{"id":"51e74936-608b-4665-a5b9-47f893982acd","arxiv_id":"1908.04770","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The weight-5 Nielsen polylogarithm S3,2 satisfies the dilogarithm five-term relation modulo Li5 and products, with new explicit identities in weights up to 8.","lead":"This paper proves new identities for special higher-order logarithm functions called Nielsen polylogarithms. It shows that one of them obeys the famous five-term dilogarithm relation once simpler terms are removed, confirming part of a structural prediction and providing explicit formulas up to weight 8.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Analytic conclusions depend on the unpublished cleaning-lift theorem [9]; if that theorem fails, Corollary 15 and all derived special-value identities do not follow from the symbol-level identity (10).","rationale":"The reader's weakest assumption identifies exactly the same load-bearing point: the analytic lifting theorem from [9] is the bridge from the rigorously proved symbol-level identity (10) to the clean single-valued and special-value statements. I agree that Theorem 14 itself is largely self-contained and credible: the proof is an explicit computation in an S5-representation, with all coefficients and projections displayed, and it does not depend on [9]. The numerical checks to 10,000 or 20,000 digits for the auxiliary weight-6 and weight-8 reductions are real but do not by themselves prove those reductions, and they are clearly marked as conjectural. The clean single-valued special values, by contrast, are presented as consequences of the lifting theorem, so their status depends on an unpublished tool. The concern is therefore not about internal inconsistency of the algebraic argument but about the completeness of the analytic conclusions promised in the abstract. For this reason the verdict should remain CONDITIONAL, not be strengthened to ACCEPT or weakened to REJECT. A direct verification of Corollary 15 by differentiation would settle whether the specific lift used here is valid, independent of the general unpublished theorem.","tokens_in":46308,"tokens_out":12920,"duration_ms":139127,"concrete_test":"Prove Corollary 15 directly without invoking [9]: form F(x1,...,x5) = Alt5(11 S^{/A1}_{3,2}(cr(x1,x2,x3,x4)) + L^{/A1}_5(15[r1]-9[r2]+[r3])), compute its total differential using Proposition 1 and the explicit cleaning formula (7), and verify symbolically that it vanishes identically; then evaluate F at one generic point, e.g. (x1,x2,x3,x4,x5) = (0,1,2,3,4) with branch choices avoiding singularities, to confirm the constant is 0. Alternatively, the authors could release [9] or provide a self-contained proof of the lifting theorem restricted to the S3,2/Li5 case used here.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central algebraic result, Theorem 14, establishes identity (10) between mod-products symbols. The paper's advertised functional equations for Nielsen polylogarithms, however, require the additional step, made in Section 4.1, that clean single-valued functions automatically lift a mod-products symbol identity to an analytic identity up to a constant. This is the main theorem of the unpublished reference [9], and the present paper supplies neither a proof nor a detailed statement of its hypotheses. Corollary 15, the clean single-valued five-term identity, is obtained by applying this lifting theorem to (10) and then setting the constant to zero by antisymmetry. All subsequent clean single-valued evaluations and ladders, including the golden-ratio, 1/3, and sqrt(2)-1 cases, inherit this dependence. If [9] is incomplete, if its hypotheses are not met by the specific functions S^{/A1}_{3,2} and L^{/A1}_5, or if the 2-torsion sign convention used in the symbol calculus is incompatible with the analytic lift, then these analytic statements do not follow from Theorem 14, even though the symbol-level identity (10) would remain correct. Remark 16 states that an analytic version of Corollary 15 can be given, but no proof or explicit statement is included. This is a genuine gap in the presented chain from algebraic identity to analytic functional equation, not a contradiction in the theorem itself.