{"id":"441bcd38-6931-4196-a7ac-7e5f16c815ab","arxiv_id":"2507.05777","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Compact smooth curved surfaces with nonzero Gaussian curvature, including hemispheres and self-intersecting curves, admit no Fourier frames.","lead":"This paper proves that many curved surfaces, including hemispheres and self-intersecting curves with nonzero curvature, cannot carry a Fourier frame. It settles an open question about the boundary case between small caps, which do admit frames, and full spheres, which do not.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Key divergence/convergence implications are imported from [6] without proof; the contradiction collapses if they fail, but the cited result supports them.","rationale":"I read the paper in good faith and checked the main proof chain: Theorem 1.1 first derives divergence of Σ|λ|^{-(d-1)} from the lower frame bound and the upper decay (via the Section 2 observation), then derives convergence when Λ⊂C_S' using a partition of unity and stationary phase (3.4)–(3.6). The stationary phase uniformity over the compact chart is justified because the Gaussian curvature is bounded away from zero and h_p depends smoothly on p. The compactness argument reducing to S'=S and the covering of all normal directions are correct. Theorem 1.3's symmetry identity (2.5) is correct and turns the full-sphere lower bound (1.2) into a lower bound for σ+. The only step I could not verify from the text alone is the Section 2 observation; this is the reader's weakest assumption and mine. I do not see a counterexample to it, and it is attributed to a published paper, so it does not change the accept verdict. I also noted that the identity \\widehat{ψdσS}=\\widehat{ψdσi} near self-intersections is informal, but for transverse crossings the extra branch contributes only rapidly decaying terms, and for tangential crossings the branch contributions do not cancel in the curve case, so this appears patchable and is not the main risk.","tokens_in":6759,"tokens_out":52097,"duration_ms":652198,"concrete_test":"Independently re-derive the two implications stated in Section 2 from the published proof of [6, Thms 1.3–1.4], keeping only the exponential-family inequalities. In particular, verify that the convergence implication for Theorem 1.3 is valid for μ=σ+ using only the integrated lower bound (1.2) (since |hat σ+| is not pointwise bounded below, cf. (1.8)). If the proof requires a pointwise lower Fourier bound or another regularity condition absent for σ+, the endpoint theorem fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing step is the observation in Section 2: for a measure satisfying the upper Fourier decay (1.1), the lower frame bound tested only on the exponentials e^{2πix·ξ} is claimed to force divergence of Σ_{λ≠0}|λ|^{-(d-1)}, while the lower integrated estimate (1.2) together with the upper frame bound tested on the same exponentials is claimed to force convergence. This observation is used in the proof of Theorem 1.1 (divergence part) and in both halves of Theorem 1.3. The paper cites [6, Thms 1.3–1.4] but does not prove the exponential-family version, and the measures considered here (ψdσ+ and σ+) have features not present for the full sphere, such as the boundary term (1.8) giving slower decay along the axis. If either implication is false or requires an extra hypothesis (e.g., a pointwise lower Fourier bound), the central contradiction collapses. This is a verifiability gap rather than an identified error: the implications are standard and likely correct.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Fourier frames for surface-carried measures on smooth (d-1)-dimensional surfaces with nonvanishing Gaussian curvature. Theorem 1.1 shows that any frame spectrum for such a surface measure must intersect the complement of the normal set of every compactly contained submanifold; Corollary 1.2 concludes that compact immersed smooth submanifolds with nonvanishing Gaussian curvature admit no Fourier frames, covering self-intersecting examples beyond the convex-body case. Theorem 1.3 settles the endpoint case for hemispheres of centrally symmetric convex bodies, answering a question of Kolountzakis and Lai. The proofs use stationary phase to obtain upper Fourier decay and a local