{"id":"0f906486-7f7f-46b5-a8e1-726ada46b2fc","arxiv_id":"2508.11108","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Using calculus of variations to optimize linear combinations of zeta derivatives, the paper claims positive proportions of critical-line zeros for arbitrarily short mollifiers and doubled proportions for modular L-functions.","lead":"This number theory paper claims a new variational construction that finds a positive proportion of zeros of the Riemann zeta function on the critical line, even when the mollifier is arbitrarily short. The abstract also reports that the same construction more than doubles known proportions for modular L-functions, using the same arithmetic inputs as prior work.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Uniform positivity of the variational weight as mollifier length →0 is unverified; without it, 'regardless of how short' is unsupported.","rationale":"The reader's weakest_assumption focuses on the validity of arithmetic moment inputs at very short mollifier lengths. I agree that this is central, but the more immediate structural concern is the uniform positivity of the variational solution across all mollifier lengths—a condition that must hold for the 'regardless of how short' claim and that is not established by anything in the received text. The reader correctly assigns UNVERDICTED because no equations, proof, or numerical data are available; the present stress-test does not detect a proven error, but it identifies a specific, load-bearing condition whose verification would settle the claim. The text-integrity mismatch between the abstract (math.NT) and the full text (a CS paper on code diffusion) reinforces the need for the actual manuscript but is not itself an argument against the mathematical claim. Given the absence of auditable content, the honest verdict remains UNVERDICTED rather than ACCEPT or REJECT. I partially agree with the reader because the concern about arithmetic inputs is closely related, but I emphasize the uniformity of the variational construction as the precise point that must be checked first.","tokens_in":4389,"tokens_out":2779,"duration_ms":36360,"concrete_test":"Obtain the actual mathematics manuscript (the current submission mismatches the abstract) and: (1) write down the variational problem and the resulting Euler–Lagrange equations; (2) for a one-parameter family of mollifiers of length L, solve these equations numerically for L = 10^{-1}, 10^{-2}, ..., 10^{-6} and evaluate the Levinson integral I(L). If I(L) is not strictly positive and bounded below by a constant independent of L, the headline claim fails. If the paper already contains a theorem bounding I(L) below, verify that the proof's error terms are uniform in L and that the arithmetic inputs are proven rather than conjectural for the short-mollifier regime.","verdict_should_be":"UNVERDICTED","load_bearing_attack":"The central claim requires that Levinson's method, with the new variational linear combination, yields a fixed positive proportion of zeros uniformly as the mollifier length tends to 0. In Levinson's framework, the recovered proportion is controlled by a positive functional of the mollifier; for very short mollifiers, standard choices make this functional tend to 0. The abstract asserts this is avoided by optimizing the linear combination, but no equations are provided in the submission to check the variational problem, its Euler–Lagrange solution, or the resulting lower bound. The key technical step must be showing that the optimal weight stays positive and yields an integral bounded below independent of the mollifier length. If the optimizer degenerates (e.g., becomes sign-changing or oscillatory) or if the moment expansions used have error terms that are not uniform for arbitrarily short mollifiers, the proportion may still vanish. The claim about modular L-functions 'more than doubles' adds a further quantitative dependence on the same arithmetic moment asymptotics, which are often only asymptotic for long enough mollifiers; their extension to the short regime is precisely where the risk lies. Thus, the load-bearing assumption—positive, uniform, sufficiently precise arithmetic main terms and a controlled variational solution—is entirely unauditable from the material received.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The submission consists of an abstract announcing a calculus-of-variations construction of linear combinations of derivatives of the Riemann zeta-function, adapted to Levinson's method, with the claimed conclusion that a positive proportion of zeros lies on the critical line 'regardless of how short the mollifier is.' It further claims the construction extends to modular L-functions, more than doubling previously known proportions while using the same arithmetic inputs as Bernard and Kühn--Robles--Zeindler, and that the resulting linear combinations provide nontrivial smooth approximations of Siegel's f-function. The body of the received manuscript, however, is an unrelated paper on code-diffusion models for last-mile program repair. No equations, definitions, theorem statements, moment asymptotics, or proofs pertaining to the abstract are present.","tokens_in":4520,"tokens_out":2860,"duration_ms":36830,"significance":"If the abstract's claims were substantiated, the result would be substantial: it would overcome a known difficulty in Levinson's method by showing that optimizing the linear combination, rather than lengthening the mollifier, can preserve a positive proportion of critical-line zeros, and it would give a concrete quantitative improvement for modular L-functions. However, the submission as received contains none of the mathematical content needed to evaluate these claims. It is effectively an abstract-only submission. The uniform 'regardless of how short' claim is historically delicate, and the modular L-function claim depends on arithmetic moment inputs whose validity at short mollifier lengths must be checked. No evidence is supplied for any of these points.","major_comments":[{"comment":"The received manuscript contains no mathematical development. There is no definition of