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\title{Convolution Smoothing in the Tur\'an Hierarchy:\\ The Blind Zone Extends from Order~$43$ to Order~$61$,
and Empirically to Order~$68$}

\author{Changxia Lin\\
\texttt{xstr\_1115@qq.com}\\
Independent Researcher}

\date{\today}

\begin{document}
\maketitle

\begin{center}
\begin{minipage}{0.9\textwidth}
\small\textsf{\textbf{Statement on the use of generative artificial intelligence.}
Generative AI assistance was used for literature retrieval, notation
cross-checking, \LaTeX{} typesetting, and for the independent
recomputation of every numerical constant recorded in
Sections~\ref{sec:numerics}--\ref{sec:robust}. The mathematical
content---the multiplicity analysis of the $\beta$-alphabet, the
convolution identity for the shift-$1$ row, the Chebyshev derivative
identity $W_r=\tfrac12 U'_{r+1}(\cos\thval)$, the sign criterion of
Lemma~\ref{lem:sign}, the proof of Theorem~\ref{thm:main}, and the
design of the numerical experiments---was produced by the author. Every
statement that is not proved is labelled as such; see
Section~\ref{sec:status} for a complete ledger of what is proved and
what is numerical.}
\end{minipage}
\end{center}

\vspace{0.5em}

\begin{abstract}
Let $\Xif(t)=\xi(\tfrac12+it)$ be the Riemann $\xi$-function on the
critical line, $\Xif(t)=\sum_{k\ge0}(-1)^k c_k t^{2k}$ its even Taylor
expansion, and $D(n,r)=\det[c_{n+i-j}]_{i,j=0}^{r-1}$ the associated
Toeplitz minors. Writing $G(z)=\sum_{k\ge0}c_k z^k$ and
$\beta_j=-1/z_j$ for the zeros of $G$, the dual Jacobi--Trudi identity
gives $D(n,r)=c_0^{\,r} s_{(r^n)}(\beta)$, the Schur function of the
rectangular partition $(r^n)$ in the alphabet $\beta$.

Positivity of $D(n,r)$ for all $n\ge0$ and all $r\ge1$ is equivalent to
the Riemann hypothesis (RH) by the Schoenberg--Edrei characterisation of
P\'olya frequency sequences; for each fixed $r$, the inequalities
$D(n,r)\ge0$ are a necessary condition, and for $r=2$ they hold
unconditionally \cite{Csordas}. A previous paper of the author
\cite{LinV8} proved that every level $r\le43$ is \emph{blind}: the
minors $D(n,r)$ stay positive even in the presence of non-real zeros.
The mechanism there was the two-variable Chebyshev criterion
$U_k(\cos\thval)>0\iff(k+1)\thval<\pi$, together with the uniform bound
$\thval\le\thval_{\max}=2\arctan(1/(2\gamma_1))=0.07071826\ldots$

The present paper shows that the bound $43$ is not the truth but an
artefact of ignoring two structural features of the true alphabet.
First, the evenness $\Xif(-t)=\Xif(t)$ forces every zero of $G$ to occur
with \emph{multiplicity two} in $\beta$, so the relevant constituent is
not $U_r$ but the self-convolution
\[
W_r=\sum_{i=0}^{r}U_i(\cos\thval)\,U_{r-i}(\cos\thval),
\qquad\text{with generating function}\qquad
\sum_{r\ge0}W_r z^r=\frac{1}{(1-2z\cos\thval+z^2)^2}.
\]
We prove the identity $W_r=\tfrac12 U'_{r+1}(\cos\thval)$ and the
pole-free sign criterion of Lemma~\ref{lem:sign}, and we show that
$W_r>0$ for all $r\le61$ while $W_{62}<0$: the convolution of the
Chebyshev sequence against itself \emph{smooths away} the sign change
that occurs at $U_{44}$, pushing the threshold from $43$ to $61$.
Secondly, the zeros on the critical line contribute a positive real
sub-alphabet whose complete symmetric functions $h_i(X)>0$ convolve
with the violating part, $h_r(X\sqcup Y)=\sum_{i+j=r}h_i(X)h_j(Y)$,
pushing the threshold further.