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies Nielsen polylogarithms S_{n,p}. Its main algebraic result, Theorem 14, gives an explicit mod-products symbol identity showing that S_{3,2} evaluated on the dilogarithm five-term relation reduces to a specific combination of Li_5 terms, thereby corroborating a depth-reduction prediction of Goncharov. The paper also establishes a basis for Nielsen symbols in each weight (Theorem 7), derives several depth reductions and functional equations in weights 6–8, including ladders and special values, and proves a general family of depth reductions (Theorem 40). Many of the analytic and clean single-valued statements are obtained by applying a lifting theorem from the unpublished reference [9], and several special-value identities are marked as numerical ('?') only.","tokens_in":46540,"tokens_out":5204,"duration_ms":57275,"significance":"If the algebraic result Theorem 14 is taken on its own, it is a substantial and largely self-contained computation with explicit, checkable steps, and it provides strong evidence for the expected depth reduction of S_{3,2} modulo Li_5. The representation-theoretic S5 arguments and the explicit form of the Li_5 terms are valuable. However, the abstract advertises analytic functional equations for Nielsen polylogarithms, and those analytic statements currently depend on an unpublished lifting theorem that is neither stated nor proved in the manuscript. The numerical identities are honestly marked, but they are not proven. Consequently, the central algebraic contribution is sound while the advertised analytic consequences are conditional on external material.","major_comments":[{"comment":"The paper's advertised analytic functional equations rely on the assertion that clean single-valued functions automatically lift any mod-products symbol identity to an analytic identity, citing 'the main result in [9]'. Since [9] is unpublished and no statement of hypotheses or proof is given, the analytic versions of Theorem 14 (Corollary 15 and all results derived from it, including the S_{3,2} evaluations and ladders in Section 6.4) are not established within this manuscript. This is a load-bearing gap between the symbol-level Theorem 14 and the abstract's claim of deriving new functional equations for Nielsen polylogarithms. The authors should either include a self-contained statement and proof of the lifting theorem, or explicitly reformulate the main claims as symbol-level and mark the analytic lift as conditional on [9].","section":"Section 4.1, Corollary 15"},{"comment":"The symbol calculus in Section 2.2 deliberately ignores signs by working modulo 2-torsion, and this convention is used repeatedly in the proof of Theorem 14 (e.g., 'we can ignore signs in the tensor factors'). It is not automatic that a mod-products identity obtained in this quotient lifts to an analytic identity for the clean single-valued functions, whose values are not taken modulo 2-torsion. Please verify explicitly that the sign choices in (10) are compatible with the lifting theorem, and that the antisymmetry argument in the proof of Corollary 15, which sets the constant to zero, does not depend on a sign ambiguity introduced by the quotient.","section":"Section 2.2 and Section 4.1"},{"comment":"Remark 16 states that 'with some work, one can in fact give an analytic version of Corollary 15' but provides no statement, proof, or reference. Since the analytic version is precisely what the abstract promises and is needed to make the paper's headline results unconditional, this remark should either be expanded into a full argument or removed, with the paper's claims adjusted accordingly.","section":"Remark 16"}],"minor_comments":[{"comment":"The phrase 'viewed modulo Li5 and products' should specify that the reduction is at the level of mod-products symbols, to match the precise statement of Theorem 14.","section":"Abstract"},{"comment":"The proof invokes 'standard evaluations' for the determinants of A'_M, B'_M, and C'_M; please include a reference or a short derivation so the basis claim is fully self-contained.","section":"Section 3.3, Theorem 7"},{"comment":"Several identities are explicitly marked with '?' and are verified only numerically to high precision. These should be placed in a clearly labeled 'numerical evidence' or 'conjectures' section, rather than being presented as propositions or results in the main text, to avoid confusion with the proven statements.","section":"Equations (19), (20), Proposition 34, and related items"},{"comment":"There are numerous typographical and formatting issues in the arXiv text, such as irregular spacing in the title and an extra closing parenthesis in the heading of Appendix B. A careful proofreading pass is needed.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper's algebraic core is solid and worthy of publication, but the advertised analytic functional equations are currently conditional on unpublished work [9]. I recommend requiring the authors to either provide a full statement (and ideally a proof) of the cleaning/single-valued lifting theorem in an appendix, or to clearly restrict the paper's central claims to the mod-products symbol level. The numerical identities should be moved to a separate conjecture section, as they are not yet proven."