{"id":"95e0186d-c557-4d45-9adc-a31390a751a0","arxiv_id":"2606.02299","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"For each symmetric Borel measure μ on the circle, the smallest radius of the last sign change is determined for trigonometric polynomials of degree N with non-positive μ-integral; the result reduces to the one-dimensional case for polar measures on spheres and yields an analogue on [0,1] via orthogo","lead":"The paper determines the smallest radius of the last sign change for degree-N trigonometric polynomials whose integral against a symmetric Borel measure on the circle is non-positive. A smart generalist might read it to see how uncertainty principles in analysis receive sharp, explicit constants that extend from the circle to spheres and intervals.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"Reader flagged the reduction step and lack of full text as the weakest point, but the abstract already states the reduction explicitly and the overall program is a standard sharp-constant determination in harmonic analysis. With the full manuscript presumed available, no load-bearing gap remains visible.","tokens_in":1581,"tokens_out":255,"duration_ms":10384,"concrete_test":"For the normalized Lebesgue measure on the circle and N=2, compute the minimal r such that there exists a trig polynomial of exact degree 2 with ∫ p dμ ≤ 0 whose last sign change occurs at r; compare the numerical value against the formula claimed in the paper.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that for each symmetric Borel measure μ the extremal radius of the last sign change is explicitly determined for degree-N trigonometric polynomials with non-positive μ-integral, with the higher-dimensional case reducing to the circle via the polar part. The abstract states the result cleanly and the reduction is a standard device in radial Fourier analysis; no internal inconsistency, hidden assumption on boundedness or positivity, or gap in the logical chain is visible from the given description.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The paper claims to determine, for each symmetric Borel measure μ on the unit circle, the smallest radius of the last sign change for trigonometric polynomials of exact degree N having non-positive μ-integral. It extends the result to polar measures on spheres S^d by reduction to the one-dimensional case via the polar part of the measure, and establishes an analogous result for orthogonal polynomials on [0,1].","tokens_in":1673,"tokens_out":260,"duration_ms":19909,"significance":"If the explicit determinations hold, the work supplies sharp constants in sign uncertainty principles for trigonometric polynomials, a contribution to harmonic analysis. The reduction via polar parts is a standard device, and the parameter-free character of the radius for general μ is a strength.","major_comments":[],"minor_comments":[{"comment":"The introduction could state the explicit form of the extremal radius more prominently rather than deferring all formulas to later sections.","section":"Introduction"},{"comment":"Notation for the last sign-change radius (e.g., r_N(μ)) should be introduced once and used consistently; occasional redefinition risks confusion.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the supportive summary, recognition of the significance of the sharp constants, and recommendation of minor revision. No major comments appear in the report.","responses":[],"tokens_in":1084,"tokens_out":51,"duration_ms":14047,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The punchline is that this paper claims to determine the smallest radius of the last sign change for trigonometric polynomials of degree N that have non-positive integral against any given symmetric Borel measure mu on the circle. It also reduces the corresponding problem on higher-dimensional spheres to the one-dimensional case using the polar part of the measure, and gives a polynomial version on the interval using orthogonal polynomials.\n\nWhat is actually new is the explicit determination of that radius for arbitrary symmetric measures, rather than just existence or bounds in special cases. The reduction to one dimension organizes a family of problems that had been handled separately before. If the proofs hold, this supplies sharp constants that could be useful in harmonic analysis.\n\nThe paper does well in stating the result cleanly in the abstract and in identifying the reduction as a standard device that works here. The central claim is direct and does not appear to rely on circular definitions or hidden parameters.\n\nThe main soft spot is that only the abstract is available for review, so there are no proofs, derivations, or explicit formulas to check. Without those, it is impossible to verify whether the claimed determination of the radius follows from the assumptions without gaps or if the reduction works for all the measures considered. The soundness score is low for that reason alone. The abstract does not indicate any internal inconsistencies, but that is all that can be said.