{"id":"58d5cedb-4df8-4701-b9ad-695b297a3d62","arxiv_id":"2608.22578","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"The radial extremal function for the sharp Nikolskii constant factors into two entire functions of exponential type 1/2 satisfying explicit differential equations, reducing the problem to a spectral problem in any dimension.","lead":"This paper derives differential equations and a factorization for the extremal function behind the sharp Nikolskii constant in L1 Paley-Wiener spaces on R^d. It gives a unified spectral approach for every dimension and computes the constant for d=2.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof is internally consistent, but the entire chain from the extremal problem to (16) depends on imported unproven facts about φ: identity (3) and τ_k ∼ π k from [2], [6].","rationale":"The reader identified the same weak spot, so I agree. I do not find a new internal flaw: the shifts via σ̃, the cancellation in §3.4, the defect argument in §3.5–3.6, and the degree/leading-coefficient determination in §4.5–4.6 are mutually consistent. The d = 2 recurrences check out, and the claimed type 1/2 follows from the coefficient asymptotics. The numerical values and Section 7.4 are ancillary to Theorem 1 and do not change the verdict. Because the central theorem's correctness is conditional on the imported extremal-function profile, the reader's CONDITIONAL verdict stands; no adjustment is needed.","tokens_in":15479,"tokens_out":29222,"duration_ms":269196,"concrete_test":"Review [6] to determine whether identity (3)/(4) and the asymptotics τ_k ∼ π k are proved for arbitrary d ≥ 1 and not only odd d. If they are, the dependency is benign. If not, supply a self-contained derivation of (3) for at least d = 2 (e.g., by checking first-order optimality of the computed φ) and recompute the leading zero-density used in §4.5; if the derivation fails, Theorem 1 must be downgraded to conditional on an unverified premise.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The weakest load-bearing point is not the internal algebra but the imported profile of the extremal function. Sections 1.1–1.2 take from [2]/[6] that (1) has a unique radial minimizer φ that is entire of exact exponential type 1, bounded by 1, has simple real zeros τ_k with τ_k ∼ π k, and satisfies the variational identity (3) and its Fourier consequence (4). Every subsequent object—σ, ν, M, R_d, K_d, (16), (18)—is derived from these premises. In particular, (3)/(4) is the only link from the original minimizer to the distribution identity that creates the polynomial P_d at (8); if (3) fails, no spectral problem exists. The zero asymptotics τ_k ∼ π k is used essentially in §4.5–4.6 to fix the degree and leading coefficient 1/4 of K_d, and in §3.6 to select a separated subsequence for the Plancherel–Pólya argument. The paper does not re-derive these facts, and [6] is a self-cited preprint; if [6]'s proof of (3) or of the zero profile is restricted to odd d, the even-dimensional theorem is not established. This is an external-dependency concern, not an internal contradiction: given (2)–(4), the subsequent proof is detailed and coherent.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the radial extremal function φ for the sharp Nikolskii constant C_d in PW_1^1(R^d). It claims Theorem 1: φ factors as Φ1Φ2, with Φ1 and Φ2 satisfying the functional identity (16) and the second-order differential equations (17)-(18); this reduces the original extremal problem to a one-dimensional spectral problem depending on at most d+1 parameters. The paper also derives a zeta interpretation of the coefficients of R_d, sign properties and a zero equilibrium condition, and it computes the case d=2 numerically. The main proof proceeds by replacing the weight |x|^{d-1} with x^{d-1}σ̃(x), forming the Cauchy transform M of ν=σ̃′, constructing Θ_d with one-sided Fourier support, proving the defect H_d vanishes by a Plancherel-Pólya argument, and then extracting the differential equations by zero-location and coefficient comparison.","tokens_in":15701,"tokens_out":10006,"duration_ms":96428,"significance":"If Theorem 1 is established, it is a substantial advance: it unifies the odd- and even-dimensional treatments, gives a concrete spectral route to the sharp constants, and provides new structural information about the extremal function (factorization, a third-order equation, and a zero equilibrium condition). The internal algebra is detailed and largely checkable: the cancellation of the defect in §3.5-3.7, the Plancherel-Pólya step, and the degree/leading-coefficient argument in §4.5-4.6 are coherent. The explicit d=2 computation is a useful concrete demonstration. The central weakness is not an internal contradiction but the paper's reliance on imported, unproved profile and extremality facts for φ, so the true scope of the theorem is conditional on the status of those inputs.","major_comments":[{"comment":"The proof of Theorem 1 is conditional on the variational extremality condition (3), the canonical product (2), and the zero asymptotics τ_k∼πk, none of which