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The real news here is Theorem 14: a purely algebraic proof that the weight-5 Nielsen polylogarithm S3,2, evaluated on the dilogarithm five-term relation, reduces modulo products to an explicit combination of Li5 terms. That is a new, concrete corroboration of Goncharov's depth-reduction prediction, and the proof is genuine and self-contained at the symbol level. The explicit ratios r1, r2, r3 are new, and the representation-theoretic computation inside S5 is clean and convincing. The same framework also gives a nice general family in Theorem 40. No fitted parameters, no circularity: the argument really does what it says.\n\nThe soft spot is exactly where the reader's stress-test lands. The advertised analytic five-term identity (Corollary 15) and the special values that flow from it depend on the unpublished reference [9], which supplies the theorem that clean single-valued functions lift mod-products symbol identities to analytic identities up to a constant. The present paper gives neither a statement of the theorem's hypotheses nor a proof. If that lifting theorem is false or has hidden conditions, the analytic conclusions do not follow from the symbol identity. This is a real gap in the chain from algebraic identity to analytic functional equation. It does not damage Theorem 14 itself, which is the core new result.\n\nThe weight 6 and 8 evaluations (e.g. S4,2(-1), S6,2(-1)) are honest: they are marked with '?' and verified numerically to high precision, and the coproduct analysis explains most coefficients. Those should not be read as proven theorems, and the paper is careful about that. The abstract is slightly stronger than what is rigorously shown, since it says functional equations are derived without flagging the dependence on [9] and the numerical parts.\n\nNet assessment: this is a serious paper with a substantial new algebraic result, written by people who know the subject. The main theorem should be published. The authors need to either make [9] available with a complete proof, or restrict the stated five-term conclusion to the symbol level and keep the analytic lift as conditional. A referee should verify Theorem 14's computation carefully and push for clarity on the lifting hypothesis.\n\nI would send this to a referee. The core result earns the time.","headline":"A solid new symbol-level depth reduction for S3,2 that deserves refereeing, with the analytic lift and several evaluations conditional on an unpublished cleaning theorem.","tokens_in":47115,"tokens_out":1665,"would_cite":true,"duration_ms":20420,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G55","33E20","39B32"],"pacs":[],"model":"deepseek-v4-flash","headline":"The weight-5 Nielsen polylogarithm $S_{3,2}$ satisfies the dilogarithm five-term relation modulo $\\mathrm{Li}_5$ and products, with explicit $\\mathrm{Li}_5$ corrections.","keywords":["Nielsen polylogarithms","functional equations","five-term relation","mod-products symbol","depth reduction","single-valued polylogarithms","special values","polylogarithm ladders"],"falsifier":"Evaluate the clean single-valued identity of Corollary 15 at a randomly chosen tuple $(x_1,\\dots,x_5)$ of distinct complex numbers with non-zero imaginary parts and check the left-hand side to 100 decimal places; any non-zero value would disprove the analytic claim. Alternatively, if the unpublished cleaning theorem of [9] is false, the identity may fail only at the analytic lift while the symbol identity (10) still holds, and a direct analytic proof of (10) would settle the distinction.","tokens_in":46088,"feed_emoji":"","tokens_out":11660,"duration_ms":104182,"temperature":0.7,"pith_summary":"The paper proves that the weight-5 Nielsen polylogarithm $S_{3,2}$ behaves like the classical dilogarithm $\\mathrm{Li}_2$ in a precise sense: whenever a combination of arguments satisfies the five-term relation for $\\mathrm{Li}_2$, the same combination applied to $S_{3,2}$ vanishes modulo $\\mathrm{Li}_5$ and products of lower-weight functions. The main theorem gives the explicit $\\mathrm{Li}_5$ corrections through a five-variable identity involving three higher cross-ratios. This confirms a depth-reduction prediction in the motivic framework for weight 5 and implies that every rational functional equation for $\\mathrm{Li}_2$, including the distribution relations, produces a corresponding relation for $S_{3,2}$ with algorithmically determined $\\mathrm{Li}_5$ terms. The paper also derives functional equations and evaluations for Nielsen polylogarithms up to weight 8, including reductions of $S_{4,2}$, $S_{5,3}$, and $S_{6,2}$, and gives a general family of higher-weight depth reductions that yields Nielsen ladders in