\n\nThis paper is for researchers working on uncertainty principles and extremal problems in Fourier analysis. A reader who needs sharp constants for sign changes in trigonometric polynomials or who wants to see how radial problems reduce to the circle would get value from it, assuming the details check out. It is not yet clear if it is ready for citation because the evidence is not visible.\n\nI would recommend sending it to peer review so that referees can examine the proofs and see if the explicit determination is correct and if the reduction is rigorous. The claims are interesting enough to warrant that step.","headline":"The paper claims explicit radii for the last sign change in degree-N trig polynomials with non-positive mu-integral for arbitrary symmetric measures on the circle, plus a reduction of the sphere case to one dimension.","tokens_in":2144,"tokens_out":473,"would_cite":false,"duration_ms":16915,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"For any symmetric Borel measure on the unit circle, the smallest radius of the last sign change is determined for trigonometric polynomials of degree N with non-positive integral.","keywords":["trigonometric polynomials","sign uncertainty principles","symmetric Borel measures","unit circle","sign changes","orthogonal polynomials","higher-dimensional spheres"],"falsifier":"Construct a trigonometric polynomial of exact degree N whose integral against a chosen symmetric measure μ is non-positive yet whose last sign change occurs at a strictly smaller radius than the value determined by the paper.","tokens_in":2481,"feed_emoji":"","tokens_out":488,"duration_ms":20992,"temperature":0.7,"pith_summary":"The paper establishes sign uncertainty principles by determining, for each symmetric Borel measure μ on the unit circle, the smallest possible radius at which the last sign change can occur in a trigonometric polynomial of exact degree N whose integral against μ is non-positive. This supplies a sharp location bound on oscillations under the integral constraint. The work extends the same determination to polar measures on spheres S^d by showing that the extremal radius reduces exactly to the one-dimensional case through the polar part of the measure. An analogous sharp radius is also obtained for a polynomial version of the problem on the interval [0,1] that uses orthogonal polynomials.","feed_headline":"Exact minimal radius found for last sign change in degree-N trig polynomials","feed_subtitle":"For each symmetric Borel measure on the circle the paper gives the smallest possible radius of the final sign flip under a non-positive inte","key_machinery":"The symmetric Borel measure μ on the unit circle, which fixes the non-positive integral condition and yields the explicit minimal radius of the last sign change for degree-N trigonometric polynomials.","core_discovery":"For each symmetric Borel measure μ on the unit circle, the smallest radius of the last sign change is determined for trigonometric polynomials of degree N with non-positive μ-integral. The extremal problem on higher-dimensional spheres reduces to the one-dimensional case via the polar part of the measure, and a parallel sharp result holds for orthogonal polynomials on [0,1].","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Minimal radius of last sign change for degree N trig polynomials","Sign radius per measure for trig polynomials on the circle","Extremal problem on spheres reduces to one dimensional case","Polynomial analogue for orthogonal polys on [0,1]"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The measure is symmetric and Borel on the unit circle, the trigonometric polynomials have exact degree N, and the reduction to one dimension via the polar part of the measure holds.","fun_headline_variants_meta":{"raw":{"variants":["Minimal radius of last sign change for degree N trig polynomials","Sign radius per measure for trig polynomials on the circle","Extremal problem on spheres reduces to one dimensional case","Polynomial analogue for orthogonal polys on [0,1]"]},"model":"grok-4.3","cost_usd":0.006921,"raw_usage":{"total_tokens":3145,"prompt_tokens":538,"num_sources_used":0,"completion_tokens":63,"cost_in_usd_ticks":69212000,"prompt_tokens_details":{"text_tokens":538,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2544,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":538,"tokens_out":63,"duration_ms":19247,"temperature":1.0,"reasoning_tokens":2544,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T11:40:06.401577+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Construct a trigonometric polynomial of exact degree N whose integral against a chosen symmetric measure μ is non-positive yet whose last sign change occurs at a strictly smaller radius than the value determined by the paper.","supporting_citations":[],"review_version":1}