gpt 6 astra ultra attempt on Math-2

The Riemann Hypothesis

Definitions and mathematical statement

For $s\in{\mathbb{C}}$ with $\Re s>1$, define \[ \zeta(s)=\sum_{n=1}^{\infty}n^{-s}=\prod_{p\text{ prime}}(1-p^{-s})^{-1}. \] Use its meromorphic continuation to ${\mathbb{C}}$, whose only pole is at $s=1$. The assertion is \[ \forall\rho\in{\mathbb{C}},\qquad \left(0<\Re\rho<1\ \text{and}\ \zeta(\rho)=0\right) \ \Longrightarrow\ \Re\rho=\tfrac12. \] The open critical strip excludes the negative even integer zeros. The assertion concerns locations, not multiplicities, of zeros.

Formulation and concept references: [S1].

Short English statement

Are all zeta zeros between the vertical lines zero and one located on the halfway line?

Research attempt

Current frontier and status distinctions.

The exact target is the zero-location assertion for $\zeta$ alone, with multiplicities unrestricted [S1]. An established computational theorem of Platt--Trudgian verifies it throughout $0<\Im\rho\le 3\cdot10^{12}$, using interval arithmetic and a count of all zeros, not merely a list of zeros found on the line [E1]. On the analytic side, Guth--Maynard's work, initiated in 2024 and published in 2026, gives the zero-density estimate \[ N(\sigma,T)\le T^{15(1-\sigma)/(3+5\sigma)+o(1)}, \qquad N(\sigma,T)=\#\{\rho:\Re\rho\ge\sigma,\ |\Im\rho|\le T\}, \] for fixed $1/2<\sigma<1$ as $T\to\infty$, with multiplicities counted [E2], Theorem 1.2. Such an upper bound does not assert that the exceptional set is empty.

A substantial recent development in the supplied bibliography must not be omitted: Lamzouri's September 2026 manuscript states the unconditional bound \[ \liminf_{T\to\infty}\frac{N_0^{\rm s}(T)}{N(T)} \ge \frac32-\frac1{\sqrt2}\cot\!\left(\frac1{\sqrt2}\right) =0.6725007\ldots, \] where $N_0^{\rm s}$ counts simple zeros on the line and $N$ counts all upper-half strip zeros with multiplicity [S3], [E9], Theorem 1.1. The associated September 3 lecture is also listed by the host institution [S4]. The theorem statement and relevant mechanism were inspected; the complete analytic proof and its formal-certificate dependencies were not independently verified in this run. It is recorded as a recent research result at manuscript stage, not as an established resolution of RH. Even an asymptotic proportion of one would not, by itself, exclude finitely many exceptions.

Two established results sharpen the positivity route below. Griffin--Ono-- Rolen--Zagier prove eventual hyperbolicity of the central Jensen polynomials for every fixed degree; the threshold can depend on that degree [E3], Theorem 1. A recent, unaudited August 2026 manuscript of Holland states hyperbolicity in the joint range $n^3\log^2(n+2)\ge Kd^5$ for an absolute $K>0$, still not for every pair $(d,n)$ [E6], Theorem 1.1. Rodgers--Tao prove that the de Bruijn--Newman constant is nonnegative; since RH is equivalent to its nonpositivity, the required endpoint is exactly zero [E4]. These are selected frontier results relevant to this attempt, not a claim that all current research is exhausted by them.

Attack map.

A heat-flow argument would have to establish real-rootedness down to the exact endpoint, not merely at positive deformation parameters [E4]. The new pair-correlation approach would need additional control that detects every possible exception, beyond a global proportion estimate [E9], Sections 2--3. A factorization argument would need to show why the functional equation fixes individual zero factors instead of permuting them; the first stress test below shows that symmetry alone cannot do this. The selected route combines local Taylor data, logarithmic derivatives, and polynomial filters that isolate hypothetical nonreal inverse zeros. Its precise missing lemma is positivity of one explicit quadratic form on all real polynomials. Unlike a numerical search over zeros, the form can be defined without assuming where any zeros lie.

