{"id":"19468ba1-a0ee-4027-a96c-d37ae368986e","arxiv_id":"2508.15938","paper_version":1,"verdict":"REJECT","confidence":"LOW","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"The abstract promises new Fourier coefficient formulas and counterexamples to the weak Shanks conjecture, but the submitted body is a different paper (a heavy-ion experiment report), so the result is not actually present in the manuscript.","lead":"The abstract claims new hypergeometric formulas for the Fourier coefficients of |f|^2 for the two-variable function f(z1,z2) = (1 - (z1+z2)/r)^(-alpha), together with new counterexamples to the weak Shanks conjecture. The manuscript body, however, is an unrelated NA61/SHINE experimental report on heavy-ion physics, so the mathematical claims cannot be checked from the submitted text.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central claim—explicit hypergeometric formulas and new counterexamples to the weak Shanks conjecture—is entirely absent from the attached body; no derivation, equation, or parameter regime appears, so the premise that the paper contains the proof is unsupported.","rationale":"The reader's rejection rests on the abstract/body mismatch: the submission announces a mathematics paper but contains only an unrelated heavy-ion conference report. My stress-test confirms this is the single load-bearing defect. The strongest claim—explicit hypergeometric formulas and additional counterexamples—cannot be evaluated because none of the supporting mathematics is present in the artifact. The reader's weakest assumption is that the bridge between the Fourier coefficients and the weak Shanks conjecture must be checkable; that bridge is not merely unverified, it is absent. This is a whole-submission claim-without-derivation problem, not a disagreement with consensus or an internal inconsistency that could be repaired by reading more carefully. If the correct full text had been attached, a mathematical review could proceed; as submitted, the only defensible verdict is unchanged rejection. No ad hominem is intended; the issue is structural and factual: the wrong document appears to be attached.","tokens_in":6134,"tokens_out":2165,"duration_ms":23819,"concrete_test":"Fetch the manuscript contents for the arXiv identifiers cited: the claimed ID 2508.15938 and the header ID 2508.15939v2. Run a full-text search for the key terms 'hypergeometric', 'Fourier', 'Shanks', 'f(z_1,z_2)', and for any displayed equation environments. If, as currently observed, the body contains none of these, the submission provides no derivation of the advertised formulas and no counterexamples. If a corrected manuscript is subsequently supplied, re-review that text for the actual coefficient formulas, the convergence/boundedness conditions on r and α, and the explicit connection to the weak Shanks conjecture.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract promises Fourier coefficients of |f|^2 in terms of hypergeometric functions and additional counterexamples to the weak Shanks conjecture. The body, however, is an unrelated NA61/SHINE heavy-ion experimental report whose header cites arXiv:2508.15939v2 [nucl-ex]. None of the promised mathematics appears anywhere in the submitted text: there is no definition of f, no Fourier coefficient formula, no hypergeometric function, no statement of the Bénéteau–Khavinson–Seco counterexample criterion, no condition linking the coefficients to that criterion, and no admissible parameter range (e.g., r > 2 for boundedness on the torus, or the α-range preserving integrability). Every component needed to support the central claim is missing. This is not a subtle gap in an otherwise present derivation; it is the total absence of the derivation. Treating every part of the manuscript as in-scope evidence only confirms the mismatch: the body's own abstract states its goal is the search for the critical point in strongly interacting matter. The claim is therefore unsupported at the level of the whole submission.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The abstract claims that the paper derives formulas for the Fourier coefficients of |f|^2 for f(z1,z2)=(1-(z1+z2)/r)^{-alpha}, expresses them in terms of hypergeometric functions, and uses them to provide additional counterexamples to the weak Shanks conjecture recently disproven by Bénéteau, Khavinson and Seco. The submitted full text, however, is an unrelated NA61/SHINE experimental heavy-ion physics report (arXiv:2508.15939v2 [nucl-ex]) on the search for the critical point in strongly interacting matter. None of the claimed mathematics appears anywhere in the body: there is no definition of f, no Fourier coefficient formula, no hypergeometric function, no statement of the weak Shanks conjecture or the Bénéteau-Khavinson-Seco counterexample criterion, and no numerical counterexamples. The central claim is therefore entirely unsupported by the submitted text.","tokens_in":6311,"tokens_out":2116,"duration_ms":24658,"significance":"If the abstract's claims were backed by a complete derivation and a verified parameter search, the paper could contribute meaningfully to the recent work on the weak Shanks conjecture, particularly by supplying a computable family of functions and explicit counterexample parameters. The paper offers no such support. It contains no equations, no convergence conditions, no proofs, no code, and no reproducible counterexample tables. As submitted, the significance cannot be assessed beyond the abstract's assertion, and the asserted result is unverified and unverifiable from the supplied manuscript.","major_comments":[{"comment":"The body of the submission is an unrelated NA61/SHINE experimental physics report. It contains no definition of f(z1,z2), no Fourier coefficient formulas, no hypergeometric functions, no mention of the weak Shanks conjecture or of Bénéteau, Khavinson and Seco, and no counterexample parameters. The abstract's central claim is thus completely absent from the manuscript. This