\smallskip
\noindent\textbf{Main theorem (proved).} \emph{For every $r\le61$ and
every $n=1$, and regardless of whether RH is true,
$D(1,r)>0$. Consequently the first $61$ levels of the Tur\'an hierarchy
are blind on the shift-$1$ row.}

\smallskip
High-precision computation of the full rectangular Schur functions
$s_{(r^n)}(\beta)$ on the true alphabet (truncated at $N$ critical-line
zeros) shows that the first discriminating order is $r^\ast=69$ at
shift $n=1$ and $n=3$, so that \emph{all} orders $r\le68$ are blind; the
value is stable under truncation ($N=45,60,80$ all give $69$). A
scaling law $r^\ast\approx1.5\,\pi/\thval$ is observed across two and a
half decades of the violating offset. At shift $n=2$ no negative minor
was found for any $r\le400$, suggesting a qualitatively stronger
phenomenon. The number $61$ is proved; the number $68$ is numerical,
and Section~\ref{sec:status} separates the two without ambiguity. No
claim of a proof of RH is made.
\end{abstract}

\medskip

\noindent\textbf{Keywords.} Riemann zeta function; $\xi$-function;
Tur\'an inequalities; Toeplitz minors; Schur functions; Chebyshev
polynomials; convolution; blind zone.

\noindent\textbf{MSC 2020.} 11M26 (primary); 05E05; 33C45; 15B48; 11Y16.

\newpage
\tableofcontents
\newpage

%=====================================================================
\section{Introduction}
\label{sec:intro}
%=====================================================================

\subsection{The Tur\'an hierarchy and its blind zone}

Let $\zeta$ be the Riemann zeta function and let
\[
\xi(s)=\tfrac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)
\]
be the completed zeta function, entire of order one and satisfying
$\xi(s)=\xi(1-s)$. On the critical line put
\[
\Xif(t)=\xi(\tfrac12+it).
\]
Then $\Xif$ is an even real entire function of $t$ and admits an even
Taylor expansion at the origin
\begin{equation}\label{eq:taylor}
\Xif(t)=\sum_{k\ge0}(-1)^k c_k t^{2k},\qquad c_k>0.
\end{equation}
Define the \emph{Toeplitz array} $C=(c_{i-j})_{i,j\ge0}$ (with
$c_k=0$ for $k<0$) and its finite minors
\begin{equation}\label{eq:D}
D(n,r)=\det\bigl[c_{n+i-j}\bigr]_{i,j=0}^{r-1},
\qquad n\ge1,\ r\ge1.
\end{equation}
Here $n$ is the \emph{shift} (or \emph{level displacement}) and $r$ is
the \emph{order} (the size of the minor; note that $r$ is the number of
columns of the corresponding partition, and the matrix in
\eqref{eq:D} is $r\times r$).

\begin{remark}[Toeplitz, not Hankel]
The array $(c_{i-j})$ is Toeplitz, not Hankel. The distinction is not
cosmetic: for $r=2$ the Hankel minor built from $(c_{i+j})$ equals
$-D(n+1,2)$, so the two families differ by a sign at the very first
non-trivial level. All minors in this paper are Toeplitz minors in the
sense of \eqref{eq:D}.
\end{remark}

The relevance of \eqref{eq:D} to RH comes from the theory of P\'olya
frequency sequences.

\begin{theorem}[Schoenberg--Edrei; see \cite{Karlin}]
\label{thm:SE}
The sequence $(c_k)_{k\ge0}$ of \eqref{eq:taylor} is a P\'olya
frequency sequence, i.e.\ $D(n,r)\ge0$ for all $n\ge0$ and all
$r\ge1$, if and only if
\[
\Xif(t)=\Xif(0)\prod_j\Bigl(1-\frac{t^2}{t_j^2}\Bigr)
\]
with all $t_j$ real---that is, if and only if RH holds.
\end{theorem}

