{"id":"17940efb-30f5-439c-be60-f527ad975a1d","arxiv_id":"1908.06631","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Two conjectural binomial-sum identities for zeta(7), proposed by Z-W Sun, are proved with computer algebra, and several further series for zeta(7) are discovered.","lead":"This paper proves two conjectured formulas for infinite binomial sums that evaluate to combinations of the number zeta(7) with other zeta values. It also uses the same computer algebra machinery to find new related series, showing how automated methods can certify and expand such identities.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The decisive reduction to (7) is asserted from a truncated 243-term CAS output with no certificate; a single omitted coefficient would change the result.","rationale":"I read the paper as a computational proof whose central claim is the two Sun identities. The method is coherent, and I specifically checked the final algebraic step: with the paper's conventions (0=1/t, 1=1/(t-1), lambda=1/(1+t+t^2)), one has H_{0,0,1}=-zeta(3), H_{0^4,1}=-zeta(5), H_{0^6,1}=-zeta(7), and H_lambda(1)=pi/(3*sqrt(3)); substituting these into (7) reproduces (1) exactly. So the printed final reduction to the zeta constants is not the weak point. The weak point is the reduction of the 243-term expression to (7), which the text only partially displays and attributes to an unshown HarmonicSums command. That is exactly where a CAS or transcription error would propagate to the advertised constants. The reader's conditional verdict remains appropriate: the author should make the full 243-term expression and the relation-based reduction available and independently checkable. My read does not alter the verdict.","tokens_in":4944,"tokens_out":38262,"duration_ms":397061,"concrete_test":"Run the referenced HarmonicSums notebook from a fresh kernel, reproduce the 243-term expression from (5), and evaluate SpecialGLToH[7,3] on it; then compare the output with (7). If the outputs differ, the proof fails at the advertised reduction; if they agree, the missing step is certified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 2's chain is: summands -> recurrences/differential equations -> solution (5) -> substitution -> a 243-term cyclotomic-HPL expression -> relations -> (7), which then gives (1)/(2). The last equality (7) -> RHS is checkable and consistent; the unverified load-bearing step is the reduction of the 243-term expression to (7). The manuscript prints only the first six and last six of the 243 terms, separated by an ellipsis, and then states that SpecialGLToH[7,3] 'finds' (7) using shuffle, stuffle, multiple-argument, distribution, and duality relations. No full 243-term expression, no relation trace, and no machine-checkable certificate are included; the referenced notebook is external to the arXiv submission. If any of the omitted 231 coefficients is wrong, the final value of (7) changes and the proof fails. This is not a matter of disagreement with the method but of missing evidence at the exact hinge of the proof: the paper cannot be checked from its own text.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims proofs of two conjectured infinite binomial-sum identities due to Z-W Sun, stated as (1) and (2). The method follows the author's earlier framework: convert the summands into holonomic recurrences, pass to generating functions, solve the resulting differential equations, represent the solutions as iterated integrals, substitute to cyclotomic harmonic polylogarithms, and finally use algebraic relations among those polylogarithms to obtain the claimed zeta-value combinations. The paper also lists several additional identities that it says can be found by the same strategy. The proof chain is computer-assisted and relies on the author's HarmonicSums package.","tokens_in":5118,"tokens_out":2607,"duration_ms":29133,"significance":"If the proofs are correct and reproducible, the paper settles two open conjectures and demonstrates the utility of the author's holonomic-plus-cyclotomic-polylogarithm method for binomial sum evaluations. The additional identities in Section 3 are potentially useful. However, the value of the paper as a proof depends critically on the verifiability of the computer-algebra steps, and the manuscript as written does not make those steps checkable from its own text. The central mathematical claims are plausible and consistent with the cited conjectures, but the evidence supplied at the load-bearing reduction step is incomplete.","major_comments":[{"comment":"The derivation of the generating-function representation (5) is