pointwise lower bound, and import from [6] an observation that the frame inequalities, tested only on the exponential family, force divergence or convergence of the series sum |lambda|^{-(d-1)}.","tokens_in":6962,"tokens_out":9853,"duration_ms":125173,"significance":"If the imported observation is supplied as a complete argument, the results are substantial: they generalize the main theorem of [6] from boundaries of convex bodies to immersed surfaces, improve the planar result of [7] from tight frames to general frames, and resolve the previously open hemisphere endpoint. The local argument in Section 3 is elegant and avoids the difficult exponential-sum integral that a direct approach would require. The paper is clearly written and includes explicit examples showing that the new class genuinely extends the convex setting. The main caveat is that the Section 2 observation is load-bearing and is stated without proof; this is a verifiability gap rather than an identified error.","major_comments":[{"comment":"The assertions that (1.1) together with the exponential-family lower inequality 1 <~ sum_lambda |hat sigma(lambda - xi)|^2 for all xi implies the divergence (2.2), and that (1.2) together with the exponential-family upper inequality implies the convergence (2.4), are stated as an observation but not proved. These assertions are load-bearing: they are used for the divergence in Theorem 1.1 and for both halves of Theorem 1.3. Please add a proof or a precise lemma statement with hypotheses, or quote the exact theorem in [6] that contains this formulation. In particular, the measures considered here, such as psi dsigma_+ and sigma_+, have features not present for the full sphere (for instance, the boundary-axis decay (1.8) for d >= 4), so it is not immediate that the hypotheses of the cited theorems in [6] hold verbatim.","section":"Section 2, Eqs. (2.1)-(2.4)"},{"comment":"The proof asserts that for each lambda in C_ij \\ {0} there is a unique p_lambda in supp psi_ij with lambda as a normal. For immersed or self-intersecting surfaces this uniqueness is not automatic and needs justification; if two points in the same small support had the same normal, the pointwise stationary phase lower bound (3.5)-(3.6) would not follow as stated. Please add a short argument using nonvanishing Gaussian curvature and the choice of the Lebesgue number to show that the Gauss map is injective on each support, or restrict the supports further so that this is explicit.","section":"Section 3, Eq. (3.4)"}],"minor_comments":[{"comment":"The word 'indclude' in the abstract should read 'include'.","section":"Abstract"},{"comment":"The phrase 'does not take use of all f' should be 'does not make use of all f'.","section":"Section 2, first paragraph"},{"comment":"The phrase 'successfully avoid dealing' should be 'successfully avoids dealing' or 'successfully avoids the need to deal'.","section":"Section 3, final paragraph"},{"comment":"The editor's name is given as 'Laba and Carol Shubin'; it should be 'Izabella Laba and Carol Shubin'.","section":"Reference [13]"},{"comment":"The stationary phase expansion gives the lower bound |widehat{psi_ij dsigma_i}(lambda)|^2 >~ |lambda|^{-(d-1)} only for sufficiently large |lambda|, not for all lambda in C_ij \\ {0} as written. Please add a sentence explaining that the finitely many small elements of the discrete spectrum Lambda can be discarded or absorbed, so that the asymptotic lower bound is enough for the convergence argument.","section":"Eq. (3.4)"}],"recommendation":"major_revision","confidential_remarks":"The main reason for major revision is the unproved observation in Section 2, which is central to both theorems. Because that observation is attributed to [6], a paper co-authored by the second author, I recommend that the editor ensure the authors supply a complete proof or a precise theorem reference. The mathematical idea appears sound and the paper is otherwise well written, so this is a fixable gap rather than a fundamental flaw."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper genuinely advances the subject. Theorem 1.3 settles the hemisphere question asked by Kolountzakis and Lai, and Corollary 1.2 extends the no-frame theorem from [6] on convex bodies to all compact immersed surfaces with nonvanishing curvature, including self-intersecting curves. It also improves [7] from tight frames to frames. The local stationary-phase argument in Section 3 is a genuine simplification: it sidesteps the global exponential sums that made the earlier approach heavy.