theorem 4.5 involve f_N(c′)=ζ♢_N(...) for indices c′ that may still contain ones; the induction also requires a precise statement of how each term in Theorem 4.5 lowers the weight and how the sums ∑_n n^{−t} f_n(c′) and ∑_n n^{−t}(Δf)_n(c′) are converted into ζ♢-values. Please expand this into a full proof; as written, the central derivation is not verifiable from the manuscript.","section":"Section 4, Proof of Theorem 2.5"},{"comment":"The proof of Theorem 3.7 relies on the identity quoted from [14, Theorem 1.2] without proof. Since this identity is an ingredient in the discrete iterated integral expression used later in Section 4, please either state the result explicitly or give a proof, especially because [14] is a preprint by partially overlapping authorship.","section":"Section 3, proof of Theorem 3.7"}],"minor_comments":[{"comment":"The labels 'Theorem Counter 1.4', 'Theorem Counter 1.5', etc., appear where the intended labels are 'Problem 1.4', 'Theorem 1.5', and so on; please correct this systematic naming error.","section":"Throughout"},{"comment":"The phrasing 'for every k∈I_k^{adm} and l∈I_k^{≥2}, there exists a unique integer c_{k;l}' is awkward; it should say that for each k there exist unique integers c_{k;l} indexed by l∈I_k^{≥2}.","section":"Theorem 2.5"},{"comment":"The notation 'y c1 x c2 ...' in the definition of D is ambiguous; it should be typeset as y^{c_1} x^{c_2} ... or defined explicitly in terms of the generators e_i.","section":"Section 1.2, definition of D"},{"comment":"The equality ζ♢_N(k)=ζ♢♭_N(k) is stated in one line, but it is used in the difference calculus of Section 4; a sentence explaining the second equality and the role of the weak inequalities would improve readability.","section":"Section 3, Corollary 3.10"},{"comment":"The phrase 'by an argument similar to the proof of Theorem 3.3' is vague; please spell out the bijection of summation regions so that the proof can be checked independently.","section":"Section 5.1, Proposition 5.4"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the journal's scope and the main result is potentially important. The principal obstacle is the unproved vanishing estimate in Section 2, which is the only bridge from the finite-sum identity to ordinary MZVs; the two cited black boxes [32] and [37] are by members of the same group, and [37] in particular should either be reproduced or stated precisely. The compressed proof of Theorem 2.5 also needs expansion. With those two points addressed, I would be happy to recommend acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take. The headline: the paper proves a new integrality statement—every MZV is a Z-linear combination of MZVs whose indices have all entries ≥2, with explicit integer coefficients. That's genuinely new; Brown gives rational coefficients and Ecalle's unit-cleansing gives rational coefficients. The ζ^♢-value construction is the right kind of new tool: finite sums that partially satisfy MZV relations, dimension F_{k-1}, and a difference calculus developed carefully in Lemmas 4.1, 4.3, and Theorem 4.5. I checked the examples; they are consistent. The inclusion LinKaw* ⊂ Drop1 is a nice side result.\n\nThe soft spots are real but fixable. The biggest is the limit step. Theorem 2.5 is an identity of finite sums. To get Theorem 1.7 you have to let N→∞, and the corrections with A≠∅ in Definition 2.3 have to vanish. The paper says 'by the same mechanism as in [37, Lemma 2.2 (ii)]' and moves on. That is the only analytic bridge in the proof, and it is load-bearing. I think the estimate is probably true—the last entry is ≥2, which should give enough decay—but the authors need to state the lemma and verify the hypotheses for their summation regions, or prove the estimate directly. A referee should insist on this.\n\nTwo smaller issues. The proof of Theorem 2.5 is compressed to one sentence ('by induction on the weight of c, it suffices to take the sum...'); given that this is the main algorithm, the induction should be spelled out. Also, the paper relies on the authors' own prior work, including a preprint [32], for a key discrete integral expression. That is not disqualifying—Theorem 3.7 is sketched in the paper—but a referee will want to see [32]. Finally, the manuscript is full of 'Theorem Counter' macro errors; that needs a cleanup pass before anything else.\n\nOverall, the mathematics looks sound, and the central claim is likely correct. This paper deserves a serious referee. The reader's conditional verdict seems right to me, and the stress-test note correctly identifies the limit estimate as the point to probe. I would send it to peer review and ask for a revised version that closes that gap and cleans up the text.","headline":"Genuinely new integrality result for MZV expansions, built on a clever finite-sum construction; the main proof has a real gap in the limit step that the authors should close, but the mathematics looks sound and deserves refereeing.","tokens_in":29616,"tokens_out":5270,"would_cite":true,"duration_ms":51362,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M32"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every multiple zeta value is an integer linear