are proved in this manuscript. These facts are taken from [2] and [6], and (3) is the only step connecting the minimization problem (1) to the distribution identity (4) that generates P_d and hence the entire spectral construction. Since the manuscript does not state these inputs as explicit hypotheses of Theorem 1, the theorem as stated is not self-contained. The paper must either prove these facts, or state them as assumptions, or give precise statements of the theorems in [6] that establish them. In particular, because [6] is titled for odd dimensions, the manuscript must specify whether (3) and the zero profile are proved in [6] for all d; if they are only proved for odd d, then the even-dimensional cases, including the d=2 computation in Section 6, are not covered by Theorem 1 as stated.","section":"§1.1-1.2, Eq. (3)"},{"comment":"The proof that deg K_d=d+1 and that its leading coefficient is 1/4 depends on the zero-counting relation n_{u1}(X)=X/(2π)+o(X), which is imported from τ_k∼πk, and on Lemma 9, which is quoted from [6]. These are parts of the same external dependency identified above. If the zero asymptotics or Lemma 9 are not available for every d, the degree and leading coefficient of K_d are not established, and part (b) of Theorem 1 does not follow. The manuscript should state these imports precisely or move them into an appendix with proofs.","section":"§4.5-4.6, Eq. (56) and Lemma 9"}],"minor_comments":[{"comment":"The numerical values for a2, β, C2, and L*(2) are quoted at 120-digit precision, but no code, stopping criterion, or rigorous error bound is supplied; the use of Newton's method and Miller's algorithm is described qualitatively. A reproducibility appendix or a statement of the convergence criterion would strengthen the computational claim.","section":"§6, Table and numerical computation"},{"comment":"The zeta interpretation is presented as a consequence, but the key asymptotic expansion (70) is asserted with 'some integer L≥0' and the analytic continuation step is sketched rather than proved in detail. Since Section 7 is advertised as a main result in the introduction, the proof of (70) and of the continuation should be written out or the section should be labeled as a sketch.","section":"§7, Eq. (70)-(72)"},{"comment":"The asymptotic statements in Section 8.3 are not theorems of this paper: L*(d)=2^{-d}(1+o(1)) is announced for a separate paper, and (78) is supported only by numerical experiments. Please label these as conjectures or future work in the text.","section":"§8.3, Eq. (78)"},{"comment":"The graph in Section 8.3 has no axis labels or caption; as printed it is not self-contained. Please add a proper caption and axis labels.","section":"§8.3, figure"},{"comment":"The abstract says the spectral problem depends on 'at most d+1 parameters', while Section 1.6 gives d+1-ϵ parameters (d parameters when d is even). Please reconcile this discrepancy.","section":"Abstract and §1.6"},{"comment":"The sentence 'Thustheidentityisprovedforeventestfunctions' contains typographical spacing errors and should read 'Thus the identity is proved for even test functions.'","section":"Lemma 1 proof"}],"recommendation":"major_revision","confidential_remarks":"The main concern is the very heavy dependence on the author's own preprint [6], both for the profile of φ (including (3) and τ_k∼πk) and for several lemmas quoted verbatim as Lemma 4-7 and Lemma 9. If [6] is not accepted or does not cover even dimensions, the central theorem of the present paper is not established. This is a scope and self-containedness issue, not an accusation of circularity. The acknowledgment of AI assistance is unusual but not a scientific objection; I would, however, ask the editor to ensure the journal's policy on such acknowledgments is respected. The paper is otherwise a good fit for math.CA and, once the dependency issue is resolved, could be a valuable contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Glad you asked. The paper is a genuine step forward for the sharp Nikolskii constant. For d=2 it gives the first exact value L*(2)=0.282145..., and the sign-twisted signature trick (replacing sigma by sign(x)sigma) is an elegant way to unify odd and even dimensions. The factorization phi = Phi1*Phi2 and the reduction to a one-dimensional spectral problem with at most d+1 parameters is a real structural contribution. If it holds up, it gives an algorithmic route to constants in any dimension.\n\nThe proof of Theorem 1 is detailed and, on its own terms, coherent. The cancellation of the defect via the Plancherel-Polya inequality and the zero-counting argument pinning deg K_d = d+1 and leading coefficient 1/4 work. The d=2 numerical computation via Miller's algorithm and Newton at 120-digit precision looks plausible, though not certified. The zeta interpretation is a nice bonus.