the variational problem, no functional being optimized, no Euler--Lagrange equation, no choice of linear combination, no statement of the moment asymptotics used, and no theorem with an error term. The body of the paper is an unrelated document on code-diffusion repair (Sections 1--4 discuss diffusion models and program repair, not zeta-functions). It is impossible to audit the central claim from this material. A journal submission must contain the actual derivation or a precise pointer to it.","section":"Abstract / Full text"},{"comment":"The quantifier 'regardless of how short the mollifier is' is load-bearing. In Levinson's framework, standard short mollifiers make the recovered proportion tend to zero because the controlling functional degenerates. To support the claimed uniformity, the paper must show that the optimized linear combination has a nondegenerate, uniformly positive weight as the mollifier length tends to zero, and that the relevant moment expansions are valid with errors uniform in that regime. None of these steps appears in the received text.","section":"Abstract, first sentence"},{"comment":"The claim that the construction 'more than doubles' the proportions for modular L-functions while 'relying on the same arithmetic inputs' as Bernard and Kühn--Robles--Zeindler requires specification of the families, the relevant moments, and whether the needed asymptotics are proven or conjectural at the short-mollifier lengths used. If a moment main term is incorrect or only conjectural for the family in question, the derived proportion does not follow. No such specification is present.","section":"Abstract, modular L-function claim"}],"minor_comments":[{"comment":"The connection to Siegel's f-function is announced but not developed. If this is a substantive observation, a precise statement and explanation are needed; otherwise it should be removed or marked as a remark.","section":"Abstract, final sentence"}],"recommendation":"reject","confidential_remarks":"The submitted file appears to be a metadata or submission error: the body is a different paper on code-diffusion repair. I recommend that the editor verify the correct manuscript before any further review. Based on the received file, the mathematical content is restricted to an abstract, and the claims are not assessable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know one thing first: the manuscript body is not the paper in the abstract. arXiv 2508.11108 is listed as a math.NT paper by Conrey, Farmer, Kwan, Lin, and Turnage-Butterbaugh, and the abstract is a clear, strong claim about short mollifiers and positive proportions of critical-line zeros. But the full text is a NeurIPS submission by Singh et al. on code diffusion models. That is not a missing appendix or a typo; it is a different paper, with different authors, different notation, and different subject. That alone kills any chance of refereeing this version.\n\nWhat the abstract actually proposes is interesting. The idea of optimizing the linear combination of derivatives inside Levinson's method, rather than the mollifier length, is a sensible conceptual shift. The reported more-than-doubling of known proportions for modular L-functions, if correct, would be a real quantitative improvement, and the connection to Siegel's f-function is the kind of byproduct that makes a paper worth reading. The authors are not obviously wrong from the abstract alone.\n\nBut the soft spots are not small. The stress-test note is on target: the claim \"regardless of how short the mollifier is\" needs a uniform lower bound on the Levinson functional as the mollifier length tends to zero, and the abstract gives no equation, no Euler-Lagrange statement, no error term, nothing to check. Standard short-mollifier choices make that functional tend to zero, so the whole result hangs on an optimizer that stays positive and yields a bounded-below integral. That is a plausible mechanism, but it is unverifiable here. The modular L-function doubling also inherits the same arithmetic moment assumptions, which are usually only established for sufficiently long mollifiers; extending them to the arbitrarily short regime is exactly where the risk lies.\n\nStill, the dominant problem is the text mismatch. You cannot credit the math because the math is not present. You cannot even tell whether the abstract is a ghost or the body is a paste error. A serious editor would desk-reject this submission and ask for the correct full text. If the correct manuscript shows up and matches the abstract, the work may well deserve careful refereeing. But this version does not.","headline":"The abstract promises a major Levinson-method result, but the full text is a completely different computer science paper, so nothing here is auditable.","tokens_in":5134,"tokens_out":2155,"would_cite":false,"duration_ms":27365,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M06","11M26","11M41"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that a calculus-of-variations choice of linear combinations of derivatives of the Riemann zeta function yields a positive proportion of zeros on the critical line even when the mollifier is arbitrarily short, and that the s","keywords":["Riemann zeta function","critical line","Levinson's method","mollifier","calculus of variations","modular L-functions","Siegel's f-function","zeros of zeta"],"falsifier":"Work out the variational equations at zero mollifier length; if the predicted positive limiting proportion is not actually achieved when plugging the known moment main terms into Levinson's formula, or if a numerical check of a specific modular L-function gives a proportion at or below the previous bound, the claim would be falsified. Concretely: compute the paper's proportion for the first few weight and level families for which the arithmetic inputs are known and compare with the doubled bound.","tokens_in":4182,"feed_emoji":"🧮","tokens_out":7089,"duration_ms":67705,"temperature":0.7,"pith_summary":"Levinson's method proves that a positive proportion of the nontrivial zeros of the Riemann zeta function lie on the critical line by mollifying the function with a short