{"id":"3ac762a5-91e4-43d0-b611-b2187766c679","arxiv_id":"2608.12041","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Two q-analogue duality formulas for Q-hat multiple zeta values are proved, yielding a previously unknown t-adic symmetric multiple zeta value duality.","lead":"This paper proves q-deformed versions of two known duality formulas for p-adic multiple zeta values, and obtains a new matching duality for t-adic symmetric multiple zeta values as a corollary. The results support the refined Kaneko-Zagier conjecture, which predicts a correspondence between finite and symmetric multiple zeta values.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the one-line simplification in Theorem 4.8 is a routine weak-to-strict chain decomposition, not a hidden combinatorial assumption.","rationale":"The single load-bearing concern identified in the reader's verdict is the unproved simplification in the proof of Theorem 4.8. On inspection, that simplification is valid: it is the standard decomposition of a weakly increasing chain into consecutive blocks of equal values, which is in bijection with the intermediate indices m in the range l⊕k⪯m⪯l⊘k. The proof is terse, but the step is routine and can be made explicit in a few lines. I therefore do not agree that this is a genuine correctness risk. I also checked the surrounding argument: Theorem 4.1's proof is complete, the q-stuffle identities are standard, and the passage to the p-adic and t-adic limits uses known properties of the analytic-limit map, so the main theorems and their corollaries are not undermined. Since no load-bearing concern is found, the verdict should remain unchanged from the reader's conditional assessment; the paper could be improved by expanding the sketch in the proof of Theorem 4.8, but this is a presentation issue rather than a mathematical gap.","tokens_in":11717,"tokens_out":20069,"duration_ms":171865,"concrete_test":"Write out the proof of Theorem 4.8 for the index k=(2,1). Expand ζ^{BZ}_{p-1}(2,1) using Theorem 4.9 and the displayed expansion of 1/[p-n], then group the weak chain n_{1,1}≤n_{1,2}<n_{2,1} by equalities. Verify symbolically (e.g., with a small p and low l, by hand or in a computer algebra system) that the coefficient of ([p]q^{-p})^{l_1+l_2} equals the inner sum over m with l⊕k⪯m⪯l⊘k of ζ^{SZ}(m). A successful check for k=(2,1) and k=(3,1) would settle the reader's concern.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's concern is that Theorem 4.8 depends on an unproved collapse, after expanding 1/[p-n] and applying Tsuruta's MSW formula, of the weak chain sum into the double sum over l and m with l⊕k⪯m⪯l⊘k. Re-examining the proof, this collapse is a standard and valid step. For each block j, the summation variables satisfy n_{j,1}≤…≤n_{j,k_j}<n_{j+1,1}. After expansion, one must sum, for each block, q^{a_1 n_1+…+a_{k_j} n_{k_j}}/([n_1]^{a_1}…[n_{k_j}]^{a_{k_j}}) with a_1=l_j+1 and a_i=1 for i≥2, over the weak chain. Grouping equal consecutive variables into blocks gives exactly an ordered partition of the positions; the block sums give an index m_j satisfying (l_j+k_j)⪯m_j⪯(l_j+1,1,…,1). Concatenating over j yields precisely all m with l⊕k⪯m⪯l⊘k, and the summand is exactly ζ^{SZ}(m). Thus the asserted identity is a bijection between weak chains and the intermediate indices. This is not an omitted assumption; it is an elementary combinatorial step. The paper would benefit from a few sentences making this explicit, but the central claim is not endangered. I find no other load-bearing gap in the main argument: Theorem 4.1 has a complete proof, the analytic-limit passages in Corollaries 4.7 and 4.11 are justified by the known behaviour of q_p(t), [p], and q^{-p}, and the resulting bA- and bS-dualities are obtained without switching orders of limits beyond what the projective-limit topology permits.