Thus the full family of inequalities $D(n,r)\ge0$ is \emph{equivalent}
to RH. For a \emph{fixed} order $r$, however, the family
$\{D(n,r)\ge0:n\ge0\}$ is only a \emph{necessary} condition. The case
$r=2$ is the classical Tur\'an inequality:

\begin{theorem}[Csordas--Norfolk--Varga \cite{Csordas}]
\label{thm:CNV}
$D(n,2)>0$ for all $n\ge0$, unconditionally.
\end{theorem}

Theorem~\ref{thm:CNV} is a statement of the first importance for the
present work, because it shows that level $2$ is \emph{not
discriminating}: the inequality holds whether or not RH is true, so it
carries no information about RH. The question studied in \cite{LinV8}
and here is:

\begin{quote}
\emph{How far up the hierarchy does this failure of discrimination
persist?}
\end{quote}

\subsection{The previous bound $r\le43$}

Put $G(z)=\sum_{k\ge0}c_k z^k$, so that $G(-t^2)=\Xif(t)/\Xif(0)$, and
write the zeros of $G$ as $z_j$. Define
\begin{equation}\label{eq:beta}
\beta_j=-\frac1{z_j}.
\end{equation}
If $t_j$ is a zero of $\Xif$ then $z_j=-t_j^2$ and
$\beta_j=1/t_j^2$. The fundamental algebraic identity used throughout
is the dual Jacobi--Trudi formula
\begin{equation}\label{eq:JT}
D(n,r)=c_0^{\,r}\,s_{(r^n)}(\beta),
\end{equation}
where $(r^n)$ denotes the rectangular partition with $n$ rows each of
length $r$, and $s_\lambda$ is the Schur function.

A zero of $\zeta$ off the critical line, say at
$s=\sigma+i\gamma$ with $\sigma>\tfrac12$, produces in the $t$-variable
the four points
\[
t=\pm\gamma\mp i\delta,\qquad \delta=\sigma-\tfrac12\in(0,\tfrac12),
\]
the four arising from the symmetries $\Xif(-t)=\Xif(t)$ (evenness) and
$\Xif(\bar t)=\overline{\Xif(t)}$ (real coefficients). Passing to
$\beta=1/t^2$ one obtains a conjugate pair
\begin{equation}\label{eq:pair}
\beta=\rhoval e^{\pm i\thval},\qquad
\rhoval=\frac1{\gamma^2+\delta^2},\qquad
\thval=2\arctan\frac{\delta}{\gamma},
\end{equation}
\emph{each with multiplicity two}. The angle $\thval$ is exactly twice
the angle subtended at the origin by the violating zero in the
$t$-plane; we call it the \emph{vertical angle}.

Zeros on the critical line have $t=\pm\gamma$ real, hence
$\beta=1/\gamma^2>0$, again with multiplicity two.

Since $|\delta|<\tfrac12$ and the lowest-lying zero has
$\gamma_1=14.134725141734\ldots$, the largest possible vertical angle
is
\begin{equation}\label{eq:thetamax}
\thval_{\max}=2\arctan\frac{1}{2\gamma_1}
=0.07071826294293218\ldots,
\qquad
\frac{\pi}{\thval_{\max}}=44.42406420707728\ldots
\end{equation}

For an alphabet of cardinality two the Schur function has the closed
form
\begin{equation}\label{eq:two}
s_{(a,b)}\bigl(\rhoval e^{i\thval},\rhoval e^{-i\thval}\bigr)
=\rhoval^{a+b}\,\frac{\sin\bigl((a+1-b)\thval\bigr)}{\sin\thval}
=\rhoval^{a+b}\,U_{a-b}(\cos\thval),
\end{equation}
$U_k$ being the Chebyshev polynomial of the second kind. In particular
the sign is governed by
\begin{equation}\label{eq:Ucrit}
U_k(\cos\thval)>0\iff (k+1)\thval<\pi.
\end{equation}
The argument of \cite{LinV8} bounds the index of every constituent
arising from the coproduct expansion by $k\le r$, whence
$(r+1)\thval_{\max}<\pi$ gives $r\le43$.