\n\nI read the proofs carefully. The Morse-lemma reduction and the uniform stationary-phase estimate are standard and appear correct. The symmetry identity (2.5) is clean and does what it needs to: it transfers the lower bound (1.2) from the full surface to the half-surface, and the bump-function trick handles the boundary issue. I found no mathematical errors in the parts that are proved in the paper.\n\nThe real soft spot is the observation in Section 2. The paper needs a pair of implications from [6]: upper decay plus a lower frame bound tested only on the exponential family forces divergence of the sum, and the integrated lower bound plus an upper frame bound on exponentials forces convergence. The authors state this as an observation and do not prove it. The rest of the paper rests entirely on it, and the measures here (ψ dσ+ and σ+) have features, like the slower boundary decay in (1.8), that are not present for the whole sphere. I believe the implications are true—the exponentials are the only test functions used in [6] anyway—but a referee should verify this by checking the proofs in [6] or asking the authors to write out the argument. This is a verifiability gap, not an identified error.\n\nMinor issues: the abstract has typos (\"indclude\"), and the last remark is a bit hand-wavy about why the old integral approach would be complicated, but that is harmless.\n\nWho should read this: anyone working on Fourier frames for singular measures or exponential bases for surface measures. It deserves a serious referee. My recommendation: send it to peer review. The referee should ask the authors to make the Section 2 dependency explicit, ideally by giving a self-contained proof or stating the exact theorem from [6] that covers the exponential-family version.","headline":"A genuinely new endpoint result for Fourier frames on curved surfaces, but the proof leans on an unproved observation from earlier work that needs referee scrutiny.","tokens_in":7448,"tokens_out":8586,"would_cite":true,"duration_ms":105686,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42C15","42B10"],"pacs":[],"model":"deepseek-v4-flash","headline":"A compact smooth surface with nonvanishing Gaussian curvature admits no Fourier frame.","keywords":["Fourier frames","surface measure","Gaussian curvature","frame spectrum","stationary phase","spherical cap","hemisphere","normal directions"],"falsifier":"Exhibit a compact smooth surface with nonvanishing Gaussian curvature and a discrete set $\\Lambda$ with finite $\\sum_{\\lambda\\ne0}|\\lambda|^{-(d-1)}$ for which both frame bounds hold, and Theorem 1.1 and Corollary 1.2 would be false; more directly, test the dichotomy in [6] on a simple measure with a known frame and see whether the predicted divergence and convergence conclusions still follow.","tokens_in":6573,"feed_emoji":"","tokens_out":7578,"duration_ms":76735,"temperature":0.7,"pith_summary":"This paper proves that compact smooth surfaces in Euclidean space with nonvanishing Gaussian curvature never admit Fourier frames, even when the surface is self-intersecting or is not the boundary of a convex body. The core of the proof is a contradiction between two consequences that any frame spectrum would force: one Fourier estimate makes a certain frequency series diverge, while another makes the same series converge. The result also settles an endpoint question from [7] by showing that a hemisphere admits no Fourier frame, placing the exact threshold between small caps, which do admit frames by [9], and larger caps. Along the way the paper shows that a spherical cap near the north pole cannot have a frame spectrum near the $x_d$-axis.","feed_headline":"Curved compact surfaces have no Fourier frames","feed_subtitle":"Any frame spectrum would have to escape the surface's normal directions in every patch, which is impossible for a compact surface.","key_machinery":"The argument rests on a divergence-convergence dichotomy extracted from the earlier analysis in [6]. For a surface-carried measure $\\mu$, an upper Fourier decay bound $|\\widehat{\\psi