{"id":"50bee7e9-0878-4549-85e0-6bc0a14db38a","arxiv_id":"2607.19192","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Intervals maximize the L²-concentration of band-limited functions: for every measurable set E, the maximal Paley–Wiener mass on E is no greater than on an interval with |I| = |E|, settling the Donoho–Stark conjecture.","lead":"Band-limited signals can concentrate a larger fraction of their total energy on a single interval than on any other measurable set of the same size. This proves a 1989 conjecture by Donoho and Stark in full generality, giving a sharp, restriction-free bound for the classic spectral-concentration problem that anchors signal recovery and time-frequency analysis.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.2 (M=N) is the load-bearing step; no flaw found under the corrected Hermitian D_N, but the printed c_{-k}=c_k makes (9) false and must be fixed.","rationale":"Reader and I agree on the weakest step. I independently rechecked the main calculations (Lemma 3.2, endpoint velocities, Hessian identity, negative second variation, transfer estimates) and found no mathematical error beyond the D_N coefficient typo, which is evidently intended as c_{-k}=overline{c_k}. Thus the central claim is not undermined as far as the written argument goes, but the paper should not be accepted without correcting the definition and adding independent verification of Lemma 3.2 given its intricacy and lack of formal verification. This is the same CONDITIONAL posture as the reader.","tokens_in":15835,"tokens_out":29202,"duration_ms":279910,"concrete_test":"Formalize Lemma 3.2 in a proof assistant (e.g. Lean), proving: (i) g±∈D_N under c_{-k}=overline{c_k}; (ii) L_μ(g±)=m0; (iii) the touching factor creates an isolated even-order zero, so Z(g±)>Z(g). If the formalization succeeds, the load-bearing step is confirmed; if it fails, the gap is real. In parallel, rerun the variational identity (9) with the corrected Hermitian definition and verify that |1+i e^{it}|^2 is admitted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"To prove Theorem 1.1 it is necessary that an extremal density saturate its zero count: Lemma 3.2. If M<N, the construction g±=w(η±ε*ψ) would yield two global minimizers with one more zero, forcing η constant and q∈V_N. I checked the perturbation: nonnegativity, degree, normalization and, with the intended condition c_{-k}=overline{c_k}, membership in D_N all hold; the touching factor cannot be identically zero because ψ has a zero and η>0, and it creates an even-order zero, so Z increases. The argument is sound as far as I can see. The genuine defect is textual: D_N is printed with c_{-k}=c_k, which excludes generic densities (e.g. |1+i e^{it}|^2), so eq. (9) is false as printed; replacing with c_{-k}=overline{c_k} restores it. Residual risk therefore concentrates in this intricate lemma, not in any detected counterexample.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that, among all measurable sets of fixed finite measure, intervals maximize the L^2-concentration of band-limited functions in the Paley–Wiener space PW(Ω). This is Donoho–Stark's 1989 conjecture, previously known only under restrictions on the time–bandwidth product |Ω||E|. The proof proceeds in two stages: first a circle-level theorem (Theorem 2.1) asserting that intervals maximize concentration for analytic trigonometric polynomials of degree N; second a universality-type limit that transfers the circle result to the real line by approximating the Paley–Wiener sinc kernel by midpoint Riemann sums of polynomial kernels on expanding circles. The circle theorem is obtained through a variational problem for the lower-tail functional L_μ on normalized nonnegative trigonometric polynomials, an intricate zero-saturation lemma (Lemma 3.2) showing that an extremal density must have the maximal number of roots, and a negative second-variation argument for relative translations of root blocks.","tokens_in":15874,"tokens_out":13731,"duration_ms":125910,"significance":"If correct, the result resolves a 35-year-old conjecture in full generality and removes the time–bandwidth restriction that has persisted since Donoho–Stark. The circle theorem is also a substantial result of independent interest. A notable strength of the paper is that the proof is self-contained and parameter-free: the only external inputs are classical (Fejér–Riesz factorization, the bathtub principle, the implicit function theorem), and the final contradiction is an explicit negative sum of squares. The rigorous kernel approximation in Section 9 is carefully quantified and does not rely on the heuristic universality formula quoted in the introduction. The manuscript is unusually transparent about the provenance of its ideas, including the role of AI-assisted exploration, which does not affect the mathematical content.","major_comments":[{"comment":"The set D_N is defined with the condition c_{-k}=c_k. For a real-valued nonnegative trigonometric polynomial, however, the Fourier coefficients satisfy c_{-k}=\\overline{c_k}; the stronger condition c_{-k}=c_k excludes legitimate densities such as |1+i e^{it}|^2, whose coefficients are c_1=i, c_{-1}=-i. As printed, eq. (9), Λ_N(μ)=min_{g∈D_N} L_μ(g), is therefore false because the minimum is taken over a proper subset of the normalized densities obtained from Fejér–Riesz factorization. This is a load-bearing definition: Lemma 2.4, Lemma 3.2, and the identification of D_N with normalized energy densities all rely on it. The fix is local and does not affect the structure of the proof: replace c_{-k}=c_k by c_{-k}=\\overline{c_k} (equivalently, state explicitly that g is real-valued). With this correction, the compactness, concavity, and zero-saturation arguments remain valid. The