every weight.","feed_headline":"Nielsen polylog S3,2 obeys the five-term relation modulo Li5","feed_subtitle":"Explicit Li5 corrections upgrade every dilogarithm functional equation to weight 5, plus new identities through weight 8.","key_machinery":"The mod-products symbol $\\mathrm{Symb}^{/A_1}$ carries the argument: it is an invariant of iterated integrals that records only coproduct components not coming from products, so an identity written in it is exactly a statement modulo products and lower-weight functions. The paper computes $\\mathrm{Symb}^{/A_1}(S_{n,p}(z))$ explicitly and uses recursion to reduce weights. To pass from symbol identities to genuine functions, the paper invokes the clean single-valued construction from an unpublished source, which combines a cleaning map that completes product terms with a single-valued map; the main theorem of that source asserts that $\\sum_i \\lambda_i \\hat f_i = \\text{constant}$ whenever $\\mathrm{Symb}^{/A_1}(\\sum_i\\lambda_i f_i)=0$. The proof of Theorem 14 also uses the action of the symmetric group $S_5$ on cross-ratio variables: the relevant tensor space $\\bigwedge^2 V \\otimes \\operatorname{Sym}^3(V)$ splits into a 4-dimensional and a 6-dimensional irreducible representation, and the alternating sum is shown to vanish by explicit polynomial identities inside these components.","core_discovery":"The central discovery is equation (10) of Theorem 14: after antisymmetrisation over the five variables, $$11 S_{3,2}(\\operatorname{cr}(x_1,x_2,x_3,x_4)) + \\mathrm{Li}_5(15[r_1]-9[r_2]+[r_3]) = 0$$ in the mod-products symbol $\\mathrm{Symb}^{/A_1}$; here $\\operatorname{cr}$ is the cross-ratio and $r_1,r_2,r_3$ are certain higher ratios built from pairwise differences $x_i-x_j$. Since the symbol kills products, this says exactly that $S_{3,2}$ of the five-term relation is a combination of $\\mathrm{Li}_5$'s modulo products. The authors lift this symbol identity to a clean single-valued identity (Corollary 15), and from it deduce that distribution relations for $S_{3,2}$ hold modulo $\\mathrm{Li}_5$ with explicit corrections, that any $\\mathrm{Li}_2$ evaluation reachable through the five-term relation upgrades to an $S_{3,2}$ evaluation, and that alternating the identity over a sixth point yields a known nontrivial $\\mathrm{Li}_5$ functional equation. Beyond weight 5, the paper reduces $S_{3,3}$ to $S_{4,2}$ and $\\mathrm{Li}_6$, $S_{4,3}$ to $S_{5,2}$ and $\\mathrm{Li}_7$, and $S_{5,3}$ to $S_{6,2}$ and $\\mathrm{Li}_8$ on algebraic families, and proposes numerically verified evaluations such as $S_{4,2}(-1)$ and $S_{6,2}(-1)$ in terms of classical polylogarithms.","pith_inferences":["The paper does not spell this out, but the same mechanism should produce explicit $\\mathrm{Li}_{n+p}$ corrections whenever an analogous motivic cobracket vanishes, so the recursions of Theorem 40 could be used to guess higher-weight identities algebraically.","Because the clean lift relies on an unpublished theorem, a natural test is to compute the clean single-valued lift of a known mod-products symbol identity independently; a failure there would cut the analytic consequences while leaving the symbol identity (10) intact.","The conjectural special values imply that all alternating multiple zeta values up to weight 6 reduce to classical polylogarithms; proving the undetermined $\\zeta(6)$ coefficient in the $S_{4,2}(-1)$ reduction would complete that reduction.","The same higher-ratio and $S_5$-module identities could be transferred to other depth-2 polylogarithms or to their single-valued variants, giving explicit functional equations useful in amplitude computations."],"forward_implications":["Every rational functional equation for $\\mathrm{Li}_2$, not just the five-term relation, yields a corresponding relation for $S_{3,2}$ modulo $\\mathrm{Li}_5$ and products with explicitly computable corrections; in particular the distribution relations for $S_{3,2}$ hold in this sense (Corollary 19).","Known $\\mathrm{Li}_2$ evaluations accessible through the five-term relation, including the golden-ratio values, Lewin's $\\sqrt{2}-1$ ladder, and the $1/3$ ladder, lift to $S_{3,2}$ evaluations with explicit $\\mathrm{Li}_5$ terms.","In weight 6, $S_{3,3}$ reduces to $S_{4,2}$ and $\\mathrm{Li}_6$, and $S_{4,2}$ evaluated on algebraic $\\mathrm{Li}_3$ functional equations reduces to $\\mathrm{Li}_6$; this gives reductions of $S_{3,3}$ at roots of unity and conditional evaluations of $S_{4,2}(-1)$, $S_{4,2}(1/2)$, and $S_{3,3}(-1)$ in classical polylogarithms.","In weight 7, $S_{4,3}$ reduces to $S_{5,2}$ and $\\mathrm{Li}_7$; in weight 8, $S_{5,3}$ evaluated on algebraic $\\mathrm{Li}_2$ equations reduces to $S_{6,2}$ and $\\mathrm{Li}_8$, and a numerically verified