{"id":"4ae5cfc9-0ac2-4bb1-8fb4-4e88cd1c66a0","arxiv_id":"2608.08682","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For the Riemann xi-function, the Jensen polynomials J^{d,n} are hyperbolic whenever n^3 log^2(n+2) ≥ K d^5, and their scaled zeros converge to Wigner's semicircle law in this joint limit.","lead":"This paper proves that Jensen polynomials of Riemann's xi-function have all real negative zeros whenever the polynomial degree grows no faster than a power of the coefficient index, and that their scaled zeros converge to the semicircle law in that joint limit. It extends prior fixed-degree results to a genuine two-parameter asymptotic region related to the Riemann hypothesis.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 9.1's derivative-ratio proof relies on an operator identity, Eq. (86), that is algebraically false; for d=1 it asserts (D+1)y/B=0.","rationale":"I read the paper in good faith and tried to verify the chain from the comparison model to the derivative-ratio bounds. The reader's weakest_assumption is Proposition 4.1, the imported complex saddle expansion; that is a legitimate concern, but the argument has a more concrete internal problem. Lemma 9.1 is the sole proof that the critical-point derivative ratios satisfy (6), a hypothesis of Proposition 2.2. Its proof hinges on the exact factorization (86), which is stated as an identity on the _3F_2 polynomial pF. Expanding (E+D)JpF directly and using the hypergeometric equation yields an extra JpF term; the d=1 case makes the failure explicit. If (86) is false, recurrence (87) does not follow, the bound (89) is unsupported, and the proof of M≤1 collapses. The main theorems are then unproven. This is a fully internal correctness risk, not a disagreement with consensus, and it is checkable by symbolic computation. I therefore recommend changing the verdict from ACCEPT to REJECT, with the caveat that a corrected version could repair the gap if the factorization can be replaced by the correct identity and Lemma 9.1 reworked.","tokens_in":16925,"tokens_out":20337,"duration_ms":199299,"concrete_test":"Check Eq. (86) symbolically for d=2. Define pF by (69) with generic parameters A,B,C,D, set ε_p=(C−D)/C, and compute (E+D)JpF + (ε_p/A)E(E−d)(E+A)pF in a computer algebra system. The result will be JpF − (ε_p y/A)E(E−d)(E+A)pF + (ε_p/A)E(E−d)(E+A)pF, which is not identically zero. Even the d=1 case pF=1−y/B gives LHS=(D+1)y/B and RHS=0, settling the concern.","verdict_should_be":"REJECT","load_bearing_attack":"The decisive weakness is not the imported saddle expansion but an internal algebraic identity. In Section 9, after defining Jp := y(1−y/A)p'' + (B−y+(d−1)y/A)p' + dp and E=yd/dy, the paper claims that the _3F_2 equation for pF factors exactly as (E+D)JpF = −(ε_p/A)E(E−d)(E+A)pF, with ε_p=(C−D)/C. Expanding the left side and using the hypergeometric equation gives instead (E+D)JpF = JpF − (ε_p y/A)E(E−d)(E+A)pF. Thus the claimed identity would require JpF = −(ε_p/A)(1−y)E(E−d)(E+A)pF, which does not hold. A direct check with d=1: pF(y)=1−y/B, JpF=y/B, and E(E−1)(E+A)pF=0, so (86) would read (D+1)y/B=0, false for D>0. This identity is the basis for recurrence (87) and the backward-substitution bound (89); without it, the estimate M≤1 in Lemma 9.1 has no proof. Consequently the derivative-ratio hypothesis (6) of Proposition 2.2 is unverified, and both Theorem 1.1 and Theorem 1.2 lose their support.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves an unconditional hyperbolicity region for the Jensen polynomials J^{d,n} attached to Riemann's xi-function: there is an absolute constant K such that n^3 log^2(n+2) >= K d^5 implies that J^{d,n} has d distinct negative real zeros. The proof constructs a comparison polynomial from a Laguerre baseline, a first Jacobi correction matching the normalized coefficients R_0 through R_3, and a second finite-free convolution factor matching R_0 through R_4; a fifth-order multiplier stability proposition transfers real-rootedness to the Jensen polynomial. The same comparison model yields a joint semicircle limit for the naturally scaled zeros as n and d tend to infinity together in the wedge.","tokens_in":17145,"tokens_out":28264,"duration_ms":262099,"significance":"If correct, this is a substantial advance: it replaces exponential thresholds d = O(log n) with a polynomial wedge d <= c n^{3/5} log^{2/5}(n+2) and permits d and n to tend to infinity simultaneously in the global semicircle limit. The proof is largely constructive and transparent: the comparison parameters are solved from coefficient-matching equations rather than fitted to hyperbolicity, and the final stability argument is quantitative. The main analytic input, the uniform sectorial saddle expansion in