authors sho","section":"§2.2, definition of D_N and eq. (9)"}],"minor_comments":[{"comment":"The symbol M_n in Δ_n = 1/M_n is undefined; from the subsequent midpoint Riemann sum construction one needs Δ_n = 1/(2n+1) = 1/L_n. Please define M_n or replace it by L_n.","section":"§9, eq. (49)"},{"comment":"The displayed limit 1/(2N+1) K_N(e^{iπt/N}, e^{iπξ/N}) = sinπ(t−ξ)/π(t−ξ) is not correct as written, since with w=e^{iπξ/N} the exponent contains t+ξ. The intended statement requires the conjugate variable, e.g. K_N(e^{iπt/N}, e^{-iπξ/N}), or an equivalent convention. The rigorous Step 2 of Section 9 uses a correctly defined midpoint Riemann sum, so this is not a gap in the proof, but the motivational formula should be corrected.","section":"Introduction, universality display"},{"comment":"The phrase “real trigonometric polynomial” is used ambiguously: it sometimes means real-valued and sometimes seems to imply real Fourier coefficients. Given the correction to D_N, the authors should consistently say “real-valued” (or “Hermitian-symmetric coefficients”) to avoid confusion.","section":"§3.1–3.2"},{"comment":"The sentence about the preliminary version “still keeps part of the interesting graphic language used by the Language Model” is informal and should be removed or rewritten in standard mathematical prose.","section":"§1, final paragraph"}],"recommendation":"minor_revision","confidential_remarks":"The mathematical argument appears sound after the D_N definition is corrected to c_{-k}=\\overline{c_k}; I checked the central identities (§3.2, §7.2, §8.2) and found no hidden gap. The remaining issues are typographical/notational. The AI-usage disclosure is transparent and does not affect the assessment. I recommend minor revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a serious paper, possibly a major one. The main theorem removes every restriction on |Ω||E| in the Donoho–Stark concentration problem. I read the proof carefully, and the structure is genuinely new: a sharp concentration theorem for analytic trigonometric polynomials on the circle, then a controlled expanding-circle limit that transfers the result to the line. The transfer is clean; the Riemann-sum estimate and the Hilbert–Schmidt bound are routine and correct. The circle theorem itself is a real result, not a stepping stone you could have gotten from prior work.\n\nThe variational engine is the most interesting part. The zero-saturation lemma (Lemma 3.2) is the load-bearing wall. I checked the two-sided perturbation argument, the concavity, the maximal-zero selection, and the contradiction. I could not find a gap. The one-hill lemma, the endpoint-velocity formulas, and the Hessian with the boundary covariance all check out. The algebraic identity in Section 8 is exact; the translation-invariant direction is handled correctly. There are no fitted constants, no hidden parameters, and the existing partial results are cited as benchmarks, not used as inputs. That is real evidence of honesty.\n\nSoft spots. First, a genuine textual error: the definition of D_N in §2.2 prints c_{−k} = c_k. That makes eq. (9) false, as the counterexample |1 + i e^{it}|^2 shows. The intended condition is c_{−k} = \\overline{c_k}. This is fixable, but it must be fixed; as printed, a reader cannot verify the claimed equivalence with normalized energy densities. Second, Lemma 3.2 is intricate enough that I would want an independent referee to walk through it line by line. The failure mode would be a subtle sign error in the perturbation or an overlooked case in the touching argument. I didn't find one, but the lemma is doing a lot of work. Third, the AI disclosure is unusually transparent, and the retained \"graphic language\" remnant should be edited out. That is a presentation issue, not a mathematical one, but it does raise the prior on subtle errors — which makes the needed referee scrutiny more important, not less.\n\nWho is this for? Harmonic analysts, people working on spectral concentration, uncertainty principles, and shape optimization for localization operators. If the proof is correct, it closes a conjecture from 1989 that has resisted the rearrangement and isoperimetric methods. The paper deserves a serious referee. I would send it to a strong journal and specifically ask referees to attack Lemma 3.2 and the D_N definition. With the typo fixed and the AI remnants cleaned, I would be comfortable citing it.","headline":"If the proof holds, this settles the Donoho–Stark interval conjecture in full generality; the circle theorem is new and the variational core is sound — but the printed D_N definition has a real typo and Lemma 3.2 deserves an independent check.","tokens_in":16587,"tokens_out":1549,"would_cite":true,"duration_ms":18909,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42A38","42C05","46E22"],"pacs":[],"model":"deepseek-v4-flash","headline":"Intervals maximize the L²-concentration of band-limited functions on the real line, for any measurable set of fixed measure.","keywords":["Paley-Wiener space","band-limited functions","concentration inequalities","optimal intervals","analytic trigonometric polynomials","reproducing kernel","universality limit","uncertainty principles"],"falsifier":"For some N≥1, take a measurable E⊂T of measure 0<µ<2π, compute the largest eigenvalue of the concentration operator for analytic polynomials of degree N on E, and compare with the eigenvalue for the circle interval of length µ. A single non-interval E with larger eigenvalue would falsify the circle theorem and, through the limit, the main theorem. A numerical search for small N and µ could look for such a counterexample.","tokens_in":15529,"feed_emoji":"📐","tokens_out":4850,"duration_ms":42529,"temperature":0.7,"pith_summary":"The paper