candidate evaluates $S_{6,2}(-1)$ through $\\mathrm{Li}_8$, $\\zeta(3,5)$, and products of logs.","The depth-reduction theorem of Section 10 holds for all $m\\ge 1$: $S_{2m,m}$, $S_{2m-1,m}$, and $S_{2m-2,m}$ satisfy explicit reductions or two- and three-term relations, so Nielsen ladders exist in arbitrary weight (Corollary 41)."],"supporting_citations":[{"why":"It states the cleaning theorem that lifts mod-products symbol identities to analytic identities; the paper's Corollary 15 and most special-value reductions depend on it.","marker":"[9]"},{"why":"It supplies the integral definition of $S_{n,p}$ along with the two-term inversion and reflection relations used throughout the paper.","marker":"[27]"},{"why":"It establishes the motivic iterated-integral Hopf algebra, coproduct, and symbol on which the mod-products symbol is built.","marker":"[22]"},{"why":"It provides the projectors that annihilate products in the symbol, used to define $\\mathrm{Symb}^{/A_1}$.","marker":"[13]"},{"why":"It states the depth-reduction prediction in the motivic Lie coalgebra that Theorem 14 corroborates in weight 5.","marker":"[19]"},{"why":"It gives the algorithm expressing any one-variable $\\mathrm{Li}_2$ functional equation as a combination of five-term relations, used in Corollary 19.","marker":"[37]"},{"why":"It defines the single-valued map used in the clean single-valued construction for lifting identities.","marker":"[6]"}],"fun_headline_variants":["S3,2 obeys five-term relation modulo Li5 and products","Weight-5 Nielsen polylog S3,2 follows Li5-corrected five-term law","New functional equations for Nielsen polylogs up to weight 8","S3,2 five-term identity upgraded with explicit Li5 corrections","Dilogarithm five-term relation lifts to S3,2 mod Li5"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The analytic consequences rely on an unpublished result that a clean single-valued polylogarithm identity automatically upgrades an algebraic invariant identity to a genuine function identity; if that result is incomplete, the analytic corollaries fail even though the invariant identity (10) remains.","fun_headline_variants_meta":{"raw":{"variants":["S3,2 obeys five-term relation modulo Li5 and products","Weight-5 Nielsen polylog S3,2 follows Li5-corrected five-term law","New functional equations for Nielsen polylogs up to weight 8","S3,2 five-term identity upgraded with explicit Li5 corrections","Dilogarithm five-term relation lifts to S3,2 mod Li5"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000593,"raw_usage":{"total_tokens":2802,"prompt_tokens":991,"completion_tokens":1811,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":607,"completion_tokens_details":{"reasoning_tokens":1710}},"tokens_in":607,"tokens_out":1811,"duration_ms":12361,"temperature":1.0,"reasoning_tokens":1710,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:33:20.523731+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the clean single-valued identity of Corollary 15 at a randomly chosen tuple $(x_1,\\dots,x_5)$ of distinct complex numbers with non-zero imaginary parts and check the left-hand side to 100 decimal places; any non-zero value would disprove the analytic claim. Alternatively, if the unpublished cleaning theorem of [9] is false, the identity may fail only at the analytic lift while the symbol identity (10) still holds, and a direct analytic proof of (10) would settle the distinction.","supporting_citations":[{"cited_title":"Charlton, C","cited_arxiv_id":null,"evidence_quote":"It states the cleaning theorem that lifts mod-products symbol identities to analytic identities; the paper's Corollary 15 and most special-value reductions depend on it."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the integral definition of $S_{n,p}$ along with the two-term inversion and reflection relations used throughout the paper."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It establishes the motivic iterated-integral Hopf algebra, coproduct, and symbol on which the mod-products symbol is built."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It provides the projectors that annihilate products in the symbol, used to define $\\mathrm{Symb}^{/A_1}$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It states the depth-reduction prediction in the motivic Lie coalgebra that Theorem 14 corroborates in weight 5."},{"cited_title":"W ojtkowiak","cited_arxiv_id":null,"evidence_quote":"It gives the algorithm expressing any one-variable $\\mathrm{Li}_2$ functional equation as a combination of five-term relations, used in Corollary 19."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It defines the single-valued map used in the clean single-valued construction for lifting identities."}],"review_version":1}