{"id":"9b6d4af4-74bb-45f0-8bda-29bd65a4f04c","arxiv_id":"2412.00025","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"Eight families of quadratic Euler sums of odd order are reduced to zeta values and polylogarithms, but order-7 reductions remain incomplete for two unresolved nonlinear sums.","lead":"This paper lists explicit formulas for eight families of quadratic Euler sums, infinite series built from products of harmonic-type numbers, in terms of zeta values and polylogarithms. The order-5 cases are claimed to be complete, but the order-7 reductions stop at two sums the authors cannot evaluate, so the advertised full family is only partly derived.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The order-seven reduction is explicitly incomplete: the paper leaves A = ∑ H_k^(2)h_k/k^4 and B = ∑ H_k h_k/k^5 unresolved, so the advertised claim of expression by zeta values and linear Euler sums only is not established.","rationale":"The most load-bearing concern is not, in the first instance, the unproved helper identities, although those are also serious. Even if every borrowed two-valued identity is correct, the paper's own Section IV.A and Section V explicitly state that the order-seven reduction is incomplete: two nonlinear sums A and B are left unresolved, and only a numerical approximation for A is offered in Eq. (164). Since the abstract and section headings claim that the eight families are expressible by zeta values and linear Euler sums only, this internal admission directly contradicts the central claim. The order-seven formulas in later subsections depend on A and B, so the repeated assertion that \"all members\" of those families have been calculated is not justified. This is a correctness risk rather than a mere presentational issue: the advertised result requires a missing identity, and no evidence is given that such an identity exists. The reader's verdict of REJECT is therefore appropriate; my check targets the missing reduction directly and, if it fails, would render the central claim unsupported regardless of the status of the helper identities.","tokens_in":29210,"tokens_out":4890,"duration_ms":51987,"concrete_test":"Compute A = ∑_{k=1}^∞ H_k^(2) h_k/k^4 and B = ∑_{k=1}^∞ H_k h_k/k^5 to 100-digit precision using an independent method (high-precision summation with convergence acceleration, or the contour-integral representation of Flajolet–Salvy). Then run PSLQ against the vector space spanned by ζ(7), ζ(2)ζ(5), ζ(3)ζ(4), ζ(3)∑ h_k/k^3, and the linear Euler sums appearing in §IV.A. If no integer relation is found, and if Eq. (164) differs from the numerical value of A by more than 2.9e-15, the central claim that these sums reduce to zeta values and linear sums is unsupported. If a relation is found, substitute it into Eqs. (42)–(49) to verify that all first-family members become zeta-plus-linear expressions.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim asserts that the eight families are expressible by zeta values and linear Euler sums only. The order-seven section does not deliver this. In §IV.A, Eqs. (42)–(49) express the first-family members in terms of zeta values, the linear sum ∑ h_k/k^3, and the two nonlinear sums A = ∑ H_k^(2)h_k/k^4 and B = ∑ H_k h_k/k^5. The text then states: \"What remains is to find a direct relation between A and B in terms of zeta values and linear Euler sums.\" Section V repeats that this is still missing and supplies only the numerical approximation Eq. (164), \"correct within 2.9e-15.\" Since A and B are quadratic products of harmonic numbers, they are not \"special types of linear Euler sums\"; the advertised reduction is therefore incomplete for the first family. Moreover §IV.B–G claim all members of the second through seventh families \"have been calculated,\" but their displayed formulas contain A and/or B (e.g., Eqs. (76), (77), (81), (83), (92), (100), etc.), so those \"all members\" claims inherit the missing reduction. Even granting every unproved two-valued helper identity (Eqs. (56), (69), (96), (106), (115)), the paper's central claim is not supported by its own text.