bibkey: holland2026jensenwedge authors: Jonathan Holland year: 2026 title: A new hyperbolicity wedge and a joint semicircle limit for Jensen polynomials of Riemann’s xi-function doi: null url: https://arxiv.org/abs/2608.08682v1 claim: Theorem 1.1 states a joint degree-shift hyperbolicity region for the actual xi coefficients; its sufficient condition excludes every positive degree at zero shift. strata_touched: [] license: citation-only triage: anchor

A joint degree-shift Jensen supplier

The primary is arXiv:2608.08682v1, Jonathan Holland, A new hyperbolicity wedge and a joint semicircle limit for Jensen polynomials of Riemann’s xi-function. The source locators below refer to this version. Its coefficient definitions, Theorem 1.1, Proposition 2.2, Lemmas 8.1 and 9.1, and the completion of the argument in section 10 were inspected. The results remain public-preprint statements; this note supplies neither a complete independent proof audit nor Lean verification. No source text is vendored.

Source coefficients and the joint region

Section 1 uses the standard completed function

$$\xi (s)=\frac{1}{2}s(s-1){\pi }^{-s/2}\Gamma (s/2)\zeta (s),\xi (1/2+z)=\sum _{k\ge 0}\frac{{\Gamma }_k}{k!}{z}^{2k}.$$

Here ${\Gamma }_k$ denotes the paper’s positive coefficient $\gamma (k)$, not a value of the Euler Gamma function. For integers $d,n\ge 0$ it defines

$$J^{d,n}(X)=\sum _{k=0}^d\left(\genfrac{}{}{0ex}{}{d}{k}\right){\Gamma }_{n+k}{X}^k.$$

Theorem 1.1 states that there is an absolute constant $K>0$ such that

$$d\ge 1,n\ge 0,n^3{\log }^2(n+2)\ge K{d}^5⟹J^{d,n}\text{ has }d\text{ distinct negative real roots}.\text{\hspace{1em}(JW)}$$

Equivalently the sufficient region has $d\le K^{-1/5}{n}^{3/5}{\log }^{2/5}(n+2)$. It permits degree and shift to grow together. The existence of $K$ does not supply a numerical threshold or a finite-input certificate. The statement is unconditional within its sufficient region; it is not a statement for every pair $(d,n)$.

The source’s proof compares the actual coefficient ratios with a positive-root model formed from Laguerre and Jacobi families and finite-free multiplicative convolution. Proposition 2.2 supplies its multiplier stability step; the comparison matches coefficients through index four. Lemma 8.1 bounds the remaining multiplier error by

$$O\left(\frac{d^{5/2}}{n^{3/2}\log (n+2)}\right),$$

on its specified complex neighborhood, and Lemma 9.1 controls derivative ratios at the comparison model’s critical points. Section 10 combines these estimates to obtain (JW). These are source results to reuse, not new polynomial identities or estimates of this repository.

The five coefficient matching conditions concern the actual xi sequence. They do not identify the five Zeckendorf inclusion modes [null,2,3,2 5,5] with these moments or supply an estimate through that encoding. Such an application would require its own parameter and inequality correspondence.

Correspondence with the actual theta coefficients

Use the original full-axis theta representation in Romik’s owning note, with $\Xi (z)=\xi (1/2+iz)$ and $\Xi (z)={\int }_{\mathbb{R}}\Phi (t)e^{izt}dt$. The actual normalized coefficients are

$$a_k=\frac{{\int }_{\mathbb{R}}{t}^{2k}\Phi (t)dt}{(2k)!\Xi (0)},a_0=1,{\Gamma }_k=\Xi (0)k!a_k.\text{\hspace{1em}(TC)}$$

The last equality uses the even Taylor expansion of $\xi (1/2+z)=\Xi (-iz)$ under the original transform. It keeps the full theta kernel and its central-value normalization. It neither replaces the source by an arbitrary positive measure nor identifies positive theta-moment Hankel matrices with matrices of actual root power sums. These are paper-level source correspondences. In the repository’s moment interface, normalization from $a_0=1$ is a conditional result; this note is not a new Lean proof discharging that hypothesis or transporting Holland’s theorem to xiReading.

For $d\ge 1$ the source polynomial and its shifted analogues are

$$\begin{array}{rl}{P}_d(X)& =\frac{J^{d,0}(X/d)}{\Xi (0)}=\sum _{k=0}^d\frac{(d{)}_k}{d^k}{a}_k{X}^k,\\ {\mathcal{P}}_{d,n}(X)& =\frac{J^{d,n}(X/d)}{{\Gamma }_n}=\sum _{k=0}^d\frac{(d{)}_k}{d^k}{b}_k^{(n)}{X}^k,\\ b_k^{(n)}& =\frac{{\Gamma }_{n+k}}{k!{\Gamma }_n},b_0^{(n)}=1.\end{array}\text{\hspace{1em}(PN)}$$

Here $(d{)}_k=d(d-1)\cdots (d-k+1)$ is the falling factorial, with $(d{)}_0=1$. Reflecting ${\mathcal{P}}_{d,n}(-X)$ at degree $d$ gives the constant

$${\beta }_{d,n}=(-1{)}^d\frac{{\Gamma }_{n+d}}{d^d{\Gamma }_n},{\beta }_{d,0}=(-1{)}^d\frac{d!}{d^d}{a}_d={\beta }_d.\text{\hspace{1em}(BC)}$$

Only $n=0$ recovers the original $a_k$ and ${\beta }_d$. Changing $n$ changes the actual coefficient window and its integration constant; it is not a coordinate change of the same $P_d$.

Reuse and the remaining zero-shift obligation

The existing canonical normalization and degree-lowering interface, source primitive and constant, source residue criterion, and total coupling budget are the corresponding repository interfaces. Their hypotheses and scope remain those of the owning sources. Equations (TC), (PN) and (BC) specify the literature parameters; they do not warrant new binding-only wrappers or a repeated derivation of these interfaces.

For every $d\ge 1$, substituting $n=0$ into (JW) gives a zero left side and a strictly positive right side. Thus this sufficient condition covers no positive degree at zero shift. This excludes an application of (JW), not real-rootedness of those polynomials. In particular, degree-growing hyperbolicity at large shifts does not furnish a bound on the original zero-shift integration constants.

For $d\ge 2$, in the existing primitive notation, $q_d$ is the degree-$d$ reflection of $P_d(-X)$, ${\alpha }_d=(d-1)/d$, and

$$q_d(x)=R_d(x)+{\beta }_d,R_d(x)={\int }_0^{x}d{\alpha }_d^{d-1}{q}_{d-1}(u/{\alpha }_d)du.$$

To extend a source polynomial upward, one still needs a joint estimate of its actual ${\beta }_d$, critical nodes, and values of $R_d$ at those nodes. Equivalently, under source normalization $a_0=1$ and the distinct-positive-preceding-root hypotheses of the existing residue criterion, the individual source residues must be nonnegative. Their total coupling budget does not establish every individual sign. Holland’s shifted region supplies no zero-shift, unbounded-degree estimate of these quantities.

The note does not assert a largest known finite-degree range or exhaustive coverage of the Jensen literature. Its reusable input is the versioned joint region (JW) with the exact source correspondence. It supplies no new RH conclusion, signed Robin-tail estimate, or quantitative moment inequality from FIB ATOM geometry.