proves that among all measurable sets E of a given finite measure, a plain interval I of the same measure captures the largest possible fraction of the L² mass of any band-limited function. This settles a conjecture from 1989 that previously had been proved only under small time-bandwidth restrictions. The proof works by first showing the analogue for analytic trigonometric polynomials on the circle — intervals are optimal there too — and then transferring the result to the real line via a universality-type limit of the polynomial reproducing kernel to the sinc kernel. If correct, it gives a complete answer to the classic concentration problem in Paley–Wiener space and sharpens quantitative uncertainty principles.","feed_headline":"Intervals maximize band-limited concentration on the line","feed_subtitle":"Full answer to a 1989 conjecture: any measurable set concentrates no more L²-mass than the interval of the same length.","key_machinery":"The universality-type identity lim_{N→∞} (1/(2N+1)) K_N(e^{iπt/N}, e^{iπξ/N}) = sin π(t−ξ)/(π(t−ξ)) for the reproducing kernel K_N(z,w)=Σ_{j=−N}^{N} (z w)^j of analytic trigonometric polynomials. This identity is used twice: it motivates the circle theorem and it supplies a controlled limit (midpoint Riemann sums of the sinc kernel) that pushes the circle result to the real line. The circle theorem itself rests on a zero-saturation lemma forcing an extremal density to consume all N available zeros, the one-hill lemma describing lower level sets, and an exact negative second-variation computation for relative translations of root blocks.","core_discovery":"On the circle, for analytic trigonometric polynomials of degree at most N, the paper establishes that the largest eigenvalue of the concentration operator is maximized by a circle interval (Theorem 2.1). The key structural fact is that an extremal density must be the square of a real amplitude with N zeros, so its lower-level set has exactly one component; any configuration with two or more root blocks can be perturbed by relative translations, yielding a negative second variation that contradicts local minimality. The real-line statement then follows by expanding circles: the Paley–Wiener sinc kernel is a uniform limit of the midpoint Riemann sums of the polynomial kernels, and the operator","pith_inferences":["The same circle-to-line transfer might apply to other reproducing kernel Hilbert spaces whose kernels admit universality limits, yielding interval-type optimizers in those spaces — the paper does not claim this.","The quantitative negative second variation suggests a stability statement: sets that nearly attain the maximal concentration must be close to intervals, but the paper does not derive such a bound.","For large degree N and later large circle radius, the construction may be viewed as an expanding-circle discretization of the classical prolate spheroidal problem; one could test numerically whether the extremal densities converge to the prolate spheroidal wave functions on intervals.","The proof's reliance on factorizing nonnegative trigonometric polynomials into squared analytic-modulus factors suggests that analogous concentration results for weighted polynomial spaces could be approached by the same zero-saturation framework."],"forward_implications":["Confirms the 1989 interval-extremal conjecture in full generality; previous restrictions on the time-bandwidth product are no longer needed.","Gives a sharp, computable bound: the best concentration on a set E is the concentration on the interval of the same measure, so interval constants are the universal benchmark.","Provides an independent optimal-concentration theorem for analytic trigonometric polynomials, useful as a finite-dimensional sharp form.","Establishes the universality-limit transfer as a proof technique for carrying extremal results from the circle to the real line in Paley–Wiener spaces."],"fun_headline_variants":["Intervals win for band-limited concentration","Optimal concentration: intervals rule for Paley-Wiener","No set beats an interval for band-limited focus","Band-limited mass: intervals maximize concentration","Paley-Wiener: intervals are concentration champs"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The entire proof depends on the claim that a global minimizer of the lower-tail functional must use every one of its N zero degrees of freedom — if a minimizer could leave even one degree unused, the subsequent one-component and negative-second-variation argument would not go through.","fun_headline_variants_meta":{"raw":{"variants":["Intervals win for band-limited concentration","Optimal concentration: intervals rule for Paley-Wiener","No set beats an interval for band-limited focus","Band-limited mass: intervals maximize concentration","Paley-Wiener: intervals are concentration champs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00071,"raw_usage":{"total_tokens":3011,"prompt_tokens":703,"completion_tokens":2308,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":447,"completion_tokens_details":{"reasoning_tokens":2246}},"tokens_in":447,"tokens_out":2308,"duration_ms":14534,"temperature":1.0,"reasoning_tokens":2246,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T13:14:53.450198+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For some N≥1, take a measurable E⊂T of measure 0<µ<2π, compute the largest eigenvalue of the concentration operator for analytic polynomials of degree N on E, and compare with the eigenvalue for the circle interval of length µ. A single non-interval E with larger eigenvalue would falsify the circle theorem and, through the limit, the main theorem. A numerical search for small N and µ could look for such a counterexample.","supporting_citations":[],"review_version":1}