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\title{The Willow Transform and the Blind Zone Theorem:\\
A Sharp Information-Theoretic Obstruction in the Tur\'an\\
Hierarchy for the Riemann xi-Function}

\author{Changxia Lin\\
\textit{Independent researcher}}

\date{\today}

\begin{document}
\maketitle

\begin{abstract}
Let $\Xi(t)=\xi(\tfrac12+it)$ be the Riemann xi-function, let
$\Xi(t)=\sum_{k\ge 0}(-1)^k c_k t^{2k}$ be its even Taylor expansion at the
origin, and let $D(n,r)=\det[c_{n+i-j}]_{i,j=0}^{r-1}$ be the Toeplitz minors
whose positivity for all $n\ge 0$, $r\ge 1$ is equivalent to the Riemann
Hypothesis (RH). We prove a sharp and quantitative obstruction in the resulting
hierarchy of criteria.

Parametrising the zeros of $G(z)=\sum c_k z^k$ by $\beta_j=-1/z_j$, we
establish by the dual Jacobi--Trudi identity the exact identity
$D(n,r)=s_{(r^n)}(\beta)$, the Schur function of the rectangular partition
$(r^n)$ in the variables $\beta$. A zero of zeta off the critical line
contributes to $\beta$ a conjugate pair $\rho e^{\pm i\theta}$ whose angle
$\theta$ is exactly the vertical angle of the violating pair of zeros, while
zeros on the critical line contribute positive real variables. For two
variables we prove the closed form
\[
s_{(a,b)}(\rho e^{i\theta},\rho e^{-i\theta})
  = \rho^{a+b}\,\frac{\sin((a+1-b)\theta)}{\sin\theta}
  = \rho^{a+b}\,U_{a-b}(\cos\theta),
\]
$U_k$ being the Chebyshev polynomial of the second kind, so that the sign is
governed by the classical criterion $U_k(\cos\theta)>0$ iff $(k+1)\theta<\pi$.

Since a Schur function in two variables can be expanded through the coproduct
$s_\lambda(X\sqcup Y)=\sum_\mu s_\mu(X)s_{\lambda/\mu}(Y)$, and since
$s_\mu(X)>0$ for positive real $X$, positivity reduces to the two-variable
factors, whose index $k=a-b$ is bounded by the number of columns $r$, not by
the shift $n$. This yields:

\noindent\textbf{Blind Zone Theorem.} If a violating pair subtends a vertical
angle $\theta<\pi/(r+1)$, then every Toeplitz minor $D(n,r')$ with $r'\le r$ is
strictly positive, notwithstanding the presence of non-real zeros.

The critical strip together with $\gamma_1=14.134725141\ldots$ forces
$\theta\le\theta_{\max}=2\arctan\!\bigl(1/(2\gamma_1)\bigr)=0.07071826\ldots$
rad, whence $\pi/\theta_{\max}=44.424\ldots$ and therefore every order
$r\le 43$ is completely non-discriminating.

We emphasise that the bound is on the order $r$ and not on the shift $n$; we
exhibit the mechanism and show that the contrary $n$-based bound is false. We
also record the refutation of a ``universal covering law'' previously
conjectured by the author, together with a floating-point artefact that nearly
produced a false refutation of the main theorem.

\smallskip
\noindent\textbf{No claim of a proof of RH is made.}
\end{abstract}

\noindent\textbf{MSC 2020:} 11M26 (primary); 05E05; 30C15; 15B48; 33C45.

\noindent\textbf{Keywords:} Riemann zeta function, xi-function, Tur\'an
inequalities, Toeplitz minors, Schur functions, Chebyshev polynomials, total
positivity, blind zone.

\section{Introduction}

The Tur\'an hierarchy for the Riemann xi-function asks for the positivity of
the Toeplitz minors $D(n,r)$ built from the Taylor coefficients $c_k$ of
$\Xi(t)=\xi(\tfrac12+it)$. Positivity of all such minors is equivalent to RH.
This paper does not advance that programme; it measures, sharply and
quantitatively, how much information the hierarchy can carry at all.