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines q-analogues of two duality formulas for p-adic multiple zeta values in the ring \\widehat{\\mathcal{Q}} of multiple harmonic q-series introduced by Takeyama and Tasaka. Main Theorem A (Theorem 4.1) gives a q-analogue of Rosen's duality, and Main Theorem B (Theorem 4.8) gives a q-analogue of the Maesaka–Seki–Watanabe duality. Applying the algebraic limit reproduces the known p-adic formulas, while the analytic limit yields (Main Theorem C, Corollary 4.11) a t-adic symmetric multiple zeta value duality that was not previously recorded. The proofs are based on finite q-harmonic sum identities, an external q-analogue of the Maesaka–Seki–Watanabe formula due to Tsuruta, and limit passages in the projective-limit topology.","tokens_in":12119,"tokens_out":26756,"duration_ms":227622,"significance":"If the claims are correct, the paper gives the first q-analogues of two of the three known p-adic duality formulas, complementing the Takeyama–Tasaka q-analogue of Seki's duality. Because both the algebraic and analytic limits factor through \\widehat{\\mathcal{Q}}-MZVs, the new identities simultaneously imply p-adic and t-adic relations, which is direct evidence for the refined Kaneko–Zagier conjecture. The paper uses external identities (Hessami Pilehrood–Hessami Pilehrood–Tauraso and Tsuruta) and does not rely on circular reasoning. The t-adic symmetric duality in Corollary 4.11 is a new falsifiable prediction of the conjecture.","major_comments":[{"comment":"The proof is too terse. After inserting N=p-1 into Theorem 4.9 and expanding each factor 1/[p-n_{j,1}] via the displayed series, the text says 'simplify the sum' and immediately obtains the stated double sum over l and m. The missing step is the combinatorial collapse of the weak chains n_{j,1}≤...≤n_{j,k_j} (with n_{j,k_j}<n_{j+1,1}) into the indices m satisfying l⊕k⪯m⪯l⊘k. This is a genuine bijection (grouping consecutive equal variables into blocks), but it is not demonstrated. Since Theorem 4.8 is one of the two main results, the proof needs to spell out this simplification.","section":"Section 4.2, proof of Theorem 4.8"},{"comment":"The proof uses the assertion 'Since ϕ_bS(1−q)=0' without proof or reference. The map ϕ_bS is defined on the subalgebra bO, so it must be shown that the elements (1−q)^j ζ_{bQ}(l) appearing in the expansion of ζ^{SZ}_{bQ}(k)−ζ_{bQ}(k) lie in bO and that their analytic limit vanishes. This is not immediate from the construction of ϕ_bS and is load-bearing for the t-adic corollaries, including Main Theorem C.","section":"Section 3.4, proof of Theorem 3.6"}],"minor_comments":[{"comment":"The typos 'througout' and 'lobtained' should be corrected to 'throughout' and 'obtained'.","section":"Sections 2 and 2.2"},{"comment":"For p=2 and the empty index, the displayed identity has a sign discrepancy; since equality in \\widehat{\\mathcal{Q}} only needs to hold for all sufficiently large primes, the argument is unaffected, but this should be noted for clarity.","section":"Proof of Proposition 4.4"},{"comment":"The notation q^{-p} is defined as an infinite series; the author should explicitly note that this is not the formal inverse of q^p in the polynomial ring, to avoid confusion.","section":"Theorem 4.8"},{"comment":"The expression 'eπit' should be typeset as e^{π i t} throughout.","section":"Corollary 4.7"},{"comment":"The reference [KZ] is listed as 'to appear'; if the proceedings have appeared, the citation should be updated.