d\\mu}(\\xi)|\\lesssim |\\xi|^{-(d-1)/2}$ together with the lower frame inequality tested only on the exponentials $\\{e^{2\\pi i x\\cdot\\xi}\\}_{\\xi\\in\\mathbb{R}^d}$ forces the series $\\sum_{\\lambda\\in\\Lambda\\setminus\\{0\\}}|\\lambda|^{-(d-1)}$ to diverge. A lower Fourier estimate $\\int_{B_1(\\xi)}|\\hat\\mu(\\eta)|^2\\,d\\eta\\gtrsim |\\xi|^{-(d-1)}$ together with the upper frame inequality tested on the same exponentials forces the same series to converge. The paper establishes both estimates on the relevant surface measures: stationary phase supplies the upper decay, and a newly localized pointwise estimate $|\\widehat{\\psi_{ij}\\,d\\sigma_i}(\\lambda)|^2\\gtrsim|\\lambda|^{-(d-1)}$ holds for $\\lambda$ along normal directions in small patches, avoiding the need to integrate over neighborhoods or to control sums of phases from multiple normal points. The divergence and convergence conclusions contradict each other.","core_discovery":"Let $S$ be a compact $(d-1)$-dimensional smooth submanifold immersed in $\\mathbb{R}^d$ with nonvanishing Gaussian curvature, with surface measure $\\sigma_S$ normalized by its multiplicity function. Theorem 1.1 states that if $\\sigma_S$ admits a frame spectrum $\\Lambda$, then for every submanifold $S'$ compactly contained in $S$, the set $\\Lambda$ must meet the complement of $C_{S'}$, the union of the one-dimensional normal subspaces of $S'$. For compact $S$ one may take $S'=S$, and because every direction is normal to $S$ at some point, $C_S=\\mathbb{R}^d$; hence $\\Lambda$ would have to contain a point outside all of $\\mathbb{R}^d$, which is impossible. Therefore $\\sigma_S$ admits no Fourier frame. Two consequences are proved for the sphere: a small cap near the north pole cannot have a frame spectrum near the $x_d$-axis, and any cap whose interior contains a closed hemisphere admits no Fourier frame at all. The endpoint case of a hemisphere is proved separately for the restricted surface measure $\\sigma_+$ on $S\\cap\\{x_d\\ge 0\\}$ when $S$ is the smooth boundary of a centrally symmetric convex body.","pith_inferences":["The pointwise lower bound along normal directions means the proof never has to compare phases from several surface points sharing the same normal, so the same strategy may extend to immersed surfaces with many-to-one normal maps, provided one point dominates in each small chart.","Compactness is used only to conclude $C_S=\\mathbb{R}^d$; for a noncompact surface whose normal set is a proper cone, the theorem would predict that any frame spectrum must have frequencies outside that cone, a statement one could test directly on model surfaces.","The reduction to exponentials suggests a broader sufficient obstruction: a measure whose Fourier transform decays like $|\\xi|^{-s}$ and whose local average energy is at least $|\\xi|^{-2s}$ cannot carry a frame spectrum. This could be checked against other singular measures with known Fourier decay, such as self-similar measures.","Because the hemisphere proof works for any centrally symmetric smooth convex boundary, the endpoint threshold is not a special sphere phenomenon; testing non-centrally symmetric caps whose boundary is only $C^2$ would indicate how much symmetry the argument really needs."],"forward_implications":["The self-intersecting planar curve $(\\cos\\theta+2\\cos2\\theta,\\sin\\theta+\\sin2\\theta)$ and its revolution surfaces in $\\mathbb{R}^d$ admit no Fourier frame, even though they are not convex and their outward boundaries are not $C^3$.","For the sphere, a frame spectrum of a small cap near the north pole must avoid a neighborhood of the $x_d$-axis, quantifying the obstruction behind the known small-cap frame construction.","A spherical cap whose interior contains a closed hemisphere admits no Fourier frame; combined with the positive result for caps compactly contained in a hemisphere, the hemisphere is the exact threshold.","The hemisphere itself admits no Fourier frame, answering the question posed at the end of [7] and extending the no-frame theorem beyond the whole sphere.","The two-sided frequency-series test gives a reusable criterion: proving nonexistence of Fourier frames for a curved measure reduces to checking one upper and one lower Fourier estimate against the exponential family."],"supporting_citations":[{"why":"It