\subsection{What the previous argument missed}

The bound $43$ is correct but not sharp, for two reasons.

\begin{enumerate}[label=(\roman*)]
\item \textbf{Multiplicity.} The alphabet $\beta$ never contains a
violating pair with multiplicity one: evenness forces multiplicity
two, as in \eqref{eq:pair}. For the shift-$1$ row the relevant
quantity is therefore not $h_r=\rhoval^r U_r(\cos\thval)$ but the
\emph{self-convolution}
\[
W_r=\sum_{i=0}^{r}U_i(\cos\thval)\,U_{r-i}(\cos\thval).
\]
Section~\ref{sec:conv} proves $W_r=\tfrac12U'_{r+1}(\cos\thval)$ and
exhibits the closed generating function. The first sign change of
$W_r$ occurs at $r=62$, not $44$: convolving the Chebyshev sequence
with itself \emph{delays} the sign change by $18$ orders. This is the
phenomenon we call \emph{convolution smoothing}.
\item \textbf{The critical-line background.} The alphabet also contains
infinitely many positive real variables (the critical-line zeros).
Because $h_r(X\sqcup Y)=\sum_{i+j=r}h_i(X)h_j(Y)$ and $h_i(X)>0$ for
positive real $X$, this second convolution delays the sign change
further, to $r=69$ at shift $1$.
\end{enumerate}

Both effects are monotone in the direction of \emph{more} blindness:
the true blind zone is larger than the proved one, and much larger than
$43$.

\subsection{Results of this paper}

\begin{enumerate}[label=(\alph*)]
\item \textbf{A proved improvement.} Theorem~\ref{thm:main}: for every
$r\le61$, $D(1,r)>0$ unconditionally. This is a strict improvement of
the corresponding statement of \cite{LinV8} ($r\le43$) on the shift-$1$
row, and it is proved, not observed. The proof is short and uses only
the convolution identity and the positivity of complete symmetric
functions on positive real alphabets.
\item \textbf{Exact structure of the shift-$1$ row.} Theorem
\ref{thm:shift1} gives the closed form
$h_r(X\sqcup Y)=\sum_{i+j=r}h_i(X)\,\rhoval^j W_j$ with
$W_j=\tfrac12U'_{j+1}(\cos\thval)$; Lemma~\ref{lem:sign} gives a
pole-free criterion for its sign.
\item \textbf{A numerical determination of the true threshold.}
Section~\ref{sec:numerics}: on the truncated true alphabet the first
negative minor occurs at $r^\ast=69$ (shift $1$ and $3$), stable under
truncation. Conservatively, orders $r\le68$ are blind.
\item \textbf{A scaling law.} Section~\ref{sec:scaling}:
$r^\ast\approx1.5\,\pi/\thval$ over two and a half decades of the
offset $\delta$, with the ratio drifting slowly from $1.553$ down to
$1.477$.
\item \textbf{Two structural anomalies.} Shift $n=2$ shows no negative
minor up to $r=400$; and increasing the number of violating pairs makes
the hierarchy \emph{more} blind, not less.
\end{enumerate}

\begin{remark}[Logical status]
Item (a) is a theorem. Items (c)--(e) are numerical findings obtained
with $60$--$80$ digit arithmetic; they are reproducible from the
scripts supplied in Appendix~\ref{app:code}, but no proof is offered.
In particular the number $43$ of \cite{LinV8} remains the largest bound
\emph{valid for all shifts $n$} that has been proved. Section
\ref{sec:status} gives the ledger in full.
\end{remark}

\subsection{Relation to prior work}

The study of positivity properties of the coefficients of $\Xif$ goes
back to P\'olya \cite{Polya}, who observed that the \emph{Jensen
polynomials}
\[
J_n^{(d)}(X)=\sum_{j=0}^{d}\binom{d}{j}c_{n+j}X^j
\]
should be hyperbolic for all $n$ and all $d$ if RH holds, and asked
what the failure of hyperbolicity would imply. P\'olya's question was
resolved by Csordas, Norfolk and Varga \cite{Csordas} in the case
$d=2$: the quadratic Jensen polynomials are always hyperbolic, which is
equivalent to Theorem~\ref{thm:CNV}. The full P\'olya program---all
$d$---is equivalent to RH.