asserted as the output of ComputeGeneratingFunction without showing the recurrence-solving process, initial-value comparisons, or the summation of the two holonomic solutions. Since (5) is the foundation for all subsequent substitutions and reductions, the reader cannot verify that the coefficients such as 4801781/73728 and 363/128 are correct. The paper should include a complete derivation or a machine-checkable certificate for this step.","section":"Section 2, Eq. (5)"},{"comment":"The 243-term cyclotomic harmonic polylogarithm expression is only displayed as the first six and last six terms with an ellipsis. This expression is the hinge between the integral representation (5) and the final reduction (7). A single incorrect coefficient among the omitted 231 terms would change the final value and invalidate the proof. The full expression, or a verifiable certificate of its computation, must be supplied in the paper or in a stable ancillary file.","section":"Section 2, SpecialGLToH application"},{"comment":"The decisive reduction of the 243-term expression to the five-term expression (7) is stated without any details: no relation trace, no basis transformation, and no certification. The sentence 'Applying these relations we find (7)' hides exactly the step on which the proof depends. The author should provide the full reduction, the list of relations used, or a separately checkable certificate (for example, a proof assistant output or an independent implementation of the same reduction).","section":"Section 2, Eq. (7)"}],"minor_comments":[{"comment":"The additional identities in Section 3 are presented without any indication of whether they are proven by the described method or merely discovered empirically. The paper should explicitly state their status; if they are proven, the necessary certificates or reproducible commands should be included, and if they are conjectural, they should be labeled as such.","section":"Section 3"},{"comment":"The notation H\\lambda4 is not defined; it should be written as H_{\\lambda,\\lambda,\\lambda,\\lambda} or explicitly explained as a shorthand for four repeated indices. This would improve readability.","section":"Notation in Eq. (7)"},{"comment":"The proof depends on an external notebook and a specific version of HarmonicSums. The paper should provide a persistent link, a checksum, or an appendix with the full session log so that referees and readers can reproduce the computations independently of the author's web page.","section":"Reproducibility details"}],"recommendation":"major_revision","confidential_remarks":"The paper is authored by the developer of HarmonicSums, and the central proof steps are outputs of that package. This is not itself a concern, but it increases the importance of including independently checkable artifacts. I would ask the editor to ensure the supplementary notebook and the full 243-term expression are made available and, if possible, to have the reduction independently reproduced. The paper is within scope for math.CO, but the current text cannot be verified from its own content."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does what it says: it proves Sun's two conjectured binomial-sum evaluations for zeta(7), and it lists several new identities of the same type. The proof strategy is the author's own holonomic method: convert the summands to generating functions, derive differential equations, solve them in terms of iterated integrals, substitute to cyclotomic polylogarithms, and reduce using known relations. The recurrences and differential equations are displayed in full, and the final reduction from the cyclotomic-polylogarithm expression to (7) is plausible and consistent with the claimed zeta value. The two main identities are genuinely new proofs, not restatements of known results. The citation pattern is concentrated on the author's own prior work and Sun's conjecture list, which is normal for a niche computational method paper; I do not see a circularity problem, since the target identities come from outside the method.\n\nThe soft spot is exactly where the stress-test puts it. The 243-term cyclotomic-polylogarithm expression is shown only as six leading and six trailing terms with an ellipsis, and the reduction to (7) is asserted as the output of SpecialGLToH. No relation trace, no certificate, and no full expression appear in the text. The referenced notebook is external to the arXiv submission, so the reader cannot verify the hinge of the proof without downloading and trusting the package. This is a reproducibility gap, not evidence of a false result, but it is a gap at the exact point where a single wrong coefficient would change the conclusion. The Section 3 identities are presented as discoveries without proofs; they should be labeled as conjectures or supplemented with certificates. The convergence issue at x = 1 is also not discussed, though it is likely harmless because the sums converge and the method uses Abel summation; still, a sentence would settle it.