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims a general method for evaluating eight families of quadratic Euler sums of odd order in terms of zeta values, polylogarithms Li_n(1/2), and linear Euler sums. The order-3 and order-5 families are presented as fully evaluated, while the order-7 families are reduced to zeta values, linear sums, and two unresolved sums A = ∑ H_k^(2)h_k/k^4 and B = ∑ H_k h_k/k^5; for A only a numerical approximation is given. The method rests on a collection of \"two-valued\" helper identities, many of which are imported from the authors' own preprints [10–12].","tokens_in":29517,"tokens_out":4707,"duration_ms":44027,"significance":"If the results were correct and complete, the paper would extend the known theory of nonlinear Euler sums to mixed products of even and odd hyperharmonic numbers and would provide a systematic reduction scheme for odd orders. The many explicit formulas in the paper could serve as useful benchmark values. However, because the central order-7 claim is explicitly incomplete and the auxiliary identities are not independently verified, the significance as a proof is substantially diminished.","major_comments":[{"comment":"The order-7 reduction is incomplete. The first-family formulas in Eqs. (42)–(49) express the sums in terms of zeta values, the linear sum ∑ h_k/k^3, and the two quadratic sums A = ∑ H_k^(2)h_k/k^4 and B = ∑ H_k h_k/k^5. Section V then states that \"it remains to find a direct relation between\" A and B in terms of zeta values and linear Euler sums, and supplies only the numerical approximation (164). Because A and B are products of harmonic and odd harmonic numbers, they are not \"special types of linear Euler sums\"; therefore the abstract's claim that the eight families are expressible by zeta values and linear Euler sums only is not established. The statements in §IV.B–G that all members have been \"explicitly calculated\" inherit this gap, since Eqs. (76), (77), (92), (100), and related formulas contain A or B.","section":"IV.A, V"},{"comment":"The approximation (164), described as \"correct within 2.9e-15\", is not a proof and is not derived in the text. No explanation is given for the rational coefficients -1559/1943 and 1469/759, and a numerical agreement to fifteen digits, however suggestive, does not establish the required identity. Since this is the only evidence offered for the missing relation between A and B, the claim of a complete reduction to zeta values and linear Euler sums is unsupported.","section":"V, Eq. (164)"},{"comment":"The derivations are largely omitted. In Section III no derivations are shown: each of Eqs. (15)–(40) is introduced with \"we get\" or \"it follows\" and no intermediate steps. In Section IV the auxiliary identities on which the reductions depend are either asserted (\"we found\") or referred to the authors' own preprints [10–12]; examples are Eqs. (56), (69), (96), (106), and (115). Since the entire chain of evaluations relies on these unproved two-valued help-function identities, the soundness of the advertised evaluations cannot be checked from the manuscript as it stands.","section":"III, IV"}],"minor_comments":[{"comment":"The text repeatedly uses \"nominator\" for \"numerator\" and \"Salvi\" for \"Salvy\" (reference [1]); the manuscript needs careful proofreading.","section":"Abstract, Introduction"},{"comment":"The term \"proper two-valued integer functions\" is never defined in the present paper; the reader is left to infer its meaning from the references. A short formal definition would make the method self-contained.","section":"I, IV"},{"comment":"The displayed identity for H^{(2)}_{k+i}/(k+i)^n appears garbled as printed; the right-hand side does not match the left-hand side as written. Please correct the formula or clarify the intended finite-difference relation.","section":"Eq. (53)"}],"recommendation":"reject","confidential_remarks":"The manuscript does not deliver the central claim: the order-7 reduction is left unfinished, with only a numerical approximation for one unresolved sum. The heavy reliance on the authors' own unpublished preprints for helper identities further reduces verifiability. I would encourage the authors to resubmit once the missing relation between A and B is established and the order-5 derivations are supplied, but in its current state the paper does not meet the standard for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, this is a genuine extension of the authors' earlier work: it handles mixed products of even and odd hyperharmonic numbers, and three denominator types, k^n, (2k-1)^n, and k(2k-1). The order-3 and order-5 sections contain many explicit evaluations, and if those are correct they are useful. Second, the central claim is not actually delivered. The abstract says eight families are expressible by zeta values and special linear Euler sums. The order-7 section leaves A = Σ H_k^(2) h_k / k^4 and B = Σ H_k h_k / k^5 unresolved, and substitutes a numerical rational approximation, Eq. (164), good to 2.9e-15, with no derivation. The summary admits the reduction is incomplete. That is a load-bearing gap.