The calculations and auxiliary proofs below were developed in this notebook. Positivity criteria and moment methods are classical; no historical novelty is asserted for the resulting equivalence. The aim is to expose a usable intermediate target and determine whether the available positivity actually establishes it.

Attempt A: source-audit obstruction, not a zeta counterexample.

Theorem 1 of the existing proof claim [S5], version 19, Section 3, passes from factor matching in its equation $5$ to same-index matching in equation $7$, invoking uniqueness of zero multiplicities. That inference is invalid: multiplicity is attached to a zero, but equal multiplicities do not distinguish different zeros. This can be tested within the infinite-product class used there, without appealing to another claimed solution. Fix $a=1/4$ and put \begin{equation}\label{wsref-0-72-61-3a-32-3a-73-79-6d-6d-65-74-72-79-2d-74-65-73-74} A(s)=\prod_{n=1}^{\infty} \left(1+\frac{(s-a)^2}{n^2}\right) \left(1+\frac{(s-(1-a))^2}{n^2}\right). \end{equation} On each compact set, the sum of the absolute deviations of these factors from one is bounded by a constant times $\sum n^{-2}$, so the product converges locally uniformly to a nonzero entire function. Away from the displayed factor zeros its product is nonzero; hence its zeros are exactly $a\pm in$ and $1-a\pm in$, all simple. Their real parts lie strictly between zero and one, and the sum of inverse squared imaginary parts is finite. Reflection $s\mapsto1-s$ exchanges the two factors at each $n$, so $A(s)=A(1-s)$, although none of its zeros lies on the critical line. Each quadratic factor is irreducible over ${\mathbb{R}}$. Thus even within this convergent, strip-confined, multiplicity-preserving setting, reflection may exchange two distinct conjugate-pair factors of the same multiplicity.

This explicitly defeats the same-index inference in the cited argument. It does not disprove RH: $A$ is not the completed zeta function, and no Euler-product identity for it is asserted. In particular, merely using an Euler product to confine zeros to a strip does not repair that inference; more arithmetic information would have to enter the proof.

Attempt B: define the moment form without locating zeros.

To avoid a clash with the $t$-variable convention in [S1], throughout this attempt write \[ \xi(s)=\tfrac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s),\qquad X(z)=\frac{\xi(1/2+z)}{\xi(1/2)},\qquad F(w)=\sum_{n=0}^{\infty}a_nw^n,\quad X(z)=F(z^2). \] The functional equation makes $X$ even, so this definition of $F$ is entire and has no square-root branch choice. Also $a_0=1$. There is no real strip zero: for $0<s<1$ the alternating series $\eta(s)=\sum_{n\ge1}(-1)^{n-1}n^{-s}$ is strictly positive (group adjacent positive differences), whereas $1-2^{1-s}<0$ and $\zeta(s)=\eta(s)/(1-2^{1-s})$. In particular $\xi(1/2)>0$. The completed-function properties and zero correspondence are as in [S1]; the growth bound $|\xi(s)|\le\exp(O(|s|\log(2+|s|)))$ is recorded in [E7], Lemma 1.

Define real numbers $m_k$ solely by the convergent expansion near zero \begin{equation}\label{wsref-0-72-61-3a-32-3a-6d-6f-6d-65-6e-74-2d-64-65-66-69-6e-69-74-69-6f-6e} \frac{F'(w)}{F(w)}=\sum_{k=0}^{\infty}(-1)^km_kw^k. \end{equation} Equating coefficients in $F'=F(F'/F)$ gives the exact recurrence \begin{equation}\label{wsref-0-72-61-3a-32-3a-72-65-63-75-72-72-65-6e-63-65} m_k=(-1)^k\left((k+1)a_{k+1} -\sum_{j=1}^{k}a_j(-1)^{k-j}m_{k-j}\right). \end{equation} For example, \[ m_0=a_1,\qquad m_1=a_1^2-2a_2,\qquad m_2=a_1^3-3a_1a_2+3a_3. \] Set $\mathcal L(t^k)=m_k$, extending linearly to ${\mathbb{R}}[t]$, and let \begin{equation}\label{wsref-0-72-61-3a-32-3a-68-61-6e-6b-65-6c} H_D=(m_{i+j})_{0\le i,j\le D}. \end{equation} Thus $c^{\mathsf T}H_Dc=\mathcal L(p^2)$ for $p(t)=\sum_{i=0}^D c_it^i$. The unresolved estimate is \begin{equation}\label{wsref-0-72-61-3a-32-3a-6d-69-73-73-69-6e-67-2d-70-6f-73-69-74-69-76-69-74-79} \mathcal L(p^2)\ge0\qquad\text{for every }p\in{\mathbb{R}}[t]. \end{equation} We next prove carefully that this estimate would address the full target, not merely a density-one or bounded-height version.