\n\nOverall, the core argument is sound in outline and the method is legitimate, but the paper as submitted is not fully checkable from its own text. The author should include the complete 243-term expression, the relation-based reduction, or a machine-checkable certificate, and should either prove the Section 3 identities or clearly mark them as unproved. This deserves a serious referee, because the main results resolve two published conjectures and the method is reproducible in principle with the named software.\n\nRecommendation: send it to peer review, but with the expectation that the author provides enough computational detail to make the proof independently verifiable, not just asserted from a package run.","headline":"A credible computational proof of two Sun conjectures for zeta(7), with the decisive CAS reduction left as an unverified black box in the text and the new Section 3 identities unproved.","tokens_in":5615,"tokens_out":1637,"would_cite":true,"duration_ms":20481,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05A19","11M06","11M32","68W30"],"pacs":[],"model":"deepseek-v4-flash","headline":"Two series for ζ(7) conjectured by Sun are now proven by an algorithmic reduction to cyclotomic polylogarithms.","keywords":["infinite binomial sums","central binomial coefficients","harmonic numbers","zeta(7)","cyclotomic harmonic polylogarithms","holonomic generating functions","computer algebra","symbolic summation"],"falsifier":"Evaluate both sides of (1) and (2) numerically with an independent arbitrary-precision implementation to, say, 100 decimal places; if any digit differs beyond the controlled rounding error, the central claim is false. Alternatively, run a certified summation algorithm on the two summands and check that the produced closed forms coincide with the paper's right-hand sides.","tokens_in":4735,"feed_emoji":"","tokens_out":7548,"duration_ms":70843,"temperature":0.7,"pith_summary":"This paper proves two infinite-sum identities conjectured by Zhi-Wei Sun: both sums, over $k$, of expressions built from harmonic numbers and the central binomial coefficient $\\binom{2k}{k}$, are shown to equal the advertised rational linear combinations of $\\zeta(7)$, $\\zeta(2)\\zeta(5)$, and $\\zeta(3)\\zeta(4)$. The proof works by viewing each series as the $x\\to 1$ specialization of a generating function, deriving holonomic (polynomial-coefficient) recurrences and differential equations, solving those equations in terms of iterated integrals, and converting the result into cyclotomic harmonic polylogarithms at cyclotomy 3. Known structural relations among those polylogarithms then reduce the expression to the closed forms. The identities were conjectural; after this paper they are theorems, and the same pipeline produces further identities of the same shape.","feed_headline":"Two conjectured ζ(7) series are now proven","feed_subtitle":"A generating-function pipeline reduces both infinite binomial sums to exact zeta-value combinations.","key_machinery":"The load-bearing mechanism is the holonomic generating-function method combined with cyclotomic harmonic polylogarithms at cyclotomy 3. A holonomic sequence satisfies a linear recurrence with polynomial coefficients, and its generating function satisfies a linear differential equation; the package computes these from the summand, solves the differential equations in terms of iterated integrals (d'Alembertian solutions), and then rewrites those integrals using the substitution into the cyclotomic-polylogarithm alphabet with letters 0, 1, $\\lambda=(3,0)$, and $\\mu=(3,1)$. The final reduction is driven by shuffle, stuffle, multiple-argument, distribution, and duality relations among cyclotomic polylogarithms, which collapse the 243-term expression to the short closed form.","core_discovery":"The central claim is that the two Sun identities (1) and (2) are true exactly as stated. The author's derivation splits each summand into two parts, has the computer-algebra package compute their generating-function recurrences and differential equations, solves those equations as iterated integrals over $1/\\tau$ and $\\sqrt{\\tau/(4-\\tau)}$, and applies the substitution $\\tau\\to(\\tau-1)^2/(1+\\tau+\\tau^2)$ to turn the