\n\nWhat the paper does well: the order-3 and order-5 evaluations are stated in closed form, and the mixed even/odd products go beyond the even-only products of the authors' 2022 paper. The method of two-valued integer functions is interesting, and the paper is honest enough to flag what it cannot do. But that honesty appears in the summary, not in the abstract or section headings.\n\nWhere the soft spots are. The order-5 section gives no derivations, only 'we get' and 'it follows.' The order-7 helper identities, e.g., Eqs. (56), (69), (96), (106), (115), are asserted as 'we found' or 'analogously to' prior preprints, with no proofs or independent checks. Since the reduction chain depends on them, this is risky. Eq. (164) is a fitted approximation, not a derivation. The abstract overstates what is established. I would not call this a crank paper; the thinking is clear. But as a research claim, it is not supported by the text. There are also minor typos, e.g., Eq. (35) has two Li4 terms, one with and one without a ln(2) factor, which makes close reading harder.\n\nWho this is for: someone working on nonlinear Euler sums and harmonic number identities. If the missing derivations are supplied and the order-7 sums resolved, it would be a real contribution. As submitted, I would not accept it; it needs major revision and independent verification. A serious referee could check the order-5 formulas numerically and the helper identities symbolically, so I would send it back to the authors with clear requests rather than desk-reject outright.","headline":"A real extension with many explicit sums, but the advertised order-7 reduction is incomplete and the order-5 results are asserted without derivations.","tokens_in":30075,"tokens_out":3509,"would_cite":false,"duration_ms":31432,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M06","11B83","33B30"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that eight families of quadratic Euler sums of odd order reduce completely to zeta values, polylogarithmic values at 1/2, and linear Euler sums, with the order-7 case left depending on only two unresolved constants.","keywords":["Euler sums","harmonic numbers","hyperharmonic numbers","zeta values","polylogarithms","two-valued integer functions","odd order"],"falsifier":"Numerically evaluate a representative asserted identity, say Eq. (56), at several integer values of i to high precision; or evaluate both sides of a stated order-7 result such as Eq. (141). A mismatch at the first unsupported identity would invalidate every order-7 result that uses it.","tokens_in":28958,"feed_emoji":"🧮","tokens_out":4524,"duration_ms":37721,"temperature":0.7,"pith_summary":"The paper claims that eight infinite families of quadratic Euler sums, built from products of even-type and odd-type hyperharmonic numbers divided by three denominator types, are expressible entirely in terms of zeta values, polylogarithmic values at 1/2, and certain linear Euler sums. For order 3 and order 5, the families are fully evaluated; for order 7, every family is reduced to explicit combinations of zeta values and linear sums, with only two unresolved sums remaining. The method rests on two-valued integer functions that turn nested sums into closed form. If correct, this gives a systematic calculus for odd-order quadratic Euler sums that earlier treatments left inaccessible.","feed_headline":"Eight Euler-sum families collapse to zeta values","feed_subtitle":"Odd-order quadratic sums with mixed even-odd harmonic products are reduced to known constants and linear sums.","key_machinery":"The engine is a collection of two-valued integer functions: identities that give closed forms for infinite sums indexed by a free integer parameter, such as sum_{k>=1} H_k^(2)/(k(k+i)) or sum_{i>=1} h_i/((i+k)^2), in terms of harmonic numbers of that parameter. Each such identity, when fed through partial-fraction decompositions and summation-order reversals, peels the quadratic sum down to zeta values and linear Euler sums. Many of these helper identities are imported from the authors' earlier preprints with phrases like 'analogously' or 'we found' rather than proved in the present text.","core_discovery":"The central discovery is a set of eight families of quadratic Euler sums of odd order — products such as H_k^(a) h_k^(b) divided by k^c, H_k^(a) H_k^(b) divided by (2k-1)^c, h_k^(a) h_k^(b) divided by k(2k-1), and similar — that can be evaluated