Our main result is a \emph{blind zone}: a violating pair of zeros whose
vertical angle is smaller than $\pi/(r+1)$ is invisible to every minor of order
at most $r$. Combining this with the critical strip and the first ordinate
$\gamma_1$, we obtain that all orders $r\le 43$ carry no discriminating power
whatsoever.

\section{Notation}

Write
\[
\Xi(t)=\xi\Bigl(\tfrac12+it\Bigr)=\sum_{k\ge 0}(-1)^k c_k t^{2k},
\qquad
G(z)=\sum_{k\ge 0}c_k z^k .
\]
Let $\beta_j=-1/z_j$ where $z_j$ runs over the zeros of $G$, and define
\[
D(n,r)=\det\bigl[c_{n+i-j}\bigr]_{i,j=0}^{r-1}.
\]

\section{The Willow correspondence}

\begin{theorem}[Dual Jacobi--Trudi]
For all $n\ge 0$ and $r\ge 1$,
\[
D(n,r)=s_{(r^n)}(\beta),
\]
the Schur function of the rectangular partition $(r^n)$ evaluated in the
variables $\beta=(\beta_j)$.
\end{theorem}

\section{Two-variable closed form}

\begin{lemma}
For integers $a\ge b\ge 0$,
\[
s_{(a,b)}(\rho e^{i\theta},\rho e^{-i\theta})
  = \rho^{a+b}\,\frac{\sin((a+1-b)\theta)}{\sin\theta}
  = \rho^{a+b}\,U_{a-b}(\cos\theta).
\]
\end{lemma}

\begin{lemma}
$U_k(\cos\theta)>0$ if and only if $(k+1)\theta<\pi$.
\end{lemma}

\section{Coproduct reduction and the index bound}

The coproduct expansion
\[
s_\lambda(X\sqcup Y)=\sum_\mu s_\mu(X)\,s_{\lambda/\mu}(Y),
\]
together with $s_\mu(X)>0$ for positive real $X$, reduces the positivity
question to the two-variable factors. The surviving index $k=a-b$ is bounded
by the number of columns $r$, and \emph{not} by the shift $n$.

\begin{remark}[Status]
The treatment of skew shapes (Lemma 7.2 in the full manuscript) and the index
bookkeeping (Lemma 7.3) are the two steps the author invites scrutiny on.
\end{remark}

\section{The Blind Zone Theorem}

\begin{theorem}[Blind Zone Theorem]
If a violating pair subtends a vertical angle $\theta<\pi/(r+1)$, then every
Toeplitz minor $D(n,r')$ with $r'\le r$ is strictly positive, notwithstanding
the presence of non-real zeros.
\end{theorem}

\section{The number 43}

For a zero $\rho=\beta+i\gamma$ off the critical line one has
$t=\gamma-i(\beta-\tfrac12)$, hence
\[
\theta = 2\arctan\frac{|\beta-\tfrac12|}{\gamma}
       \le 2\arctan\frac{1}{2\gamma_1}
       = \theta_{\max},
\]
using $|\beta-\tfrac12|<\tfrac12$ and $\gamma\ge\gamma_1=14.134725141\ldots$.
Numerically
\[
\theta_{\max}=0.07071826\ldots\ \text{rad}=4.051858\ldots^{\circ},
\qquad
\frac{\pi}{\theta_{\max}}=44.424\ldots
\]
so $r=43$ satisfies $(43+1)\theta_{\max}<\pi$ while $r=44$ does not. Hence
every order $r\le 43$ is completely non-discriminating.

\section{Refuted conjecture and a numerical artefact}

The author previously conjectured a ``universal covering law'' for the
hierarchy; this was refuted and is recorded as such. A floating-point artefact
in the numerical exploration nearly produced a false refutation of the main
theorem; it is recorded for reproducibility.

\section{Disclosure}

This work was carried out with the assistance of generative AI tools. Every
step stated as proved was independently checked by the author, and the steps
that are not proved are flagged explicitly above.

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\end{thebibliography}

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