{"id":"6c3e1bc7-07ab-4eaa-b84a-58015ef3f159","arxiv_id":"1908.05061","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Generalized Jacobi-Trudi determinants for regularized Schur multiple zeta values show that alternating 1-3 checkerboard Schur multiple zeta values are polynomials in Riemann zeta values.","lead":"This mathematics paper develops new determinant formulas for Schur multiple zeta values, which are sums that generalize multiple zeta values. It uses the formulas to prove that checkerboard-pattern sums with entries 1 and 3 can always be written as polynomials of single zeta values.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 4.4 asserts without proof that subribbons of an F-type stair are also F-type; Proposition 4.10 shows subribbons can be S/S* for an A-type ribbon, so the purity conclusions in Corollaries 4.5 and 4.8 rest on an unverified combinatorial closure property.","rationale":"The reader's weakest assumption identifies exactly the same unproved claim in Proposition 4.4 that I find most load-bearing. The main theorem of the paper, Theorem 4.3, does not depend on this claim: its proof uses a column ribbon and the four column families, which are handled by Proposition 4.2 and known results. However, the paper's advertised 'conditions on the shape' for purity in odd or even Riemann zeta values (Corollaries 4.5 and 4.8) rest directly on Proposition 4.4. The nearby Proposition 4.10 illustrates that subribbons of a single F-type ribbon are not always F-type, so the assertion in Proposition 4.4 is not a trivial consequence of the definition; it is a real combinatorial closure property that needs proof. The correct verdict is therefore CONDITIONAL: the main determinant formula and the broad 1-3 polynomial result appear sound, but the purity conditions are not fully justified. Since the reader already assigned CONDITIONAL with MODERATE confidence, my stress-test does not change that verdict. The proposed concrete test would settle whether the gap is merely expository or a genuine mathematical failure; until such a check is run, the conditional status is appropriate.","tokens_in":15662,"tokens_out":17344,"duration_ms":164323,"concrete_test":"Implement the greedy outside-decomposition construction of Corollary 2.10 in a short script for each of the four F-types and for every skew shape of up to 8 boxes that satisfies the respective pure-F tessellation condition from Corollary 4.5 and Example 4.6; verify that every subribbon R_Theta(i,j) is again an F-type stair. If any counterexample appears, recompute the determinant in Corollary 3.4 for that shape numerically: if a non-F entry contributes a zeta value outside Q[F(n)], then the purity claim for that shape is false. If all small cases pass, the concern reduces to a missing proof rather than a counterexample.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central determinant formula Theorem 3.3 is plausible and well-supported by the Hamel-Goulden framework. The load-bearing gap is in Proposition 4.4, where the proof states: 'The construction of the outside decomposition Theta in the proof of Corollary 2.10 then assures that all the subribbons R_Theta(i,j) are also F-type stairs.' No proof of this closure property is given. The determinant in Corollary 3.4 then has entries claimed to be F_{a,b}(n), yielding zeta_reg(k) in Q[F_{a,b}(n)] and hence the odd/even purity conditions of Corollaries 4.5 and 4.8. But Proposition 4.10 itself shows that with an A-type ribbon R, the subribbons R_Theta(i,j) can include S-type and S*-type stairs: the determinant displayed there is [[A, S], [S*, B]]. This does not directly refute the pure-F case, but it demonstrates that the type of a subribbon depends on the cut points and is not an automatic property of an F-type ribbon. If a shape tessellated purely by F-type stairs produced even one non-F subribbon, the determinant would contain a different zeta value, and the claimed purity conclusion for that shape would fail. The abstract's secondary claim about conditions for purely odd or purely even Riemann zeta values depends entirely on this unproved combinatorial assertion.