Unconditional zero expansion. Let the distinct zeros of $F$ be $w_j$, with multiplicities $d_j$, and put $\lambda_j=-1/w_j$. Equivalently, $w_j=(\rho-1/2)^2$, with each pair $\{\rho,1-\rho\}$ represented once. The standard zero count $O(T\log T)$ implies $\sum_jd_j|\lambda_j|<\infty$; finitely many bounded zeros cause no problem because $F(0)=1$. Hadamard factorization applied to $X$ pairs its zeros $b,-b$ into $(1-z/b)e^{z/b}(1+z/b)e^{-z/b}=1-z^2/b^2$. Absolute convergence of the canonical product permits this pairing. The remaining exponential is $e^{Bz}$, and evenness forces $B=0$. Consequently \begin{equation}\label{wsref-0-72-61-3a-32-3a-70-72-6f-64-75-63-74} F(w)=\prod_j(1+\lambda_jw)^{d_j},\qquad m_k=\sum_jd_j\lambda_j^{k+1}. \end{equation} The factorization theorem and growth/zero-count prerequisites are in [E7], Section 1; see also [E8], Section 3. For $|w|\sup_j|\lambda_j|<1/2$, both the logarithmic-derivative sum and its geometric expansions converge absolutely, bounded by a constant times $\sum_jd_j|\lambda_j|$. This justifies the coefficient interchange in (ra:2:product). It does not assume RH.

The $\lambda_j$ are closed under conjugation, with equal multiplicities. They have no accumulation point except zero. Since a nontrivial zeta zero has nonzero imaginary part, elementary squaring gives \begin{equation}\label{wsref-0-72-61-3a-32-3a-6c-61-6d-62-64-61-2d-6c-6f-63-61-74-69-6f-6e} \lambda_j\in{\mathbb{R}}\quad\Longleftrightarrow\quad \Re\rho=\tfrac12, \qquad\text{and then }\lambda_j=(\Im\rho)^{-2}>0. \end{equation} For every fixed polynomial, absolute convergence yields \begin{equation}\label{wsref-0-72-61-3a-32-3a-4c-7a-65-72-6f} \mathcal L(p^2)=\sum_j d_j\lambda_jp(\lambda_j)^2. \end{equation} The square here is not an absolute square unless its argument is real. Mistaking these expressions would assume precisely the desired conclusion.

Lemma 2.1 (polynomial isolation of a hypothetical off-axis pair). If some $\lambda_*$ in (ra:2:product) is nonreal, there is a real polynomial $q$ for which $\mathcal L(q^2)<0$.

Proof. Put $r=|\lambda_*|>0$, and let $S$ consist of all distinct $\lambda_j$ with $|\lambda_j|\ge r$, except $\lambda_*$ and $\overline{\lambda_*}$. This is a finite conjugation-invariant set. Therefore \[ P(t)=\prod_{\lambda\in S}(t-\lambda)\in{\mathbb{R}}[t], \qquad P(\lambda_*)\ne0. \] An empty product is one. Choose a complex number $\eta$ with $\lambda_*\eta^2=-1$. For each integer $k\ge0$ choose real $u_k,v_k$ solving \[ u_k+v_k\lambda_* =\frac{\eta}{\lambda_*^kP(\lambda_*)}, \qquad q_k(t)=t^kP(t)(u_k+v_kt). \] The real-linear map $(u,v)\mapsto u+v\lambda_*$ is invertible because $\Im\lambda_*\ne0$. Its inverse has fixed finite norm, so $|u_k|+|v_k|\le C_*r^{-k}$, with $C_*$ independent of $k$. We have $q_k(\lambda_*)=\eta$ and $q_k(\overline{\lambda_*})=\overline\eta$. The contribution of this pair to (ra:2:Lzero) is exactly $-2d_*$, where $d_*$ is its common multiplicity. All terms in $S$ vanish.