\n\nThe soft spot is the one the stress-test note flags: the entire construction is built from premises imported from [2] and especially [6], a self-cited preprint. Specifically, the variational identity (3) and the zero asymptotics tau_k ~ pi k are load-bearing. The paper says [6] covered the odd-dimensional case; it is not clear that the proof of those facts in [6] also covers even d. If it does not, the even-dimensional theorem is not yet established. This is not an internal contradiction given (2)-(4), but it is a real external dependency that a referee will need to chase.\n\nMinor points: the numerical value lacks an error bound, and the Section 7.4 argument for eta(0) in odd d has a limit-interchange that is sketched too quickly. Also no code is shipped, though the algorithm is described well enough to reproduce.\n\nOverall: worth sending to a serious referee. The referee should demand a precise statement of which facts come from [6], a proof or reference for the even-dimensional case, and a cleaner treatment of 7.4. If those are supplied, the paper is a solid contribution to approximation theory.","headline":"New even-dimensional machinery and the first d=2 value, but the proof leans on imported facts about the extremal function that still need checking.","tokens_in":16323,"tokens_out":2045,"would_cite":false,"duration_ms":18831,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":["41A17","41A44","30D15","34A30"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that the sharp Nikolskii constant in every dimension reduces to a one-dimensional spectral problem, via a factorization of the extremal function into two entire factors of half the exponential type.","keywords":["Nikolskii constant","Paley-Wiener space","extremal function","entire functions of exponential type","spectral problem","Cauchy transform","zeta function","equilibrium condition"],"falsifier":"Compute the zeros $\\tau_k$ for $d=2$ by an independent numerical method, form $\\Phi_1,\\Phi_2$ from (14), and evaluate identity (16) at a regular point such as $z=1.7$. Any difference between the two sides beyond round-off error would falsify the central claim.","tokens_in":1854,"feed_emoji":"📐","tokens_out":2362,"duration_ms":114943,"temperature":0.7,"pith_summary":"The paper targets the sharp Nikolskii constant $C_d$ for the Paley--Wiener class $\\mathrm{PW}_1^1(\\mathbb R^d)$ --- the largest possible ratio between the $L^1$ norm and the supremum norm of a bandlimited function in $d$ dimensions. It proves that the unique radial extremal function $\\varphi$ factorizes as $\\varphi=\\Phi_1\\Phi_2$, with $\\Phi_1,\\Phi_2$ entire functions of exponential type $1/2$ that satisfy an exact functional identity and a second-order differential system with polynomial coefficients. The polynomial data in that system are determined by at most $d+1$ parameters, so the original infinite-dimensional optimization collapses to a one-dimensional spectral problem for a single ordinary differential equation on the half-line. That matters because it makes the sharp constant, in principle, computable in every dimension. The paper demonstrates the computation for $d=2$, obtaining the normalized constant $L^*(2)=0.28214506418970912412\\ldots$.","feed_headline":"A factorization turns the Nikolskii constant into a spectral problem","feed_subtitle":"The radial extremal function splits into two half-type factors, so the sharp constant follows from a d+1-parameter ODE.","key_machinery":"The load-bearing mechanism is the Cauchy transform of the derivative of a modified sign function. One sets $\\tilde\\sigma(x)=\\mathrm{sign}(x)^{\\epsilon}\\sigma(x)$ (with $\\epsilon=0$ for odd $d$ and $1$ for even $d$) so that $|x|^{d-1}\\sigma=x^{d-1}\\tilde\\sigma$ is a monomial; then $\\nu=\\tilde\\sigma'$ is a discrete measure supported at the zeros $\\tau_k$. Its Cauchy transform $M(z)=\\mathrm{p.v.}\\int d\\nu(t)/(z-t)$ has boundary values whose Fourier transforms vanish on $(-1,1)$ after a carefully chosen polynomial correction $R_d$. The function $\\Theta_d(z)=z^dM(z)-2\\epsilon z^{d-1}+4R_d(z)/z$ therefore has a spectral gap, and the defect $H_d=\\varphi\\Theta_d-4a_d/z$, being entire with spectrum concentrated at the origin, is a polynomial; a classical sampling inequality for bandlimited functions forces it to vanish identically. That vanishing is exactly the functional identity (16), and differentiating it pins down the differential system.","core_discovery":"The central discovery is Theorem 1: for every $d\\ge1$, the radial extremal function $\\varphi$ of the extremal problem splits as $\\varphi=\\Phi_1\\Phi_2$, where the factors are entire of exponential type $1/2$ and, for all $z\\in\\mathbb C$, satisfy $z^{d+1}(\\Phi_1'(z)\\Phi_2(z)-\\Phi_1(z)\\Phi_2'(z))-2R_d(z)\\Phi_1(z)\\Phi_2(z)=-2a_d$, with $a_d=1/(2v_d C_d)$ and $R_d$ an even polynomial built from the distributional data of the sign of $\\varphi$. The same identity, differentiated, yields the second-order system (18) for $\\Phi_1,\\Phi_2$; after an exponential gauge change the system becomes a single one-dimensional equation $u''+V_d(x)u=0$ on the positive half-line, with a rational potential $V_d$ fixed by at most $d+1$ parameters. Thus the sharp constant is encoded in the spectral data of one ordinary differential equation, and the paper works out the new case $d=2$ numerically.","pith_inferences":["The