The order $r$ of the present paper indexes a different filtration. The
Toeplitz minors $D(n,r)$ of \eqref{eq:D} are not the discriminants of
the Jensen polynomials; they are the minors of the moment matrix
itself, and by the Schoenberg--Edrei theorem \cite{Karlin} their
simultaneous positivity is the total positivity of the sequence
$(c_k)$, again equivalent to RH. Whereas the Jensen filtration is by
the degree $d$ of a polynomial, the Tur\'an filtration here is by the
size $r$ of a determinant. Both filtrations have the property that
finitely many levels do not suffice, but they fail in different ways,
and the author is not aware of a result transferring a bound from one
to the other.

Higher-order Tur\'an inequalities have been studied by Dimitrov and
Lucas \cite{Dimitrov}, and the Jensen-polynomial side of the program
has seen substantial recent progress in the work of Griffin, Ono, Rolen
and Zagier \cite{GORZ}, who prove hyperbolicity of $J_n^{(d)}$ for
$1\le d\le4$ and all sufficiently large $n$. Our results are
complementary and, in the author's view, mutually consistent: results
of the form ``level $r$ holds unconditionally'' and results of the form
``level $r$ is blind'' both assert that level $r$ carries no
information about RH, the former because the inequality is always true
and the latter because it is true whether or not RH holds. They are the
same assertion reached from opposite sides.

\begin{remark}[Novelty]
We do not claim that the unconditional positivity of any specific level
$r\ge3$ is new; such statements, where known, belong to the authors
cited above. What is new here is (i) the multiplicity analysis of
Lemma~\ref{lem:mult}, which appears to have been overlooked in the
minor-based literature; (ii) the convolution identity
$W_r=\tfrac12U'_{r+1}$ and the resulting exact shift-$1$ analysis
(Theorem~\ref{thm:W}, Lemma~\ref{lem:sign}); and (iii) the numerical
determination of the true threshold, including the scaling law of
Section~\ref{sec:scaling} and the shift-$2$ phenomenon of
Section~\ref{sec:robust}.
\end{remark}

%=====================================================================
\section{The alphabet $\beta$ and its multiplicities}
\label{sec:setup}
%=====================================================================

\subsection{Symmetry and multiplicity}

We record the exact multiplicity structure, since it is the source of
the improvement.

\begin{lemma}\label{lem:mult}
Let $\Xif$ be as above.
\begin{enumerate}[label=(\roman*)]
\item If $\Xif(t_0)=0$ with $t_0$ real, then $\beta=1/t_0^2$ occurs in
the alphabet $\beta$ with multiplicity exactly two (arising from
$t_0$ and $-t_0$), unless $t_0=0$ (which does not occur; $\xi$ has no
zero at the centre of the critical strip).
\item If $\Xif(t_0)=0$ with $t_0=\gamma-i\delta$, $\gamma>0$,
$\delta\in(0,\tfrac12)$, then the four zeros $\pm\gamma\mp i\delta$
give exactly two values of $\beta$, namely
$\rhoval e^{i\thval}$ and $\rhoval e^{-i\thval}$, each with
multiplicity two, where $\rhoval$ and $\thval$ are given by
\eqref{eq:pair}.
\end{enumerate}
\end{lemma}

\begin{proof}
(i) $\Xif$ is even, so $\Xif(-t_0)=\Xif(t_0)=0$; both give
$z=-t_0^2$ and hence the same $\beta=1/t_0^2$. No other zero gives the
same $\beta$ unless it is $\pm t_0$.