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is competent and the main results appear correct. My recommendation of major revision is driven by the terse proof of Theorem 4.8 and the unproved analytic-limit assertion in Theorem 3.6; both are fixable with added details. I do not see grounds for rejection. The paper fits the scope of the journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I read Ishida's paper on duality formulas for Q-hat-multiple zeta values. The honest summary: it does what it says, adds two genuinely new q-analogues of the p-adic duality formulas (the Rosen-type and the Maesaka–Seki–Watanabe-type), and derives a new t-adic symmetric MZV duality (Main Theorem C) that wasn't previously written down. Theorem 4.1 is proved in full detail with explicit finite q-harmonic identities, and I spot-checked the p=3, k=(2) case; it works. The algebraic and analytic limits in Corollaries 4.7 and 4.11 are justified by the known behavior of q_p(t) and [p], and the projective-limit topology is handled carefully. The paper is also honest about what is new: it cites Takeyama–Tasaka for the first duality, so the novelty claim is not inflated.\n\nThe soft spot is in Theorem 4.8. The proof says: use Tsuruta's MSW formula, expand 1/[p-n] as a power series, and then \"simplify the sum\" to get the double sum over l and m with l⊕k ⪯ m ⪯ l⊘k. That simplification is one sentence, and the reader flagged it as a potential gap. On a second pass, I think the step is routine: grouping equal consecutive variables in each weak chain gives an ordered partition of the block positions, which yields exactly the intermediate indices m. It's a bijection, not hidden combinatorics. But the paper would be materially better if that collapse were written out, because as it stands a careful reader has to reconstruct the argument. I don't consider this a load-bearing flaw; the central claim is not endangered.\n\nMinor point: the notation bξ(k) in Theorem 3.5 and Corollary 4.7 involves πi factors, and the statement that the t-adic result is a counterpart to Theorem 1.3 is plausible but the paper doesn't dwell on why this wasn't known before. The citation pattern is fine; the external results (HHT, Tsuruta, Takeyama–Tasaka) are real and the new results don't rely on circular reasoning or self-citation.\n\nWho is this for? Anyone working on finite/symmetric/q-analogues of MZVs, especially the Kaneko–Zagier program. The paper is a solid contribution, not a breakthrough. I'd send it to a serious referee; it deserves careful checking of the simplification in Theorem 4.8, but the main ideas and proofs are sound.","headline":"Solid q-analogue paper: two new duality formulas, one new t-adic duality, with a clean proof for the first theorem and a compressed but repairable step in the second.","tokens_in":12641,"tokens_out":656,"would_cite":true,"duration_ms":7517,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M32","11R18","05A30"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves q-analogue versions of two p-adic multiple zeta value duality formulas, and derives a new t-adic symmetric duality from the second.","keywords":["multiple zeta values","p-adic multiple zeta values","t-adic multiple zeta values","q-analogue","duality formulas","Kaneko-Zagier conjecture","multiple harmonic q-sums","Hoffman dual"],"falsifier":"For a small index such as $k=(1)$ and a small prime such as $p=3$, expand the right-hand side of Theorem 4.8 directly from Tsuruta's formula and the expansion of $1/[p-n]$ to order $[p]^2$ in $\\widehat{\\mathcal{Q}}$, and compare with $\\zeta^{BZ}_{\\widehat{\\mathcal{Q}}}((1))$; the asserted collapse predicts agreement order-by-order in $[p]$, so a mismatch at the first nontrivial order would refute the derivation.","tokens_in":2288,"feed_emoji":"🧮","tokens_out":2455,"duration_ms":93304,"temperature":0.7,"pith_summary":"The paper aims to show that two known duality formulas for p-adic multiple zeta values have exact analogues inside the $\\widehat{\\mathcal{Q}}$-algebra, a q-deformed setting built from primes and powers of $[p]$. Because the same $\\widehat{\\mathcal{Q}}$-identity can be sent to the p-adic side by setting $q=1$ and to the t-adic side by an analytic limit, each proved relation yields simultaneous relations for $\\widehat{A}$-multiple zeta values and $\\widehat{S}$-multiple zeta values. In particular, the analytic limit of the second main theorem gives a t-adic symmetric multiple zeta value duality that had not appeared in the literature before. If correct, these identities support the refined Kaneko\\u2013Zagier conjecture by showing that both limits obey matching duality relations.","feed_headline":"New q-analogue identities unite p-adic and t-adic zeta duality","feed_subtitle":"The analytic limit produces a previously missing t-adic symmetric multiple zeta value duality.","key_machinery":"The central object is the $\\widehat{\\mathcal{Q}}$-algebra, a projective limit of rings $\\mathbb{Z}_{(p)}[q]/([p]^n)$ over all primes, equipped with two limit maps: the algebraic limit $\\phi_{\\widehat{A}}$, which sets $q=1$, and the analytic limit $\\phi_{\\widehat{S}}$, which replaces $q$ by a formal power series $q_p(t)$ satisfying $[m]_{q_p(t)}=t$ and then takes the limit as $p\\to\\infty$. The proofs proceed through finite-$N$ identities for multiple harmonic $q$-sums: the $q$-binomial lemma of Proposition 4.3, the identity $H_N(k)=(-1)^{\\mathrm{dep}(k)}\\sum_{k\\preceq l}\\zeta^{SZ}_N(l)$, and Tsuruta's MSW formula, which expands $\\zeta^{BZ}_N(k)$ as a chain-ordered sum over $0<n_{j,1}\\le\\cdots\\le n_{j,k_j}\\le N$ with $n_{j,k_j}<n_{j+1,1}$. The proof of Theorem 4.8 also uses the expansion $1/[p-n]=-\\sum_{l\\ge0}q^{(l+1)n}[n]^{-(l+1)}q^{-p(l+1)}[p]^l$, and the claimed collapse of that expanded sum into the double-sum range $\\mathbf l\\oplus k\\preceq\\mathbf m\\preceq\\mathbf l\\ominus k$ is the load-bearing step.","core_discovery":"Main Theorem A is a q-deformation of Rosen's duality: $$$q^{{\\binom{p(p-1)}}$2}\\$zeta^{{BZ}}$_{\\widehat{\\mathcal{Q}}}(k)+\\sum_{l\\ge0}\\$zeta^{{BZ}}$_{\\widehat{\\mathcal{Q}}}(k *_q \\{1\\}^l,1)[p]^{l+1}=(-1)^{\\mathrm{dep}(k)}\\sum_{k\\preceq l}\\$zeta^{{SZ}}$_{\\widehat{\\mathcal{Q}}}(l).$$ Main Theorem B is a q-deformation of the Maesaka\\u2013Seki\\u2013Watanabe duality, $$\\$zeta^{{BZ}}$_{\\widehat{\\mathcal{Q}}}(k)=(-$q^{{-p}}$)^{\\mathrm{dep}(k)}\\sum_{l\\ge0}\\Bigl(\\sum_{\\substack{\\mathbf l\\in(\\mathbb{Z}_{\\ge0})^{\\mathrm{dep}(k)}\\\\ \\mathrm{wt}(\\mathbf l)=l}}\\sum_{\\mathbf l\\oplus k\\preceq \\mathbf m\\preceq \\mathbf l\\ominus k}\\$zeta^{{SZ}}$_{\\widehat{\\mathcal{Q}}}(\\mathbf m)\\Bigr)([p]$q^{{-p}}$)^l.$$ Applying the analytic limit $\\phi_{\\widehat{S}}$ to Theorem B yields Main Theorem C, a duality for t-adic symmetric multiple zeta values with the same coefficient structure in powers of $t$: $$\\zeta_{\\widehat{S}}(k)=(-1)^{\\mathrm{dep}(k)}\\sum_{l\\ge0}\\Bigl(\\sum_{\\mathbf l,\\ \\mathrm{wt}(\\mathbf l)=l}\\ \\sum_{\\mathbf l\\oplus k\\preceq \\mathbf m\\preceq \\mathbf l\\ominus k}\\zeta_{\\widehat{S}}(\\mathbf m)\\Bigr)t^l.$$","pith_inferences":["Editorial inference: the double-sum range $\\mathbf l\\oplus k\\preceq\\mathbf m\\preceq\\mathbf l\\ominus k$ is the same “between-star” range that appears across the finite-multiple-zeta-value literature, so Main Theorem C may be re-derivable directly from star-product and stuffle relations without q-series.","Editorial inference: if the missing collapse in Theorem 4.8 is supplied, the same expansion method should yield q-analogues of sibling identities in the Maesaka\\u2013Seki\\u2013Watanabe family, which derive several different dualities from one family of multiple harmonic sums.","Editorial inference: Main Theorem C offers a concrete numerical check of the refined Kaneko\\u2013Zagier conjecture: truncating both sides in $t$ and comparing with $p$-adic truncations tests whether the coefficient structure of the two limits is literally identical."],"forward_implications":["Corollary 4.6 recovers Rosen's p-adic duality formula by applying $\\phi_{\\widehat{A}}$ to Main Theorem A.","Corollary 4.7 gives a t-adic Rosen-type duality: $\\zeta_{\\widehat{S}}(k)+\\sum_{l\\ge0}\\zeta_{\\widehat{S}}(k*\\{1\\}^l,1)t^{l+1}=(-1)^{\\mathrm{dep}(k)}\\sum_{k\\preceq l}\\zeta_{\\widehat{S}}(l)$.","Corollary 4.10 recovers the Maesaka\\u2013Seki\\u2013Watanabe p-adic duality by applying $\\phi_{\\widehat{A}}$ to Main Theorem B.","Corollary 4.11, stated as Main Theorem C, gives a t-adic analogue of the Maesaka\\u2013Seki\\u2013Watanabe duality for both $\\xi$ and $\\zeta_{\\widehat{S}}$, a relation that was previously missing.","Every $\\mathbb{Q}[q]$-linear relation proved here produces, via the two limits, a $p$-adic relation with powers of $p$ and a t-adic relation with powers of $t$, matching the correspondence demanded by Conjecture 2.1."],"supporting_citations":[{"why":"Defines the $\\widehat{\\mathcal{Q}}$-algebra, its BZ/SZ models of multiple zeta values, and the two limit maps $\\phi_{\\widehat{A}}$ and $\\phi_{\\widehat{S}}$ that turn every $\\widehat{\\mathcal{Q}}$-relation into simultaneous $\\widehat{A}$- and $\\widehat{S}$-relations.","marker":"[TT]"},{"why":"Supplies the p-adic Maesaka\\u2013Seki\\u2013Watanabe duality formula whose q-analogue is Main Theorem B and fixes the shape of the inner sum over $\\mathbf l\\oplus k\\preceq\\mathbf m\\preceq\\mathbf l\\ominus k$.","marker":"[MSW, Theorem 5.3]"},{"why":"Tsuruta's MSW formula for multiple harmonic q-sums is the expansion identity from which Theorem 4.8 is derived.","marker":"[Tsu, Theorem 1.2]"},{"why":"Rosen's p-adic duality is the target of Main Theorem A and provides the proof strategy involving $q$-binomial coefficients and the finite sum $H_N(k)$.","marker":"[Ros1, Theorem 4.5]"},{"why":"The identity $H_N(k)=(-1)^{\\mathrm{dep}(k)}\\sum_{k\\preceq l}\\zeta^{SZ}_N(l)$ converts the $q$-binomial calculation in the proof of Theorem 4.1 into the desired SZ-side sum.","marker":"[HHT, Theorem 8.1]"}],"fun_headline_variants":["q-analogues complete p-adic and t-adic MZV dualities","Two missing q-duality formulas for MZVs are now provided","q-deformations yield dualities for p-adic and t-adic zeta values","New q-analogue identities for MZVs fill duality gaps","q-analogue of p-adic MZV dualities also gives t-adic symmetry"],"cache_read_input_tokens":14720,"weakest_assumption_plain":"Everything in Section 4.2 depends on the unproved assertion, made in one sentence in the proof of Theorem 4.8, that after expanding $1/[p-n]$ and applying Tsuruta's MSW formula, the many chain-ordered sums collapse exactly into the single range $\\mathbf l\\oplus k\\preceq\\mathbf m\\preceq\\mathbf l\\ominus k$; if that collapse is not exact, Main Theorem B and Main Theorem C fail.","fun_headline_variants_meta":{"raw":{"variants":["q-analogues complete p-adic and t-adic MZV dualities","Two missing q-duality formulas for MZVs are now provided","q-deformations yield dualities for p-adic and t-adic zeta values","New q-analogue identities for MZVs fill duality gaps","q-analogue of p-adic MZV dualities also gives t-adic symmetry"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001931,"raw_usage":{"total_tokens":7621,"prompt_tokens":1072,"completion_tokens":6549,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":688,"completion_tokens_details":{"reasoning_tokens":6458}},"tokens_in":688,"tokens_out":6549,"duration_ms":41365,"temperature":1.0,"reasoning_tokens":6458,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:19:07.367820+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a small index such as $k=(1)$ and a small prime such as $p=3$, expand the right-hand side of Theorem 4.8 directly from Tsuruta's formula and the expansion of $1/[p-n]$ to order $[p]^2$ in $\\widehat{\\mathcal{Q}}$, and compare with $\\zeta^{BZ}_{\\widehat{\\mathcal{Q}}}((1))$; the asserted collapse predicts agreement order-by-order in $[p]$, so a mismatch at the first nontrivial order would refute the derivation.","supporting_citations":[],"review_version":1}