author leaves implicit that the spectral reduction can be used to test the conjectured large-dimension asymptotics $L^*(d)=2^{-d}(1+o(1))$ and the empirical refinement $2^dL^*(d)=\\pi/2-cd^{-1/3}$ with $c\\approx0.777$ by solving equation (19) numerically across a range of $d$.","A natural testable extension is whether the same factorization and spectral gap appear when the weight $|x|^{d-1}$ is replaced by other monomial-compatible weights; the construction in Section 3 uses only the polynomial character of the weight after the signature trick.","The zeta formula $\\eta_{\\tau,d}(-d+2j)=r_j$ suggests that the dependence of the constant on $d$ might be studied by analytic continuation in the dimension, a route the paper does not itself pursue."],"forward_implications":["For every fixed $d$, the sharp constant $C_d$ is determined by the one-dimensional spectral problem (19); no search over the whole space $\\mathrm{PW}_1^1(\\mathbb R^d)$ remains.","The factors $\\Phi_1,\\Phi_2$ have exact exponential type $1/2$ and decay as $|x|^{-(d+1-\\epsilon)/2}$ and $|x|^{-(d+1+\\epsilon)/2}$, so the extremal function itself decays like $|x|^{-(d+1)}$.","The coefficients of $R_d$ are regularized alternating power sums over the zeros: $\\eta_{\\tau,d}(-d+2j)=r_j$, with $\\eta_{\\tau,d}(0)=1/2$.","The zero equilibrium condition $a_d=\\tau_k^d\\prod_{j\\ne k}|1-\\tau_k^2/\\tau_j^2|$ gives an explicit finite-dimensional algorithm for the zeros and hence for $C_d$, valid uniformly in $d$.","For $d=2$ the method yields the new numerical value $L^*(2)=2/a_2=0.28214506418970912412\\ldots$."],"supporting_citations":[{"why":"Establishes the d=1 case and the original zeta-function relation that the present work generalizes.","marker":"[1]"},{"why":"Supplies existence of the unique radial extremal function and the two-sided bounds on the normalized constant used as inherited input.","marker":"[2]"},{"why":"Supplies the backward-recurrence algorithm used in the d=2 numerical solution.","marker":"[3]"},{"why":"Gives the distribution identities for boundary values of 1/(z-t) used in the spectral-gap argument.","marker":"[4]"},{"why":"The companion odd-dimensional paper whose extremality framework and lemmas the present proof extends to arbitrary d.","marker":"[6]"}],"fun_headline_variants":["Factorization turns Nikolskii constant into a spectral problem","Arbitrary dimension Nikolskii constant via one ODE","Sharp Nikolskii in any dimension from a spectral split","Two half-type factors reduce Nikolskii to a spectral problem","A single ODE captures the Nikolskii constant for all d"],"cache_read_input_tokens":18304,"weakest_assumption_plain":"Everything rests on the previously established profile of the extremal function: it is the unique radial minimizer, entire of exact exponential type one, bounded by one on the real line, with simple positive zeros spaced roughly by $\\pi$, and it satisfies a prescribed balance condition after Fourier transformation; the present paper uses this input but does not prove it.","fun_headline_variants_meta":{"raw":{"variants":["Factorization turns Nikolskii constant into a spectral problem","Arbitrary dimension Nikolskii constant via one ODE","Sharp Nikolskii in any dimension from a spectral split","Two half-type factors reduce Nikolskii to a spectral problem","A single ODE captures the Nikolskii constant for all d"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001092,"raw_usage":{"total_tokens":4563,"prompt_tokens":951,"completion_tokens":3612,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":567,"completion_tokens_details":{"reasoning_tokens":3528}},"tokens_in":567,"tokens_out":3612,"duration_ms":23464,"temperature":1.0,"reasoning_tokens":3528,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-27T18:27:15.376203+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the zeros $\\tau_k$ for $d=2$ by an independent numerical method, form $\\Phi_1,\\Phi_2$ from (14), and evaluate identity (16) at a regular point such as $z=1.7$. Any difference between the two sides beyond round-off error would falsify the central claim.","supporting_citations":[{"cited_title":"(2026), To appear","cited_arxiv_id":null,"evidence_quote":"Establishes the d=1 case and the original zeta-function relation that the present work generalizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies existence of the unique radial extremal function and the two-sided bounds on the normalized constant used as inherited input."},{"cited_title":"Gautschi,Computational aspects of three-term recurrence relations, SIAM Rev.9(1967), 24–82","cited_arxiv_id":null,"evidence_quote":"Supplies the backward-recurrence algorithm used in the d=2 numerical solution."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the distribution identities for boundary values of 1/(z-t) used in the spectral-gap argument."},{"cited_title":"The Nikolskii constant in odd dimensions","cited_arxiv_id":"2608.15674","evidence_quote":"The companion odd-dimensional paper whose extremality framework and lemmas the present proof extends to arbitrary d."}],"review_version":1}