The remaining inverse zeros have modulus at most some $R<r$: otherwise infinitely many would accumulate at modulus $r>0$. If there are no remaining zeros the proof is already complete. On $|t|\le R$, the fixed polynomial $P$ is bounded, hence \[ |q_k(t)|\le C'_* (R/r)^k. \] It follows that the absolute value of the remaining contribution is at most \begin{equation}\label{wsref-0-72-61-3a-32-3a-66-69-6c-74-65-72-2d-74-61-69-6c} (C'_*)^2(R/r)^{2k}\sum_{\lambda_j\notin S\cup\{\lambda_*,\overline{\lambda_*}\}} d_j|\lambda_j|\ \longrightarrow\ 0. \end{equation} Thus $\mathcal L(q_k^2)=-2d_*+o(1)<0$ for all sufficiently large $k$. All operations used a finite polynomial and an absolutely summable tail. $\square$

Consequence (an exact positivity reduction). RH is equivalent to $H_D\succeq0$ for every $D\ge0$. If RH holds, then (ra:2:lambda-location) and (ra:2:Lzero) give positivity term by term. In fact every $H_D$ is positive definite, since there are infinitely many distinct positive $\lambda_j$ and a nonzero polynomial has only finitely many roots. Conversely, if RH fails, Lemma 2.1 supplies a finite-degree polynomial violating positivity of one $H_D$. This argument does not impose simplicity of any zero. Its usefulness remains conditional: the proof constructs the negative polynomial assuming a nonreal inverse zero has already been specified; it neither produces such a zero for $\zeta$ nor proves positivity from arithmetic.

Attempt C: can the positive theta kernel prove the missing estimate?

In the normalization of [E4], equations $1$--$3$, set \[ \Phi(u)=\sum_{n\ge1} (2\pi^2n^4e^{9u}-3\pi n^2e^{5u})e^{-\pi n^2e^{4u}},\qquad u\ge0. \] Each summand is positive because its prefactor is $\pi n^2e^{5u}(2\pi n^2e^{4u}-3)>0$. The cited Fourier representation gives \begin{equation}\label{wsref-0-72-61-3a-32-3a-74-68-65-74-61} X(z)=\frac{\int_0^\infty\Phi(u)\cosh(2zu)\,du} {\int_0^\infty\Phi(u)\,du},\qquad a_n=\frac{2^{2n}}{(2n)!} \frac{\int_0^\infty u^{2n}\Phi(u)\,du} {\int_0^\infty\Phi(u)\,du}. \end{equation} The super-exponential decay of $\Phi$ dominates $e^{2|z|u}$ on every compact $z$-set, so differentiation and expansion under the integral are justified. In particular all $a_n$ are positive. The attempted bridge is to combine ordinary moment positivity of the measure $\Phi(u)du$ with recurrence (ra:2:recurrence) to prove positivity of $\mathcal L$. But the factorial reweighting and the logarithmic derivative are nonlinear operations; ordinary moment positivity does not automatically survive them.

Here is an exact falsification of that bridge in its proposed generality. With the rescaled integration variable $v=2u$, take the positive probability measure with masses $3/4$ at $v=0$ and $1/4$ at $v=1$. Its normalized transform $\int\cosh(v\sqrt w)\,d\mu(v)$ is \[ F_{\rm toy}(w)=\tfrac34+\tfrac14\cosh\sqrt w =1+\tfrac18w+\tfrac1{96}w^2+\tfrac1{2880}w^3+\cdots. \] Its coefficients are nonnegative and its square-root notation again denotes an entire power series. The same recurrence gives \[ m_0=\tfrac18,\qquad m_1=-\tfrac1{192},\qquad m_2=-\tfrac7{7680},\qquad \det H_1=-\tfrac{13}{92160}<0. \] Indeed, the equation $\cosh z=-3$ has the nonimaginary solutions $z=\pm\operatorname{arcosh}(3)+(2\ell+1)\pi i$. This is not a counterexample to RH, nor a replacement of the actual theta kernel. It proves that positivity of a cosh-integrating measure alone cannot justify the proposed step. Even smooth nonnegative kernels cannot rescue that general statement: replace the two atoms by narrow nonnegative smooth bumps of total masses $3/4$ and $1/4$. Their first six moments converge to those of the atoms, so the displayed strictly negative determinant persists. Any successful use of (ra:2:theta) must therefore exploit an additional, specific property of this theta kernel.