supplies the divergence-convergence dichotomy on which the contradiction rests, and it proves the no-frame theorem for convex boundaries that Corollary 1.2 extends.","marker":"[6]"},{"why":"It gives the Herz formula for Fourier transforms on convex boundaries, the prior class of surfaces to which the stationary-phase estimates applied.","marker":"[5]"},{"why":"It provides the tight-frame-only theorem for piecewise smooth curves and poses the hemisphere question that Theorem 1.3 answers.","marker":"[7]"},{"why":"It establishes the positive small-spherical-cap frame result whose threshold the paper sharpens.","marker":"[9]"},{"why":"It shows every set of finite Lebesgue measure admits Fourier frames, marking why the singular surface-carried measures are the delicate case.","marker":"[11]"},{"why":"It supplies the uniform stationary-phase estimate used for the upper Fourier decay of $\\psi d\\sigma_+$ and $\\psi d\\sigma_S$.","marker":"[12]"},{"why":"It supplies the Morse-lemma normalization and stationary-phase asymptotics used for the pointwise lower bound in normal directions.","marker":"[13]"}],"fun_headline_variants":["Curved compact surfaces reject Fourier frames","No Fourier frames on curved compact surfaces","Nonvanishing curvature forbids Fourier frames","Fourier frames impossible on curved compact manifolds","Curved surfaces have no frame spectra"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on the dichotomy from [6] that frame inequalities tested only on exponentials force the series $\\sum|\\lambda|^{-(d-1)}$ to diverge under the upper Fourier decay and to converge under the lower Fourier estimate; if that dichotomy fails for these surface measures, the contradiction collapses.","fun_headline_variants_meta":{"raw":{"variants":["Curved compact surfaces reject Fourier frames","No Fourier frames on curved compact surfaces","Nonvanishing curvature forbids Fourier frames","Fourier frames impossible on curved compact manifolds","Curved surfaces have no frame spectra"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000168,"raw_usage":{"total_tokens":1289,"prompt_tokens":1003,"completion_tokens":286,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":619,"completion_tokens_details":{"reasoning_tokens":221}},"tokens_in":619,"tokens_out":286,"duration_ms":2972,"temperature":1.0,"reasoning_tokens":221,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:21:56.990207+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a compact smooth surface with nonvanishing Gaussian curvature and a discrete set $\\Lambda$ with finite $\\sum_{\\lambda\\ne0}|\\lambda|^{-(d-1)}$ for which both frame bounds hold, and Theorem 1.1 and Corollary 1.2 would be false; more directly, test the dichotomy in [6] on a simple measure with a known frame and see whether the predicted divergence and convergence conclusions still follow.","supporting_citations":[{"cited_title":"Iosevich, C.-K","cited_arxiv_id":null,"evidence_quote":"It supplies the divergence-convergence dichotomy on which the contradiction rests, and it proves the no-frame theorem for convex boundaries that Corollary 1.2 extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It gives the Herz formula for Fourier transforms on convex boundaries, the prior class of surfaces to which the stationary-phase estimates applied."},{"cited_title":"Non-spectrality of some piecewise smooth curves and unions of line segments","cited_arxiv_id":"2507.00581","evidence_quote":"It provides the tight-frame-only theorem for piecewise smooth curves and poses the hemisphere question that Theorem 1.3 answers."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It establishes the positive small-spherical-cap frame result whose threshold the paper sharpens."},{"cited_title":"Nitzan, A","cited_arxiv_id":null,"evidence_quote":"It shows every set of finite Lebesgue measure admits Fourier frames, marking why the singular surface-carried measures are the delicate case."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the uniform stationary-phase estimate used for the upper Fourier decay of $\\psi d\\sigma_+$ and $\\psi d\\sigma_S$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the Morse-lemma normalization and stationary-phase asymptotics used for the pointwise lower bound in normal directions."}],"review_version":1}