Proposition 4.1, is imported from published work and is clearly identified. However, one algebraic identity used in the root-localization lemma is false as written, so the manuscript is not yet in publishable form.","major_comments":[{"comment":"The displayed expression for the difference between the diagonal entry (75) and V is not correct. Subtracting V from (75) and simplifying gives the numerator 2k(k+H+1)(U-2V) + V H(d-1) + 2k(k+1)(1-U), not 2k(k+H+1)(U-2V) + V H(d-1). For example, with U=20, V=10, d=5, k=1, the omitted term changes the value from 560/288 to 484/288. This identity is used to prove the diagonal bound |D_k - V| <= 4d that underlies Lemma 7.3's root localization and, through it, the proof of Lemma 11.1. The missing term is O(d^2/U) after division by the denominator, and under (57) it is O(d/K_r) for both factors, so the intended bound is likely repairable; however, as written the proof of the central real-rootedness claim has a gap that must be fixed.","section":"Section 7, Eq. (78)"}],"minor_comments":[{"comment":"I checked the factorization (E+D)JpF = -(epsilon_p/A) E(E-d)(E+A)pF; it is correct. The apparent d=1 counterexample disappears because for pF(y)=1-y/B one has JpF=0, so both sides vanish.","section":"Section 9, Eq. (86)"},{"comment":"The proof's main analytic weight is carried by the imported sectorial saddle expansion from [3, Section 3], including derivative bounds through order five derived by Cauchy estimates. This is a legitimate use of a published theorem, but the paper should state explicitly which formula or theorem in [3] is being imported, since the derivative bounds are not quoted verbatim.","section":"Section 4, Proposition 4.1"},{"comment":"Reference [7] contains a typographical error in the title: 'BemerkungÜber' should be 'Bemerkung über'.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The algebraic slip in Lemma 7.3 is localized and appears easily repairable, and I found no other load-bearing error. The central theorem is likely correct, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe headline is simple: this is a real step forward in the Jensen-polynomial program. It proves an unconditional hyperbolicity wedge with degree growing polynomially in n—d up to a constant times n^{3/5} log^{2/5} n—and a joint semicircle limit for the zeros as n and d go to infinity together. That is a genuine improvement over the earlier fixed-degree Hermite limit and the exponential thresholds of Griffin, Ono, Rolen, Thorner, Tripp, and Wagner.\n\nThe proof is well engineered and internally coherent. The sign structure is the core: after the Laguerre baseline, the R3 defect is positive, the Jacobi correction has a negative R4 defect, and a second finite-free Jacobi factor absorbs it to match five coefficients exactly. The parameter matching in Lemma 6.1 is a real contraction argument with explicit constants; the error orders scale as d^{5/2}/(n^{3/2} log n), which is exactly what the wedge needs. The multiplier stability argument is sound.\n\nI looked carefully at the stress-test objection to Lemma 9.1. It is a red herring. For d=1 the polynomial Jp is identically zero, not y/B, so the alleged failure is a miscalculation. I checked the factorization (86) explicitly for d=2—it holds. So that concern does not land.\n\nThe genuine soft spot is Proposition 4.1, the uniform sectorial saddle expansion for log M_z, quoted from [3] with a proof sketch and a normalization correction. The derivative estimates through fifth order are load-bearing for the signs of the defects and for the final multiplier bound. An expert referee should verify this against [3, Section 3]. The rest of the proof is internal, and I found no gaps, no circularity, and no fitted parameters controlling the main claim. The paper is appropriately modest: it makes no claim about local GUE statistics and explicitly notes it is not a route to RH.\n\nThis is for specialists in analytic number theory and orthogonal polynomials. It deserves a serious referee, not a desk rejection. Send it to two referees: one to certify the saddle input, one for the algebraic and comparison machinery. I would accept after that check.","headline":"A sound and substantial polynomial-wedge hyperbolicity theorem; the only real vulnerability is the imported complex saddle expansion, and the identified algebraic objection does not hold up.","tokens_in":17770,"tokens_out":8989,"would_cite":true,"duration_ms":82096,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M06","11M26","26C10","33C45","60B20"],"pacs":[],"model":"deepseek-v4-flash","headline":"There is an absolute constant K such that whenever n^3 log^2(n+2) ≥ K d^5, the Jensen polynomial J^{d,n} has d distinct negative real zeros, and its scaled zeros follow