is not a derivation gap that can be filled by local revision; the submission as it stands contains none of the claimed mathematics.","section":"Full text"},{"comment":"The abstract states that formulas are derived and that counterexamples are provided, but the full text is a different document with its own abstract and references (the header shows arXiv:2508.15939v2 [nucl-ex]). There are no equations in the body to check, no statements of theorems, and no numerical results. Every load-bearing element of the claimed paper is missing.","section":"Abstract vs. full text"},{"comment":"Even taking the abstract as the entire mathematical content, the parameter range for which the formulas and counterexamples are valid is not stated. For the function f(z1,z2)=(1-(z1+z2)/r)^{-alpha} to be defined and integrable on the unit torus, conditions such as r > 2 and suitable restrictions on alpha are needed; the formulas for Fourier coefficients or their use in a weak-Shanks counterexample criterion require such hypotheses. Without these, no counterexample can be certified.","section":"Parameter regime and convergence conditions (abstract)"},{"comment":"The abstract says the formulas 'allow for (numerical) optimization over the parameters alpha and r' and that the paper provides 'additional counterexamples', but no counterexamples, parameter values, coefficient formulas, or verification procedures are actually presented. A claim of counterexamples to a conjecture requires at least the specific functions and a check of the criterion; none appears.","section":"Numerical counterexamples (abstract)"}],"minor_comments":[{"comment":"The submitted manuscript is labeled with the NA61/SHINE experimental paper ID (arXiv:2508.15939v2 [nucl-ex]) and an entirely different set of authors and title. This is a clear mismatch with the abstract and with the arXiv listing for 2508.15938.","section":"Full text header"},{"comment":"The title promises a Fourier analysis proof/disproof of the weak Shanks conjecture, but the document's content is about heavy-ion collision measurements. The title does not match the body.","section":"Title and content"}],"recommendation":"reject","confidential_remarks":"The submitted document appears to be a different paper than the one described in the abstract. Even setting that aside, the abstract alone provides no derivation, no theorem statement, no convergence conditions, and no specific counterexample data. The central mathematical claim is therefore entirely unsupported in the manuscript as submitted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe short version: this submission does not contain the paper it claims to be. The abstract announces explicit hypergeometric formulas for Fourier coefficients of |f|^2 and new counterexamples to the weak Shanks conjecture. The body is an NA61/SHINE heavy-ion experimental report on the search for the QCD critical point. Different topic, different arXiv number embedded in the header. There is nothing here to referee on the merits.\n\nTo give credit where it is earned: the abstract itself is plausible, and the claimed formulas could be a real technical contribution if a derivation were attached. The BKS work already disproved the weak Shanks conjecture, so “additional counterexamples” is an extension, not a headline result. The possible value lies in the explicit formulas and the parameter optimization they enable. But none of that is present. No definition of f beyond the abstract, no Fourier coefficient formula, no hypergeometric function, no statement of the BKS criterion, no parameter regime (e.g., r > 2 for boundedness on the torus), no convergence conditions, no numerical search. The only equation in the body is the kaon ratio R_K.\n\nThis is not a subtle derivation gap; it is total absence of the derivation. The stress-test note is accurate: every part of the manuscript confirms the mismatch. The body’s own abstract says the main goal is the critical point in strongly interacting matter. Under the rule that all manuscript text is in-scope evidence, there is zero support for the central claim. If the correct file was uploaded, that is an honest mistake, but a referee cannot fix it by reading more carefully.\n\nThe reader’s scores track my own: soundness is near zero because there are no equations to check; novelty and significance are conditional on the abstract being true. I would reject as submitted. If the authors resubmit the actual mathematics paper, then it deserves a serious look, because explicit formulas for this family would be useful and the BKS connection gives it context. But this version should be desk-rejected.\n\nReading group? No, there is nothing to chew on. Cite it? No.\n\nMy recommendation: reject without external review, with a clear message that the correct manuscript would be welcome.","headline":"Abstract promises Fourier-analysis counterexamples to the weak Shanks conjecture, but the attached text is an unrelated heavy-ion experiment; the submission is unassessable.","tokens_in":6862,"tokens_out":2752,"would_cite":false,"duration_ms":26215,"reading_group":"no","serious_thinker":"unclear","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper aims to derive hypergeometric formulas for the Fourier coefficients of |f|^2 and use them to find new counterexamples to the weak Shanks conjecture.","keywords":["weak Shanks conjecture","Fourier coefficients","hypergeometric functions","two-variable rational functions","counterexamples","|f|^2","parameter optimization","complex analysis"],"falsifier":"Take a specific parameter pair that the paper says is a counterexample and compare the advertised hypergeometric value of the relevant Fourier coefficient with direct high-precision numerical integration of |f|^2 over the unit torus. If they disagree, the formula fails. Also check the parameter regime: for positive alpha and r < 2, f is not bounded on the torus, so any claimed counterexample there would not satisfy the conjecture's