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a generalized Jacobi-Trudi determinant formula for regularized Schur multiple zeta values. Theorem 2.8 establishes a determinant identity for a general weighted sum S^f_M over semi-standard Young tableaux with an outside decomposition, via a Gessel-Viennot path argument in the style of Hamel-Goulden; Theorem 3.3 specializes this to regularized Schur multiple zeta values zeta_reg. The applications target checkerboard-style tableaux with alternating entries: Theorem 4.3 shows that every 1-3 checkerboard Schur multiple zeta value lies in Q[pi^4, zeta(3), zeta(5), ...][T], Proposition 4.4 and Corollaries 4.5 and 4.8 give purity conditions, and Proposition 4.10 evaluates a gluing product. The determinant identity is the central tool; the purity results depend on Corollary 3.4 and on the claim in Proposition 4.4 that subribbons of an F-type stair are again F-type.","tokens_in":15838,"tokens_out":11712,"duration_ms":117723,"significance":"The determinant theorem is a genuine and useful extension of the Jacobi-Trudi formulae in [NPY], and the regularization argument via [IKZ] is standard and sound. The paper also gives explicit evaluations, including Bernoulli-polynomial expressions, and answers questions raised in [BY]. The Gessel-Viennot proof of Theorem 2.8 is plausible, and Theorem 3.3 is well supported. However, the abstract's claims about purely odd or purely even zeta values rest on Proposition 4.4, whose key closure assertion is unproved. The significance is therefore conditional; with a proof of that combinatorial assertion, the paper would be a solid contribution to the subject.","major_comments":[{"comment":"Proposition 4.4 is the load-bearing step for the purity statements in Corollaries 4.5 and 4.8, but its proof contains the unsupported sentence: 'The construction of the outside decomposition Theta in the proof of Corollary 2.10 then assures that all the subribbons R_Theta(i,j) are also F-type stairs.' No argument is given that the cut points produced by the construction preserve the stair type. This is not a formal consequence of the determinant theorem; Proposition 4.10 shows that subribbons of a ribbon of one type can be of other types when the outside decomposition contains a different stair type. A proof is needed that a pure F-tessellation forces all theta_i and hence every R_Theta(i,j) to be F-type. If this fails, the determinant in Corollary 3.4 would contain entries outside Q[F_{a,b}(n)] and the conclusions of Corollaries 4.5 and 4.8 would be false.","section":"§4, Proposition 4.4"},{"comment":"The final step of Lemma 4.7 says only that the desired identity 'follows directly by the harmonic product formula ... since terms in the second summation telescope.' Since Lemma 4.7 is used to prove Corollary 4.8(iii), the telescoping should be displayed or replaced by a short induction; as written, the proof is not checkable from the text.","section":"§4, Lemma 4.7"}],"minor_comments":[{"comment":"The phrase 'tessellated purely by F-type stairs' is used without a formal definition; a precise definition, for example in terms of an outside decomposition all of whose parts are translates of a fixed stair shape, would make the statement and the proof checkable.","section":"§4, Proposition 4.4 and Corollaries 4.5/4.8"},{"comment":"The symbol zeta*({1,3}^n) is not defined; if it denotes the zeta-star value or a star-regularized value, please clarify.","section":"§4, Proposition 4.2"},{"comment":"The path construction is described briefly; adding a precise statement of the bijection between path systems and semi-standard Young tableaux, or a small weighted example, would help the reader verify the modified edge weight f(j, i-1).","section":"§2, proof of Theorem 2.8"},{"comment":"The displayed definition of G_{1,3}(n) contains stray TeX artifacts such as 'bracehtipdownleft' and 'bracehtipupright'; the formula should be typeset cleanly.","section":"§4, definition of G_{1,3}(n)"},{"comment":"The proof states 'One can check directly' for the generating-function identity involving Bernoulli polynomials; a few intermediate manipulations would improve verifiability.","section":"§4, Lemma 4.9"}],"recommendation":"major_revision","confidential_remarks":"The missing proof of Proposition 4.4 is the main obstacle. If the authors can supply a proof, or a precise counterexample, the paper's central contribution is sound. The paper's heavy use of [BY], coauthored by the first author, is legitimate here because the determinant theorem is new and the cited evaluations have their own proofs; I do not see a circularity issue."