Wigner's semicircle law as n and d grow together.","keywords":["Jensen polynomials","Riemann xi-function","Riemann hypothesis","hyperbolicity","Wigner semicircle law","finite free convolution","saddle-point asymptotics","real-rooted polynomials"],"falsifier":"Find integers d and n with d ≤ c $n^{{3/5}}$ $log^{{2/5}}$(n+2) for some fixed positive c (for instance c = 1/2) such that $J^{{d,n}}$ has a non-real zero, or two roots that coalesce; the theorem asserts that no such pair exists. A less direct check is to compute the second-difference quantity D_n = log(R_3/$R_3^{{(L)}}$) for large n, since Lemma 4.3 predicts D_n ~ 2/($n^{2}$ log n) > 0 and the opposite sign would identify the analytic input as false.","tokens_in":16649,"feed_emoji":"📈","tokens_out":6752,"duration_ms":72151,"temperature":0.7,"pith_summary":"The Riemann hypothesis is equivalent to requiring that every Jensen polynomial $J^{{d,n}}$ built from Riemann's xi-function is hyperbolic, meaning all its zeros are real. This paper proves an unconditional region in the (d,n)-plane: there is an absolute constant K such that whenever $n^{3}$ $log^{2}$(n+2) ≥ K $d^{5}$, the degree-d Jensen polynomial has d distinct negative real zeros. Equivalently, hyperbolicity holds for degrees up to a fixed multiple of $n^{{3/5}}$ $log^{{2/5}}$(n+2). The paper also shows that in this same wedge, as n and d tend to infinity together, the naturally centered and scaled zeros converge to Wigner's semicircle law, giving a simultaneous degree-derivative version of the fixed-degree Hermite limit.","feed_headline":"Jensen polynomials of Riemann's xi hit a real-rooted wedge","feed_subtitle":"When n^3 log^2 n ≥ K d^5, all zeros are real and scale to Wigner's semicircle law.","key_machinery":"The proof is organized by quotient invariants q_k = R_{k+1}^2/(R_k R_{k+2}), which are unchanged by rescaling every coefficient and therefore capture all coefficient information. Starting from a Laguerre family, the author matches the first two normalized coefficients; a complex saddle expansion for the logarithmic moment function, with Lambert-type scale L_x solving x = L_x(π $e^{{L_x}}$ + 3/4), shows a positive defect at the third coefficient. A Jacobi deformation absorbs that defect, and a second Jacobi factor inserted through finite-free multiplicative convolution absorbs the opposite-sign defect at the fourth coefficient, so the model matches R_0 through R_4 exactly. Hyperbolicity is preserved because finite-free multiplicative convolution keeps positive real roots, and the remaining factor is a holomorphic multiplier c_F whose deviation from 1 is small exactly in the stated wedge; a fifth-order multiplier stability principle transfers real-rootedness from the model to $J^{{d,n}}$.","core_discovery":"The central claim is Theorem 1.1: for an absolute constant K > 0, the condition $n^{3}$ $log^{2}$(n+2) ≥ K $d^{5}$ forces all d zeros of $J^{{d,n}}$ to be real, distinct, and negative. The proof constructs a comparison polynomial with explicit simple positive roots, matches the first five normalized coefficients exactly, and then shows that the remaining coefficient multiplier is a small holomorphic perturbation; the wedge condition is precisely what makes this perturbation small. Theorem 1.2 then identifies the empirical distribution of the centered and scaled zeros of the actual Jensen polynomial with Wigner's semicircle law, uniformly along every sequence of pairs (n,d) staying in the wedge.","pith_inferences":["If the saddle expansion can be differentiated further and more coefficients matched, the same mechanism suggests the exponent on d in the wedge condition might be improved; this is not claimed in the paper.","The sign pattern of the defects, positive at R_3 and negative at R_4, is what makes the two-stage Jacobi construction work; a natural testable extension is whether the same pattern persists for moments of other L-functions with similar theta kernels.","The joint semicircle theorem invites numerical checks of local statistics, such as spacing distributions or edge behavior, in the same wedge; the paper explicitly does not claim such local universality."],"forward_implications":["For any fixed degree d, the theorem yields hyperbolicity as soon as n grows beyond a polynomial scale in d, roughly d^{5/3} up to log factors, rather than an exponential scale.","The result verifies one infinite family of Jensen polynomials unconditionally, but it does not cover the remaining region of the (d,n)-plane, so it does not prove the Riemann hypothesis.","In the joint limit n,d → ∞ inside the wedge, the scaled zeros of J^{d,n} converge to Wigner's semicircle law, matching the random-matrix prediction in a simultaneous degree-derivative limit.","Matching five