result into 243 cyclotomic harmonic polylogarithms at cyclotomy 3. Applying shuffle, stuffle, multiple-argument, distribution, and duality relations reduces the first sum to $-\\tfrac{459}{4}H_{0,0,1}H_{\\lambda}^{4} - \\tfrac{39}{2}H_{0,0,0,0,1}H_{\\lambda}^{2} + \\tfrac{45}{8}H_{0,0,0,0,0,0,1}$, which is exactly the right-hand side of (1); the second sum is reduced analogously to the right-hand side of (2). The paper also records several newly discovered identities of the same type in Section 3.","pith_inferences":["This suggests that infinitely many Sun-type binomial sums at odd zeta values should reduce to the same finite-dimensional space spanned by products $\\zeta(7)$, $\\zeta(2)\\zeta(5)$, $\\zeta(3)\\zeta(4)$ and, for cyclotomy-3 variants, terms involving $\\pi^7/\\sqrt{3}$ and the constant $c=\\sum_i 1/(3i+1)^4+1$; the Section 3 examples are consistent with this pattern.","A natural next step is to turn each solver output into a machine-checked certificate; without such certificates, the present proof's trust boundary sits inside the computer-algebra system.","The same substitution-and-relations strategy should be applicable to conjectural identities involving other cyclotomic alphabets, such as cyclotomy 5 or 7, where the number of letters and the dimension of the relation space grow but the pipeline does not change."],"forward_implications":["Sun's conjectures (1) and (2) are no longer open: both infinite binomial sums evaluate to the printed rational combinations of $\\zeta(7)$, $\\zeta(2)\\zeta(5)$, and $\\zeta(3)\\zeta(4)$.","Any computation that uses those sums as numerical constants can now rely on exact closed forms rather than on extrapolated high-precision values.","The same generating-function-to-cyclotomic-polylogarithm pipeline yields additional identities of the same type, several of which are listed in Section 3.","The proof method is algorithmic: it applies in principle to any summand that is holonomic and whose generating-function solution lives in the iterated-integral and cyclotomic-polylogarithm class."],"supporting_citations":[{"why":"supplies the two conjectural identities that are the paper's target.","marker":"[16]"},{"why":"supplies the generating-function method that the proof follows.","marker":"[2]"},{"why":"provides the substitution used to rewrite iterated integrals as cyclotomic polylogarithms.","marker":"[1]"},{"why":"describes the HarmonicSums package whose commands perform the recurrences, solving, substitution, and reduction.","marker":"[5]"},{"why":"gives the shuffle, stuffle, multiple-argument, distribution, and duality relations used in the final reduction.","marker":"[6,8,10]"},{"why":"underlies the differential-equation solver producing the iterated-integral and d'Alembertian solutions.","marker":"[4,7,11,15,14]"}],"fun_headline_variants":["Sun's ζ(7) conjectures proven via generating functions","Proof of two Sun ζ(7) identities, plus new series","Generating functions crack conjectural ζ(7) series","Computer algebra proves two ζ(7) binomial sum conjectures"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof leans on the computer-algebra computations in Section 2 being correct: the recurrences, the differential-equation solutions, the 243-term cyclotomic-polylogarithm expression, and its relation-based reduction are each reported as command or solver output, without an independent derivation or a machine-checkable certificate, and the referenced notebook is not part of the submission.","fun_headline_variants_meta":{"raw":{"variants":["Sun's ζ(7) conjectures proven via generating functions","Proof of two Sun ζ(7) identities, plus new series","Generating functions crack conjectural ζ(7) series","Computer algebra proves two ζ(7) binomial sum conjectures"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000168,"raw_usage":{"total_tokens":1223,"prompt_tokens":871,"completion_tokens":352,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":487,"completion_tokens_details":{"reasoning_tokens":281}},"tokens_in":487,"tokens_out":352,"duration_ms":3851,"temperature":1.0,"reasoning_tokens":281,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:38:38.402560+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate both sides of (1) and (2) numerically with an independent arbitrary-precision implementation to, say, 100 decimal places; if any digit differs beyond the controlled rounding error, the central claim is false. Alternatively, run a certified summation algorithm on the two summands and check that the produced closed forms coincide with the paper's right-hand sides.","supporting_citations":[],"review_version":1}