hypotheses.","tokens_in":5919,"feed_emoji":"📐","tokens_out":6777,"duration_ms":74436,"temperature":0.7,"pith_summary":"The paper's announced goal is to establish explicit hypergeometric formulas for the Fourier coefficients of |f|^2, where f(z1,z2)=(1-(z1+z2)/r)^(-alpha), and then use those formulas to produce additional counterexamples to the weak Shanks conjecture. That conjecture concerns how Taylor coefficients of rational functions in two complex variables behave, and it was recently shown to fail. If the formulas hold, coefficient checking becomes a closed-form computational task: one can scan alpha and r numerically and locate parameter values where the conjecture's predicted coefficient inequalities break. A sympathetic reading takes the derivation to be the contribution: it turns a hard coefficient-ratio problem into explicit special-function expressions. The attached body, however, is a different text about heavy-ion collisions, so the announced derivation is not present in the submitted manuscript and cannot currently be checked.","feed_headline":"Hypergeometric formulas yield new weak Shanks counterexamples","feed_subtitle":"Explicit coefficients for a two-variable power family would let anyone scan alpha and r for violations.","key_machinery":"The central object is the two-variable function f(z1,z2)=(1-(z1+z2)/r)^(-alpha) and the Fourier coefficients of its squared modulus on the unit torus. The advertised machinery is a hypergeometric-function representation of those coefficients, which is supposed to turn a check of the weak Shanks conjecture into numerical evaluation and optimization over alpha and r. Without that representation, the counterexample search would require high-dimensional coefficient computations with no closed form.","core_discovery":"On the paper's own terms, the central discovery is that the Fourier coefficients of |f|^2 for f(z1,z2)=(1-(z1+z2)/r)^(-alpha) are not just computable in principle but expressible in closed form in terms of hypergeometric functions. That converts a coefficient-ratio conjecture about two-variable rational functions into a parameterized special-function problem in alpha and r. The paper says this yields new counterexamples to the weak Shanks conjecture, supplementing the recent disproof. In the version of the paper supplied here, none of these formulas, parameter values, or counterexamples appear; the body is an unrelated experimental physics report. The claim is therefore a stated result witho","pith_inferences":["My inference: if the formulas are valid for the boundedness regime r > 2 and an integrability range of alpha, they likely connect these coefficients to known orthogonal polynomials or Appell-type functions, which could give a uniform derivation for a wider class of rational functions than the single power family.","My inference: the absence of the derivation from the body means the advertised counterexamples cannot be reproduced from the submitted text; a reader would need the full derivation or a corrected manuscript before treating them as established.","My inference: the same coefficient formulas might be used to test stronger ratio conjectures, such as monotonicity or log-concavity of coefficient sequences, not just the weak Shanks failure."],"forward_implications":["The hypergeometric formulas, if valid, make |f|^2 coefficient computation for this family exact, so counterexample searches do not rely on numerical Fourier integration.","They give a continuous two-parameter family (alpha, r) in which to look for weak Shanks violations, going beyond isolated examples.","Each new violation found this way strengthens the recent disproof of the weak Shanks conjecture.","The same formulas may allow researchers to test nearby coefficient-ratio conjectures on the same family with little extra work.","Because the formulas are differentiable in the parameters, one can optimize to find the most extreme violations."],"supporting_citations":[],"fun_headline_variants":["New hypergeometric formulas add weak Shanks counterexamples","Fourier coefficients via hypergeometrics hit weak Shanks conjecture","Closed-form coefficients yield more weak Shanks counterexamples","Weak Shanks disproved further by explicit two-variable formulas","Hypergeometric analysis expands weak Shanks counterexample set"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The load-bearing premise is that violation of the weak Shanks conjecture for this family reduces to a checkable inequality on the Fourier coefficients of |f|^2 and that the hypergeometric formulas are valid for the r and alpha values used; the submitted body contains no derivation, so this bridge is currently unverified.","fun_headline_variants_meta":{"raw":{"variants":["New hypergeometric formulas add weak Shanks counterexamples","Fourier coefficients via hypergeometrics hit weak Shanks conjecture","Closed-form coefficients yield more weak Shanks counterexamples","Weak Shanks disproved further by explicit two-variable formulas","Hypergeometric analysis expands weak Shanks counterexample set"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000377,"raw_usage":{"total_tokens":1775,"prompt_tokens":605,"completion_tokens":1170,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":349,"completion_tokens_details":{"reasoning_tokens":1087}},"tokens_in":349,"tokens_out":1170,"duration_ms":12524,"temperature":1.0,"reasoning_tokens":1087,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T17:38:41.127417+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a specific parameter pair that the paper says is a counterexample and compare the advertised hypergeometric value of the relevant Fourier coefficient with direct high-precision numerical integration of |f|^2 over the unit torus. If they disagree, the formula fails. Also check the parameter regime: for positive alpha and r < 2, f is not bounded on the torus, so any claimed counterexample there would not satisfy the conjecture's hypotheses.","supporting_citations":[],"review_version":1}