coefficients is enough: the stability lemma shows that a multiplier whose first five values are 1 and whose deviation is small in a neighborhood changes the zero count in any interval by at most one."],"supporting_citations":[{"why":"Supplies the sectorial saddle expansion of the moment function and the derivative bounds through order five that determine the defect signs and the residual multiplier bound.","marker":"[3]"},{"why":"Establishes the fixed-degree Hermite limit that previously yielded the semicircle consequence through two separate limits; the present paper improves this to a joint limit.","marker":"[2]"},{"why":"Provides the positivity and logarithmic-mesh properties of finite-free multiplicative convolution that keep the comparison model real-rooted and simple.","marker":"[4]"},{"why":"Gives the largest-root inequality for finite-free convolution used to localize the model roots.","marker":"[5]"},{"why":"Records the Jensen criterion, making hyperbolicity of all Jensen polynomials equivalent to the Riemann hypothesis.","marker":"[7]"},{"why":"Supplies the Jacobi matrix form and tridiagonal estimates used for root localization and the semicircle computation.","marker":"[8]"}],"fun_headline_variants":["Real zeros for ξ-Jensen polynomials under a new wedge","Joint semicircle limit for ξ-Jensen zeros in a new wedge","Real-rooted Jensen polynomials and Wigner semicircle in one wedge","Wedge forces real Jensen zeros and semicircle scaling","Semicircle law for ξ-Jensen zeros when n^3 log^2 n ≥ K d^5"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the quoted asymptotic expansion of the logarithmic moment of Riemann's xi-function in a complex sector, including bounds on its first five derivatives; if that expansion or its derivative bounds failed, the argument would collapse.","fun_headline_variants_meta":{"raw":{"variants":["Real zeros for ξ-Jensen polynomials under a new wedge","Joint semicircle limit for ξ-Jensen zeros in a new wedge","Real-rooted Jensen polynomials and Wigner semicircle in one wedge","Wedge forces real Jensen zeros and semicircle scaling","Semicircle law for ξ-Jensen zeros when n^3 log^2 n ≥ K d^5"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001959,"raw_usage":{"total_tokens":7631,"prompt_tokens":891,"completion_tokens":6740,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":507,"completion_tokens_details":{"reasoning_tokens":6645}},"tokens_in":507,"tokens_out":6740,"duration_ms":46550,"temperature":1.0,"reasoning_tokens":6645,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:28:26.284627+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find integers d and n with d ≤ c $n^{{3/5}}$ $log^{{2/5}}$(n+2) for some fixed positive c (for instance c = 1/2) such that $J^{{d,n}}$ has a non-real zero, or two roots that coalesce; the theorem asserts that no such pair exists. A less direct check is to compute the second-difference quantity D_n = log(R_3/$R_3^{{(L)}}$) for large n, since Lemma 4.3 predicts D_n ~ 2/($n^{2}$ log n) > 0 and the opposite sign would identify the analytic input as false.","supporting_citations":[{"cited_title":"Jensen Polynomials for the Riemann Xi Function","cited_arxiv_id":"1910.01227","evidence_quote":"Supplies the sectorial saddle expansion of the moment function and the derivative bounds through order five that determine the defect signs and the residual multiplier bound."},{"cited_title":"Jensen polynomials for the Riemann zeta function and other sequences","cited_arxiv_id":"1902.07321","evidence_quote":"Establishes the fixed-degree Hermite limit that previously yielded the semicircle consequence through two separate limits; the present paper improves this to a joint limit."},{"cited_title":"Real roots of hypergeometric polynomials via finite free convolution","cited_arxiv_id":"2309.10970","evidence_quote":"Provides the positivity and logarithmic-mesh properties of finite-free multiplicative convolution that keep the comparison model real-rooted and simple."},{"cited_title":"Finite free convolutions of polynomials","cited_arxiv_id":"1504.00350","evidence_quote":"Gives the largest-root inequality for finite-free convolution used to localize the model roots."},{"cited_title":"P´ olya, Bemerkung¨Uber die Integraldarstellung der Riemannschen ξ-Funktion,Acta Math.48(1926), 305–317","cited_arxiv_id":null,"evidence_quote":"Records the Jensen criterion, making hyperbolicity of all Jensen polynomials equivalent to the Riemann hypothesis."},{"cited_title":"Szeg˝ o,Orthogonal Polynomials, 4th ed., American Mathematical Society Colloquium Publications, vol","cited_arxiv_id":null,"evidence_quote":"Supplies the Jacobi matrix form and tridiagonal estimates used for root localization and the semicircle computation."}],"review_version":1}