Dirichlet polynomial; classically, the proved proportion shrinks as the mollifier gets shorter. This paper shows that if one replaces the usual mollifier by an optimized linear combination of derivatives of zeta—chosen through the calculus of variations—the proportion need not go to zero, no matter how short the mollifier is. The same construction applies to modular L-functions, and with Levinson's original mollifier it more than doubles the previously known zero proportions for those functions. The paper attributes the gain to optimizing the linear combination rather than to refining the mollifier, and notes a surprising connection to Siegel's $\\mathfrak{f}$-function in the Riemann–Siegel formula.","feed_headline":"Positive share of zeta zeros survives arbitrarily short mollifiers","feed_subtitle":"A variational choice of derivative combinations keeps the recovered zero share positive as the mollifier shrinks.","key_machinery":"The central object is a sequence of linear combinations of the derivatives $\\zeta^{(k)}(s)$ (and similarly for the L-functions), with coefficients determined by a calculus-of-variations optimization. This combination plays the role of Levinson's mollifier; its behavior under the relevant moments is what Levinson's method needs. The variational principle selects coefficients that make the mollified function's mean square and related averages have the right positivity, and it is this optimization that sustains a positive zero proportion even when the mollifier is arbitrarily short.","core_discovery":"The central claim is that there exists a sequence of short mollifiers, built from linear combinations of derivatives of $\\zeta$, for which Levinson's method recovers a positive proportion of critical-line zeros uniformly as the mollifier length tends to zero. The coefficients of the linear combination are chosen as the solution of a variational problem, and the paper argues that this optimization, not the shape of the mollifier, is what keeps the proportion bounded away from zero. For modular $L$-functions, the same construction gives proportions that more than double the earlier results of Bernard and Kühn–Robles–Zeindler while using the same arithmetic moment inputs. The paper also observe","pith_inferences":["A natural test is whether the same variational optimization can extend to other automorphic L-functions once analogues of the required moment asymptotics are known; if the only input is those moments, the method might generalize broadly.","The link to Siegel's $\\mathfrak{f}$-function suggests that the optimized combinations might be interpreted as approximate functional equations, which could be used to build numerical test functions for locating zeros.","If the positive-proportion claim is robust, it may imply that the obstacle to extending Levinson's method to longer ranges is not the mollifier length but the accuracy of the moment expansions; the variational viewpoint gives a way to measure how much precision is needed.","Because the stated proportions for modular L-functions depend on the same arithmetic inputs as previous work, an independent verification of the doubling would settle the method's merit without relying on the variational details."],"forward_implications":["If correct, Levinson-type methods no longer suffer from the usual 'short mollifier' limitation: a positive share of critical-line zeros can be established without needing a long mollifier.","The same construction applies to modular L-functions, doubling previously known critical-line zero proportions while relying only on the established arithmetic input (moment asymptotics).","The result redirects attention from mollifier design to optimizing the linear combinations of derivatives, a comparatively neglected component of Levinson's method.","The connection to Siegel's $\\mathfrak{f}$-function provides a new analytic handle on the Riemann–Siegel formula, potentially linking the variational construction to classical approximations of $\\zeta$."],"supporting_citations":[],"fun_headline_variants":["Variational derivatives keep zeta zeros even with short mollifiers","Short mollifiers still yield positive zero share via optimized derivatives","Optimized derivative mix doubles L-function zero proportions","Zeta zeros survive arbitrarily short mollifiers with variational twist","Derivative combination beats mollifier shape for zeta zeros"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"The conclusion depends on the moment asymptotics for the zeta function and for the modular L-functions holding at very short mollifier lengths with the precision that the variational calculation requires; if any of those main terms is wrong or merely conjectural for the families used, the derived proportions would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Variational derivatives keep zeta zeros even with short mollifiers","Short mollifiers still yield positive zero share via optimized derivatives","Optimized derivative mix doubles L-function zero proportions","Zeta zeros survive arbitrarily short mollifiers with variational twist","Derivative combination beats mollifier shape for zeta zeros"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000151,"raw_usage":{"total_tokens":999,"prompt_tokens":669,"completion_tokens":330,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":413,"completion_tokens_details":{"reasoning_tokens":250}},"tokens_in":413,"tokens_out":330,"duration_ms":4097,"temperature":1.0,"reasoning_tokens":250,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T20:08:54.280122+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Work out the variational equations at zero mollifier length; if the predicted positive limiting proportion is not actually achieved when plugging the known moment main terms into Levinson's formula, or if a numerical check of a specific modular L-function gives a proportion at or below the previous bound, the claim would be falsified. Concretely: compute the paper's proportion for the first few weight and level families for which the arithmetic inputs are known and compare with the doubled bound.","supporting_citations":[],"review_version":1}