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The generalized Jacobi-Trudi determinant (Theorem 3.3) is the real contribution, and it's solid. The paper adapts Hamel-Goulden's outside-decomposition framework to the functions S^f_M, with edge weights that depend on content, then passes to regularized Schur multiple zeta values via the IKZ asymptotics. This genuinely extends the row and column formulae in NPY, and the proof is a standard Gessel-Viennot variation. I didn't find circularity here. Corollary 3.4, which says any such value can be written as a polynomial in subribbon values for any chosen ribbon R, is a nice and useful tool.\n\nThe application to 1-3 checkerboards is mostly there. Theorem 4.3, that all regularized 1-3 checkerboards lie in Q[pi^4, zeta(3), zeta(5), ...][T], follows by choosing a long column and using the prior BY evaluations. That logic is sound and not circular. Proposition 4.10's gluing formula is also a natural consequence of the determinant, and Lemma 4.9's Bernoulli evaluation can be checked directly.\n\nNow the soft spot, and I think the stress-test concern lands. Proposition 4.4 asserts that if a checkerboard shape is tessellated purely by F-type stairs and you choose R to be that same F-type stair, then every subribbon R_Theta(i,j) from the Corollary 2.10 outside decomposition is also an F-type stair. The proof says 'the construction ... assures' but gives no argument. This is load-bearing: Corollaries 4.5 and 4.8 claim odd/even purity precisely because every determinant entry is F_{1,3}(n) or F_{1,2}(n). Proposition 4.10 is a useful counterpoint: with an A-type ribbon, the 2x2 determinant has off-diagonal entries S and S*, so subribbon type is not automatic from the ribbon being of one type. That doesn't refute the pure-F case, but it shows the claim needs proof rather than assertion. If even one subribbon came out as a different stair type, the determinant would contain other zeta values and the purity conclusion would fail.\n\nThis is a genuine gap, but it is fixable. The rest of the paper holds together. Minor issues: Lemma 4.7's telescoping is terse, and the proof of Theorem 2.8 is compressed, but neither is damaging.\n\nWho is this for? People working on Schur multiple zeta values and symmetric-function determinants. The determinant formula deserves to be cited on its own. I would send this to a serious referee, with the instruction to push for a proof or a repaired statement of Proposition 4.4. A moderately revised version should be publishable.","headline":"Solid determinant generalization; checkerboard purity claims rest on an unproved subribbon-closure assertion that needs a fix.","tokens_in":16494,"tokens_out":2278,"would_cite":true,"duration_ms":22301,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M32","05E05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves a generalized Jacobi-Trudi determinant identity for regularized Schur multiple zeta values, and derives that every checkerboard-style value with alternating entries 1 and 3 is a polynomial in Riemann zeta values.","keywords":["Schur multiple zeta values","Jacobi-Trudi determinants","multiple zeta values","regularization","checkerboard tableaux","ribbon decompositions","outside decompositions","Riemann zeta values"],"falsifier":"Construct the outside decomposition of Corollary 2.10 for an admissible 1-3 checkerboard tableau satisfying a pure-tessellation condition, with the stair as the chosen ribbon, and inspect every sub-ribbon: if any sub-ribbon is not of the same stair type, the purity claim fails. A high-precision numerical evaluation of such a tableau that lands outside the claimed polynomial ring