Attempt D: test whether finite positivity plus zero symmetry could suffice.

Consider the explicitly defined model \[ F_*(w)=\prod_{j\ge1}(1+w/j^2). \] It is entire, and its inverse zeros are the distinct positive numbers $j^{-2}$. Its moment matrices are all positive definite by the same finite-polynomial argument used above. The compact-uniform convergence also shows that its Taylor coefficients are positive. Fix a degree $D$ and let $\delta_D>0$ be the smallest eigenvalue of its $H_D$. For $0<a<1/2$, choose $T>a$ and set \begin{equation}\label{wsref-0-72-61-3a-32-3a-70-65-72-74-75-72-62} \lambda=-\frac1{(a+iT)^2},\qquad \widetilde F(w)=F_*(w)(1+\lambda w)(1+\overline\lambda w). \end{equation} Then $\Re\lambda>0$ and $(1+\lambda w)(1+\overline\lambda w) =1+2\Re\lambda\,w+|\lambda|^2w^2$ has positive coefficients. The additional zeros of $\widetilde F(z^2)$ are exactly $z=\pm a\pm iT$. Thus the translated $s$-function remains entire, real under conjugation, symmetric under $s\mapsto1-s$, and has all zeros in the open critical strip, but it has an off-line quartet.

Its moment perturbation is exactly \[ \Delta m_k=2\Re(\lambda^{k+1}). \] For $v=(1,\lambda,\ldots,\lambda^D)^{\mathsf T}$, this gives \[ \Delta H_D=\lambda vv^{\mathsf T} +\overline\lambda\,\overline v\,\overline v^{\mathsf T}, \qquad \|\Delta H_D\|_2\le 2|\lambda|\sum_{i=0}^D|\lambda|^{2i} \le\frac{2|\lambda|}{1-|\lambda|^2}\quad(|\lambda|<1). \] Since $|\lambda|=(T^2+a^2)^{-1}\to0$, choose $T$ so that this last bound is less than $\delta_D$. The perturbed $H_D$ and all its leading submatrices remain positive definite despite the off-line quartet. Moreover the perturbation of any prescribed finite list of moments can be made arbitrarily small. This is an auxiliary construction, not a modified $\zeta$ Euler product. It shows exactly why a finite group of successful tests plus these generic symmetries is inadequate; it does not rule out a finite argument using a stronger arithmetic invariant.

The polynomial-isolation proof explains the complementary phenomenon: it eventually detects the quartet, but the degree and coefficient sizes of its filter depend on the unobserved inverse zeros and on the separation $R<r$ in (ra:2:filter-tail). No uniform small-degree bound was obtained. Farmer's analysis of information lost under repeated differentiation is a related warning about extrapolating from Jensen-polynomial asymptotics [E5], Sections 2--4; it is not a theorem prohibiting all positivity methods.

Computational checks and reproducibility.

Exact rational checks in the accompanying script verify the toy fractions. They also confirm that, for the finite base $\prod_{j=1}^5(1+w/j^2)$ and the perturbation $a=1/4$, $T=10^6$, every leading determinant through the $4\times4$ moment matrix is positive both before and after adding the off-axis quartet. The corresponding zeros are specified algebraically, not inferred numerically. In the same finite model, take $P(t)=\prod_{j=1}^5(t-j^{-2})$ and solve $u+v\lambda=(a+iT)/P(\lambda)$ for rational $u,v$. Then $q(t)=P(t)(u+vt)$ annihilates every base inverse zero and satisfies $\lambda q(\lambda)^2=-1$. Thus $\mathcal L_{\rm model}(q^2)=-2$, as the code independently verifies using the rational moment matrix and polynomial coefficients. The polynomial has degree six: the same model passes the displayed low-degree tests and fails at a higher degree.