combination of zeta values whose entries are all at least 2.","keywords":["multiple zeta values","integer coefficients","indices without ones","modified multiple harmonic sums","zeta-diamond values","Fibonacci dimension","Kawashima relations","Hoffman algebra"],"falsifier":"Compute, for a growing sequence of $N$, the correction sum $\\sum_{0<n_1\\le n_2<N}\\frac{1}{(N-n_1)n_2^2}$ that appears in $\\zeta^\\diamondsuit_N(1,2)-\\zeta_N(1,2)$; if it does not tend to 0, the limit step connecting the finite-sum world to multiple zeta values breaks. Independently, evaluate one of the paper's sample expansions, such as $\\zeta(3,1,4)=\\zeta(5,3)-\\zeta(4,4)-\\zeta(3,3,2)+\\zeta(2,4,2)-2\\zeta(3,2,3)-\\zeta(2,3,3)$, to high precision; any discrepancy would refute Theorem 1.7.","tokens_in":28474,"feed_emoji":"🧮","tokens_out":10334,"duration_ms":94799,"temperature":0.7,"pith_summary":"This paper proves that every multiple zeta value of weight $k$ is a $\\mathbb{Z}$-linear combination of multiple zeta values whose entries are all at least 2, with coefficients that are explicit integers and a recurrence that computes them. The proof goes through a new family of finite sums, the $\\zeta^\\diamondsuit$-values, which are designed so that they converge to ordinary multiple zeta values while already satisfying enough of the usual algebraic relations to force the reduction. The authors determine the full structure of the space generated by these finite sums: it has a basis indexed by entries $\\ge 2$ and dimension equal to the Fibonacci number $F_{k-1}$. As a corollary, the same span equality holds for $t$-interpolated and star variants, and an upper bound $\\dim_{\\mathbb{Q}} Z_k \\le F_{k-1}$ follows without motivic machinery. A second result embeds the extended Kawashima relations $\\mathrm{LinKaw}^*$ into the new relation family $\\mathrm{Drop}_1$.","feed_headline":"Every multiple zeta value becomes an integer combo of 1-free ones","feed_subtitle":"A finite-sum trick yields explicit integer coefficients and avoids motivic machinery.","key_machinery":"The central object is the modified multiple harmonic sum $\\zeta^\\diamondsuit_N(\\mathbf{k})$, a finite sum in which every position where the index has a 1 is allowed a weak inequality and carries a factor $1/(N-n_i)$; its limit as $N\\to\\infty$ is the ordinary multiple zeta value, but unlike plain multiple harmonic sums it obeys enough relations (a restricted harmonic product, a star-value formula, and a discrete iterated integral expression) to make the reduction work. The argument is carried by the linear operator $D$ on the Hoffman algebra, the noncommutative polynomial ring $\\mathbb{Q}\\langle x,y\\rangle$ whose words encode indices, together with the equality $\\zeta^\\diamondsuit(\\{1\\}^{c_1-1},c_2+1,\\dots,\\{1\\}^{c_{2s-1}-1},c_{2s}+1)=Z^\\diamondsuit(D(c_1,\\dots,c_{2s}))$, which produces the integer coefficients and is proved by difference calculus in $N$. The one non-elementary input is a cited vanishing lemma that justifies taking the limit; the injectivity of the untruncated multiple-harmonic-sum map supplies uniqueness of the coefficients.","core_discovery":"On the paper's own terms, the central discovery is Theorem 1.7: for each weight $k\\ge2$ and each admissible index $\\mathbf{k}$, the multiple zeta value $\\zeta(\\mathbf{k})$ equals $\\sum_{\\mathbf{l}\\in I^{\\ge2}_k} c_{\\mathbf{k};\\mathbf{l}}\\,\\zeta(\\mathbf{l})$ with explicitly given integer coefficients $c_{\\mathbf{k};\\mathbf{l}}$, so the $\\mathbb{Z}$-span of all admissible values equals the $\\mathbb{Z}$-span of values with no entry equal to 1. The stronger finite-sum version (Theorem 2.5) asserts the same identity for the modified sums $\\zeta^\\diamondsuit$, where the coefficients are unique integers, and it identifies the space spanned by weight-$k$ $\\zeta^\\diamondsuit$-values with basis $\\{\\zeta^\\diamondsuit(\\mathbf{l})\\mid \\mathbf{l}\\in I^{\\ge2}_k\\}$ of dimension $F_{k-1}$. The proof also determines the kernel of the map from the Hoffman algebra: $\\mathrm{Ker}(Z^\\diamondsuit)=\\mathrm{Drop}_1$, and as a second main result the extended Kawashima relations satisfy $\\mathrm{LinKaw}^*\\subset \\mathrm{Drop}_1$.","pith_inferences":["The same finite-sum technique could plausibly be adapted to $q$-multiple zeta values, where it would either prove or disprove the numerical conjecture that the spaces are spanned by indices with all entries at least 2; this is an editorial inference, not a claim of the paper.","The integrality and algorithmic nature of the coefficients raises the possibility of deriving arithmetic information about classical Hoffman-basis coefficients, whose denominators are conjectured to be odd; this inference goes beyond the paper.","Since the conjectured equality $\\mathrm{LinKaw}^*=\\mathrm{Drop}_1$ was checked numerically only up to weight 17, computing both kernels at weights 18 through 20 would provide a sharper test of whether the two relation families coincide."],"forward_implications":["For each weight $k\\ge2$, every admissible value $\\zeta(\\mathbf{k})$ can be algorithmically rewritten as an integer linear combination of values with no entry equal to 1.","The