would also refute the result.","tokens_in":15332,"feed_emoji":"🧮","tokens_out":17546,"duration_ms":148562,"temperature":0.7,"pith_summary":"This paper establishes a determinant identity for regularized Schur multiple zeta values: for a skew Young diagram with constant diagonal entries, the value equals a determinant of regularized values attached to sub-ribbons of a fixed ribbon (Theorem 3.3). This generalizes the classical Jacobi-Trudi formulae, which are recovered when the fixed ribbon is a row or a column. The identity is then used to prove that every checkerboard-style Schur multiple zeta value with alternating entries 1 and 3 is a polynomial in $\\pi^4$ and odd Riemann zeta values (Theorem 4.3). The same determinant yields shape conditions under which such values are pure polynomials in $\\pi^4$ only, or in odd zeta values only, and analogous conditions for alternating entries 1 and 2 (Corollaries 4.5 and 4.8). A reader should care because the paper turns an infinite family of complicated nested sums into a determinant whose entries are known constants.","feed_headline":"One determinant evaluates all 1-3 checkerboard zeta values","feed_subtitle":"Every Schur multiple zeta value with alternating 1s and 3s is a polynomial in ordinary zeta values.","key_machinery":"The load-bearing mechanism is an outside decomposition of a skew diagram into ribbons: a way of cutting the diagram into ribbon-shaped pieces, each starting on the left or bottom border and ending on the right or top border. From such a decomposition one forms a containing ribbon $R_\\Theta$ and its sub-ribbons $R_\\Theta(i,j)$ defined by intervals of diagonal content; the generalized sum $S_M^f(k)$ over semi-standard tableaux with weight $f(m,d)$ specializes to truncated Schur multiple zeta values via $f(m,d)=m^{-d}$. Theorem 2.8 proves the determinant identity for $S_M^f$ by a lattice-path argument whose step weights depend on horizontal position, and Lemma 3.1 transfers the identity to the regularized setting, producing Theorem 3.3. Every later checkerboard result is a deduction from this single determinant identity.","core_discovery":"The paper's central claim is that regularized Schur multiple zeta values satisfy a generalized Jacobi-Trudi determinant formula (Theorem 3.3): $$\\zeta_{\\mathrm{reg}}(k) = \\det\\!\\left(\\zeta_{\\mathrm{reg}}(R_{\\Theta}^{k}(i,j))\\right)_{1\\le i,j\\le n}$$ for an edge-connected skew diagram $\\lambda/\\mu$ with outside decomposition $\\Theta=(\\theta_1,\\dots,\\theta_n)$ and a Young tableau $k$ with constant diagonal entries; the matrix entry is $0$ when the sub-ribbon $R_{\\Theta}(i,j)$ is undefined. Taking the fixed ribbon to be a row or a column recovers the earlier Jacobi-Trudi formulae for Schur multiple zeta values. Applied to checkerboard-style tableaux with alternating 1 and 3 entries, the identity proves that every such regularized value lies in $\\mathbb{Q}[\\pi^4,\\zeta(3),\\zeta(5),\\dots][T]$ (Theorem 4.3). It further shows that when the shape is tessellated purely by one stair type, the value is pure: $\\mathbb{Q}[\\pi^4]$ for S- and $S^\\star$-type stairs, $\\mathbb{Q}[\\zeta(4n+1)\\mid n\\ge1]$ for A-type stairs, and $\\mathbb{Q}[\\zeta(4n+3)\\mid n\\ge0]$ for B-type stairs (Corollary 4.5), with parallel purity conditions for alternating 1 and 2 entries (Corollary 4.8).","pith_inferences":["Editorial extension: the same content-dependent lattice-path argument should yield determinant identities for any family of tableaux sums whose weights factor by diagonal content, potentially giving evaluations for entry patterns beyond the one-three and one-two checkerboards.","Editorial extension: the purity pattern suggests the general principle that if all building-block stair values of a tiling lie in some subring, then every shape tessellated by those stairs lies in that subring; the testable combinatorial core is the claim that all sub-ribbons in the outside decomposition inherit the stair type.","Editorial extension: the interpolation remark in the paper points to a concrete experiment—add a $t$-deformation to the weights and check whether the determinant identity survives; if