For the actual $\xi$, the script computes its Taylor coefficients at $1/2$ through order $14$ and applies (ra:2:recurrence). It does not compute these moments from a list of presumed on-line zeros, which would silently omit any exceptions. Runs at $70$ and $90$ decimal working digits agreed to the displayed precision. Illustrative values are \[ \begin{aligned} m_0&\approx0.02310499311541897079,\\ m_1&\approx3.717259928526968616\cdot10^{-5},\\ m_2&\approx1.441739314009732797\cdot10^{-7}. \end{aligned} \] For numerical conditioning, also form the diagonal congruence $\widehat H_D=(m_{i+j}/\sqrt{m_{2i}m_{2j}})_{i,j=0}^D$; all required $m_{2i}$ were positive in these computations. The observed results are \[ \begin{array}{c|c|c} D&\det H_D&\lambda_{\min}(\widehat H_D)\\ \hline 0&2.3104993115\cdot10^{-2}&1\\ 1&1.9493355548\cdot10^{-9}&0.3559395401\\ 2&2.1728105105\cdot10^{-22}&0.01549146869\\ 3&1.1775817160\cdot10^{-41}&0.0002184026834 \end{array} \] These are ordinary high-precision floating-point calculations, not interval certificates. Repetition at higher precision is a diagnostic, not a proof of any determinant sign. Even certified signs for this finite table would not establish the universal condition (ra:2:missing-positivity).

Stress test and exact remaining gap.

The strongest step supplied here is the full finite-witness implication in Lemma 2.1, including the tail estimate. No compactness of the unknown zero set beyond its proved discreteness and inverse-zero summability was assumed. The positivity claim itself was not supplied. Positivity of the theta measure proves a different moment inequality, and the toy example refutes the proposed automatic transfer to logarithmic moments. Replacing $p(\lambda)^2$ by $|p(\lambda)|^2$, identifying two reflected factors without justification, or inferring all-degree positivity from the table would each introduce an invalid essential step. In general, nonnegative leading principal determinants alone are not a test for positive semidefiniteness; the condition used here is nonnegativity of every quadratic form of each finite matrix.

There is no dependency on GRH, the simplicity conjecture, or a conjectural spectral operator. Nor does this argument establish them: the explicit nonvanishing on the real strip and the zero correspondence used above are specific to the chosen zeta problem. A complete proof along this route still requires an arithmetic/theta identity or an inequality proving (ra:2:missing-positivity) for every polynomial, with no degree-dependent exceptions. A disproof would require a rigorously certified negative value for this actual moment functional, not for either model function.

\Needspace{11\baselineskip}

Outcome.

Outcome: Exploratory approach with unresolved gap.

The notebook supplies an exact moment-positivity reduction with a constructive negative-polynomial witness under a hypothetical off-line zero, a quantitative finite-test perturbation obstruction, and explicit failures of two tempting positivity/symmetry arguments. It also isolates a specific incorrect inference in one of the catalogue's cited proof claims. These results are proved or qualified locally; they are not presented as historically new theorems about zeta zeros. The unresolved step is all-degree positivity of the actual logarithmic-moment functional. No complete proof or counterexample candidate for the Riemann Hypothesis has been obtained in this batch.

Sources

• S1 Enrico Bombieri, Problems of the Millennium: the Riemann Hypothesis, Section I and the prime-counting equivalence. https://www.claymath.org/wp-content/uploads/2022/05/riemann.pdf. Catalogue formulation source. Consulted 24 September 2026: Section I, definition and zero-location statement.
• S2 Riemann Hypothesis. https://www.claymath.org/millennium/riemann-hypothesis/. Catalogue research/status reference; not a proof endorsement.
• S3 A new proof that more than 2/3 of the zeros of the Riemann zeta function are simple and on the critical line. https://arxiv.org/abs/2609.02882. Catalogue research/status reference; not a proof endorsement.
• S4 Lamzouri lecture at Max Planck Institute for Mathematics, September 3, 2026. https://math-events.uni-bonn.de/event/1560/. Catalogue research/status reference; not a proof endorsement.
• S5 On the Extended, Generalized, and Grand Riemann Hypotheses: A Unified Approach Based on the General Properties of L-Functions — Weicun Zhang, v19. https://www.preprints.org/manuscript/202506.0481. Catalogue research/status reference; not a proof endorsement.