in terms of zeta values, Li_n(1/2), and linear Euler sums of the form sum h_k^(n)/k^m. The order-3 case is expressed in zeta values and ln 2; the order-5 case in zeta values, Li_4(1/2), and Li_5(1/2); the order-7 case reduces every sum to zeta values and linear sums plus the two unresolved constants A = sum H_k^(2) h_k / $k^{4}$ and B = sum H_k h_k / $k^{5}$. The authors state that all members of each order-5 and order-7 family have been explicitly calculated in this sense.","pith_inferences":["Because the helper identities are asserted rather than derived, an independent computer-algebra verification of even a handful of them would either certify or refute the whole reduction chain.","The two unresolved constants A and B may admit closed forms in terms of multiple zeta values or polylogarithms at 1/2; the paper's approximation to A is already accurate to about 3e-15.","If the scheme generalizes as suggested, the same two-valued-function toolkit could be applied to even-order quadratic sums or to higher-degree products."],"forward_implications":["Order-3 and order-5 quadratic Euler sums of the eight families have fully explicit evaluations in zeta values and Li_n(1/2).","For order-7, the eight families collapse to zeta values, linear odd-type sums, and the two unresolved sums A and B.","Completing the reduction of A and B, likely with Li_7(1/2), would close the order-7 case and open order 9.","The same two-valued-function scheme is announced as extendable to ternary Euler sums, with sample evaluations listed."],"supporting_citations":[{"why":"The classic study that set up the contour-integral method for evaluating Euler sums and the framework this work generalizes.","marker":"[1]"},{"why":"The compilation of linear odd-type Euler-sum evaluations on which the paper relies as known input.","marker":"[5]"},{"why":"The source of summation identities for generalized harmonic numbers used repeatedly in the order-7 reductions.","marker":"[8]"},{"why":"An earlier preprint that first computed one of the order-5 sums and supplies related power-series results.","marker":"[10]"},{"why":"The source of odd-type Euler-sum lemmas (2a, 2b, 3a, 3b, 4a, 4b) and several two-valued help functions used throughout the order-7 calculations.","marker":"[11]"},{"why":"The authors' prior treatment of even-order nonlinear Euler sums, providing identities and lemmas used in the order-3 and order-5 cases.","marker":"[12]"}],"fun_headline_variants":["Odd-order quadratic sums reduced to zeta and linear sums","Quadratic Euler sums: eight families resolved","Eight families of odd-order Euler sums tamed","Nonlinear Euler sums cracked: eight families","Mixed even-odd harmonic products yield to zeta"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire chain depends on a large set of two-valued summation identities that are asserted without proof, mostly borrowed from earlier preprints; if any of them is wrong or merely approximate, the advertised reductions and all constants derived from them fail.","fun_headline_variants_meta":{"raw":{"variants":["Odd-order quadratic sums reduced to zeta and linear sums","Quadratic Euler sums: eight families resolved","Eight families of odd-order Euler sums tamed","Nonlinear Euler sums cracked: eight families","Mixed even-odd harmonic products yield to zeta"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000307,"raw_usage":{"total_tokens":1790,"prompt_tokens":1013,"completion_tokens":777,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":629,"completion_tokens_details":{"reasoning_tokens":705}},"tokens_in":629,"tokens_out":777,"duration_ms":7583,"temperature":1.0,"reasoning_tokens":705,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T18:19:27.498665+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Numerically evaluate a representative asserted identity, say Eq. (56), at several integer values of i to high precision; or evaluate both sides of a stated order-7 result such as Eq. (141). A mismatch at the first unsupported identity would invalidate every order-7 result that uses it.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The classic study that set up the contour-integral method for evaluating Euler sums and the framework this work generalizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The compilation of linear odd-type Euler-sum evaluations on which the paper relies as known input."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The source of summation identities for generalized harmonic numbers used repeatedly in the order-7 reductions."}],"review_version":1}