it does, the checkerboard evaluations should interpolate between the regularized and shuffle-regularized families."],"forward_implications":["Every checkerboard-style Schur multiple zeta value with alternating entries 1 and 3 can be written as a polynomial in $\\pi^4$ and odd Riemann zeta values, so this whole family of infinite sums is reducible to classical constants.","Any such tableau tessellated purely by S- or $S^\\star$-type stairs evaluates into $\\mathbb{Q}[\\pi^4]$; for A-type stairs it evaluates into $\\mathbb{Q}[\\zeta(4n+1)\\mid n\\ge1]$, and for B-type stairs into $\\mathbb{Q}[\\zeta(4n+3)\\mid n\\ge0]$.","The same determinant mechanism yields purity conditions for alternating entries 1 and 2: S-type stairs give $\\mathbb{Q}[\\zeta(3n)\\mid n\\ge1]$, $S^\\star$-type stairs give $\\mathbb{Q}[\\zeta(3n)\\mid n\\text{ odd}]$, and A-type stairs give $\\mathbb{Q}[\\zeta(3n+1)\\mid n\\ge1]$.","The explicit determinant evaluation of a 3-by-3 square in the paper shows that the formula is a practical route from a shape to a closed polynomial expression.","The observed product-minus-gluing phenomenon in checkerboard work becomes a direct determinant consequence: $B_{1,3}(n-1)A_{1,3}(n)-G_{1,3}(n)$ is a rational multiple of $\\pi^{8n}$ with an explicit coefficient."],"supporting_citations":[{"why":"Provides the outside-decomposition determinant formula for Schur functions that Theorem 2.8 adapts by making lattice-path weights depend on content.","marker":"[HG]"},{"why":"Defines Schur multiple zeta values and proves the row and column Jacobi-Trudi cases that Theorem 3.3 generalizes.","marker":"[NPY]"},{"why":"Introduces the checkerboard-style Schur multiple zeta values, supplies the basic A, B, S, S-star stair evaluations, and records the questions answered in Section 4.","marker":"[BY]"},{"why":"Supplies the harmonic regularization of multiple zeta values used to define regularized Schur multiple zeta values and to pass from truncated to regularized determinants.","marker":"[IKZ]"},{"why":"Provides the nonintersecting lattice-path determinant principle used in the proof of the generalized identity.","marker":"[GV]"},{"why":"Gives the generating-function evaluations for the 4-block zeta values used to express the S and S-star stair values.","marker":"[HI]"}],"fun_headline_variants":["Generalized Jacobi-Trudi solves Schur zeta checkerboard cases","Determinant formula proves all 1-3 zeta values are zeta polynomials","Checkerboard stairs reveal odd/even purity in Schur zeta values","One elegant determinant resolves all alternating 1-3 Schur zetas"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes, without a separate argument, that when a shape is tiled by one kind of stair, the determinant construction produces only sub-stairs of that same kind; if any other kind appeared, the determinant would introduce extra zeta values and the clean purity conclusions would fail.","fun_headline_variants_meta":{"raw":{"variants":["Generalized Jacobi-Trudi solves Schur zeta checkerboard cases","Determinant formula proves all 1-3 zeta values are zeta polynomials","Checkerboard stairs reveal odd/even purity in Schur zeta values","One elegant determinant resolves all alternating 1-3 Schur zetas"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000716,"raw_usage":{"total_tokens":3225,"prompt_tokens":959,"completion_tokens":2266,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":575,"completion_tokens_details":{"reasoning_tokens":2184}},"tokens_in":575,"tokens_out":2266,"duration_ms":16839,"temperature":1.0,"reasoning_tokens":2184,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:27:21.742284+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct the outside decomposition of Corollary 2.10 for an admissible 1-3 checkerboard tableau satisfying a pure-tessellation condition, with the stair as the chosen ribbon, and inspect every sub-ribbon: if any sub-ribbon is not of the same stair type, the purity claim fails. A high-precision numerical evaluation of such a tableau that lands outside the claimed polynomial ring would also refute the result.","supporting_citations":[],"review_version":1}