• E1 Dave Platt and Tim Trudgian, The Riemann hypothesis is true up to $3\cdot10^{12}$, arXiv:2004.09765v1 (21 April 2020), Theorem 1 and Section 2; published in Bulletin of the London Mathematical Society $2021$. https://arxiv.org/pdf/2004.09765v1. Research reference: rigorous bounded-height verification, including completeness of the zero count. Consulted 24 September 2026.

• E2 Larry Guth and James Maynard, New large value estimates for Dirichlet polynomials, Annals of Mathematics 203 $2026$, no. 2, 623--675, DOI: 10.4007/annals.2026.203.2.6, Theorem 1.2. https://annals.math.princeton.edu/2026/203-2/p06. Author manuscript: arXiv:2405.20552v2 (7 April 2026), https://arxiv.org/pdf/2405.20552v2. Research reference: zero-density estimate; publication metadata and the stated theorem were checked. Consulted 24 September 2026.

• E3 Michael Griffin, Ken Ono, Larry Rolen and Don Zagier, Jensen polynomials for the Riemann zeta function and other sequences, arXiv:1902.07321v2 (31 March 2019), Section 1, Theorems 1--2; published in Proceedings of the National Academy of Sciences $2019$. https://arxiv.org/pdf/1902.07321v2. Research reference: the fixed-degree, sufficiently-large-shift quantifiers, not all-degree hyperbolicity. Consulted 24 September 2026.

• E4 Brad Rodgers and Terence Tao, The de Bruijn--Newman constant is non-negative, Forum of Mathematics, Pi 8 $2020$, e6, DOI: 10.1017/fmp.2020.6. Author manuscript arXiv:1801.05914v5 (3 July 2021), Section 1, equations $1$--$4$, and the main theorem. https://arxiv.org/pdf/1801.05914v5. Research reference: the theta-kernel normalization and the endpoint obstruction. Consulted 24 September 2026.

• E5 David W. Farmer, Jensen polynomials are not a plausible route to proving the Riemann Hypothesis, arXiv:2008.07206, Sections 2--4. https://arxiv.org/pdf/2008.07206. Research reference: differentiation and information loss; the paper's methodological assessment is not used as an impossibility theorem. Consulted 24 September 2026.

• E6 Jonathan Holland, A new hyperbolicity wedge and a joint semicircle limit for Jensen polynomials of Riemann's $\xi$-function, arXiv:2608.08682v1 (9 August 2026), Theorem 1.1. https://arxiv.org/html/2608.08682v1. Research reference: recent manuscript, statement inspected, full proof not independently audited here; not used as a premise of any lemma. Consulted 24 September 2026.

• E7 Kiran S. Kedlaya, More on the zeroes of $\zeta$, MIT 18.785: Analytic Number Theory, spring 2007, Section 1, Lemma 1 and Theorems 2--3. https://kskedlaya.org/18.785/zeroes.pdf. Research reference: growth of the completed function, zero counting, and Hadamard factorization. Consulted 24 September 2026.

• E8 Paul Garrett, 12. Weierstrass and Hadamard products, author lecture notes, 3 February 2021, Section 3 (Hadamard products). https://www-users.cse.umn.edu/~garrett/m/complex/notes_2020-21/12_Hadamard_products.pdf. Research reference: canonical products and the finite-order factorization theorem. Consulted 24 September 2026.

• E9 Youness Lamzouri, A new proof that more than $2/3$ of the zeros of the Riemann zeta function are simple and on the critical line, arXiv:2609.02882v2 (8 September 2026), Theorem 1.1, Proposition 2.1 and Sections 2--3. https://arxiv.org/html/2609.02882v2. Version-specific research supplement to [S3]: the stated bound and method were inspected. Neither a complete analytic proof audit nor a formal-certificate audit was performed. Consulted 24 September 2026.