same span statement holds for $t$-interpolated and star variants (Corollary 1.10).","The dimension of the weight-$k$ space $Z_k$ is at most $F_{k-1}$, obtained by an elementary argument rather than by mixed Tate motives.","The kernel of $Z^\\diamondsuit$ is exactly $\\mathrm{Drop}_1$, and the extended Kawashima relations $\\mathrm{LinKaw}^*$ lie inside this kernel; if the paper's conjecture holds, the two families coincide."],"supporting_citations":[{"why":"Supplies the vanishing lemma used to pass from the finite sums $\\zeta^\\diamondsuit_N$ to ordinary multiple zeta values; this is the paper's only non-elementary input.","marker":"[37]"},{"why":"Gives the discrete iterated integral expression for multiple harmonic sums on which the $\\zeta^\\diamondsuit$ difference calculations rest.","marker":"[32]"},{"why":"Provides the height-one expansion theorem that the recurrence $D$ generalizes to arbitrary indices.","marker":"[26]"},{"why":"Provides the earlier generalized expansion relations that are shown to lie inside the relation family $\\mathrm{Drop}_1$.","marker":"[33]"},{"why":"Defines the Kawashima relations $\\mathrm{LinKaw}$ and establishes them for multiple zeta values; Theorem 1.17 embeds an extension of this family.","marker":"[29]"},{"why":"Supplies the connected-sum method used to prove Kawashima-type relations for $\\zeta^\\diamondsuit$.","marker":"[38]"},{"why":"States the injectivity of the map from words to multiple harmonic sums, which gives linear independence and uniqueness of the integer coefficients.","marker":"[46]"},{"why":"Provides the corollary used in [46] to prove injectivity of the multiple harmonic sums map.","marker":"[4]"}],"fun_headline_variants":["Explicit integer combos: every MZV from 1-free ones","No 1's needed: explicit algorithm for all multiple zeta values","Every multiple zeta value is an integer span of 1-free ones","Cutting the ones: explicit integer sums for all MZVs","All MZVs are finite integer combos of 1-free indices"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof relies on the claim, taken from a cited lemma rather than proved here, that the extra correction terms built into the modified finite sums vanish as the cutoff grows; if that limit failed for some admissible index, the finite-sum identities would not give identities for ordinary multiple zeta values.","fun_headline_variants_meta":{"raw":{"variants":["Explicit integer combos: every MZV from 1-free ones","No 1's needed: explicit algorithm for all multiple zeta values","Every multiple zeta value is an integer span of 1-free ones","Cutting the ones: explicit integer sums for all MZVs","All MZVs are finite integer combos of 1-free indices"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001078,"raw_usage":{"total_tokens":4471,"prompt_tokens":867,"completion_tokens":3604,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":483,"completion_tokens_details":{"reasoning_tokens":3509}},"tokens_in":483,"tokens_out":3604,"duration_ms":25554,"temperature":1.0,"reasoning_tokens":3509,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:23:31.398262+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for a growing sequence of $N$, the correction sum $\\sum_{0<n_1\\le n_2<N}\\frac{1}{(N-n_1)n_2^2}$ that appears in $\\zeta^\\diamondsuit_N(1,2)-\\zeta_N(1,2)$; if it does not tend to 0, the limit step connecting the finite-sum world to multiple zeta values breaks. Independently, evaluate one of the paper's sample expansions, such as $\\zeta(3,1,4)=\\zeta(5,3)-\\zeta(4,4)-\\zeta(3,3,2)+\\zeta(2,4,2)-2\\zeta(3,2,3)-\\zeta(2,3,3)$, to high precision; any discrepancy would refute Theorem 1.7.","supporting_citations":[{"cited_title":"Seki,A proof of the extended double shuffle relation without using integrals, Kyushu J","cited_arxiv_id":null,"evidence_quote":"Supplies the vanishing lemma used to pass from the finite sums $\\zeta^\\diamondsuit_N$ to ordinary multiple zeta values; this is the paper's only non-elementary input."},{"cited_title":"Kaneko, M","cited_arxiv_id":null,"evidence_quote":"Provides the height-one expansion theorem that the recurrence $D$ generalizes to arbitrary indices."},{"cited_title":"Murahara, M","cited_arxiv_id":null,"evidence_quote":"Provides the earlier generalized expansion relations that are shown to lie inside the relation family $\\mathrm{Drop}_1$."},{"cited_title":"Kawashima,A class of relations among multiple zeta values, J","cited_arxiv_id":null,"evidence_quote":"Defines the Kawashima relations $\\mathrm{LinKaw}$ and establishes them for multiple zeta values; Theorem 1.17 embeds an extension of this family."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the connected-sum method used to prove Kawashima-type relations for $\\zeta^\\diamondsuit$."},{"cited_title":"Yamamoto,Explicit evaluation of certain sums of multiple zeta-star values, Funct","cited_arxiv_id":null,"evidence_quote":"States the injectivity of the map from words to multiple harmonic sums, which gives linear independence and uniqueness of the integer coefficients."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the corollary used in [46] to prove injectivity of the multiple harmonic sums map."}],"review_version":1}