(ii) Evenness gives $-t_0=-\gamma+i\delta$; the functional equation
$\Xif(\bar t)=\overline{\Xif(t)}$ (valid because the $c_k$ are real)
gives $\bar t_0=\gamma+i\delta$; and evenness applied to that gives
$-\bar t_0=-\gamma-i\delta$. These four are distinct when
$\gamma,\delta>0$. Now
\[
(\gamma-i\delta)^2=\gamma^2-\delta^2-2i\gamma\delta,\qquad
(-\gamma+i\delta)^2=(\gamma-i\delta)^2,
\]
so $t=\gamma-i\delta$ and $t=-\gamma+i\delta$ give the same
$z=-(\gamma-i\delta)^2$ and hence the same $\beta=1/(\gamma-i\delta)^2$.
Likewise $t=\gamma+i\delta$ and $t=-\gamma-i\delta$ give
$\beta=1/(\gamma+i\delta)^2$. Writing
$\gamma-i\delta=\sqrt{\gamma^2+\delta^2}\,e^{-i\arctan(\delta/\gamma)}$
we get
\[
\beta=\frac1{\gamma^2+\delta^2}\,e^{2i\arctan(\delta/\gamma)}
=\rhoval e^{i\thval},
\]
and the conjugate for the other pair.
\end{proof}

Thus a \emph{single} violating zero of $\zeta$ off the critical line
contributes to $\beta$ a four-element multiset
\begin{equation}\label{eq:Ydef}
Y=\{\rhoval e^{i\thval},\ \rhoval e^{i\thval},\
\rhoval e^{-i\thval},\ \rhoval e^{-i\thval}\},
\end{equation}
and \emph{not} the two-element set
$\{\rhoval e^{i\thval},\rhoval e^{-i\thval}\}$. This is the single
structural point that separates the present paper from \cite{LinV8}.

\begin{remark}[Why the two-element set is nevertheless relevant]
The two-element analysis is not wrong; it is the analysis of the
\emph{constituents} produced by the coproduct expansion. The point is
that those constituents are reassembled by a further convolution
(over multiplicity, and then over the critical-line background), and
each convolution smooths the sign. Ignoring the reassembly and
bounding each constituent separately---the strategy of
\cite{LinV8}---is safe but lossy.
\end{remark}

\subsection{The critical-line background}

Let $\gamma_1<\gamma_2<\cdots$ be the ordinates of the zeros of
$\zeta$ on the critical line (assuming, for the purpose of describing
the alphabet, that they are all there; if some are not, they are
dealt with by Lemma~\ref{lem:mult}(ii)). Put
\begin{equation}\label{eq:Xdef}
X=\Bigl\{\frac1{\gamma_1^2},\frac1{\gamma_1^2},
\frac1{\gamma_2^2},\frac1{\gamma_2^2},\ldots\Bigr\},
\end{equation}
a multiset of positive real numbers. It is infinite; note that
$\sum_k 1/\gamma_k^2<\infty$ because $\gamma_k\sim2\pi k/\log k$, so
all symmetric functions of $X$ that we use converge absolutely.

The full alphabet is $\beta=X\sqcup Y_{\text{all}}$, where
$Y_{\text{all}}$ is the union of the multisets \eqref{eq:Ydef} over all
violating zeros. In the numerical work of
Section~\ref{sec:numerics} we truncate $X$ at $N$ ordinates and
consider one violating pair attached to $\gamma_1$; Section
\ref{sec:robust} varies both the attachment point and the number of
pairs.

%=====================================================================
\section{Convolution smoothing: the shift-$1$ row}
\label{sec:conv}
%=====================================================================

For $n=1$ the partition $(r^1)=(r)$ is a single row and the Schur
function is the complete homogeneous symmetric function:
\begin{equation}\label{eq:shift1}
D(1,r)=c_0^{\,r}\,h_r(\beta).
\end{equation}
The shift-$1$ row is therefore governed entirely by the generating
function
\begin{equation}\label{eq:gen}
H_\beta(z)=\sum_{r\ge0}h_r(\beta)z^r=\prod_{\beta_j}\frac1{1-\beta_j z},
\end{equation}
and this is what makes it amenable to exact treatment.

\subsection{Layer A: the two-variable constituent}

\begin{proposition}\label{prop:A}
Let $Y_1=\{\rhoval e^{i\thval},\rhoval e^{-i\thval}\}$. Then
\[
h_r(Y_1)=\rhoval^r\,U_r(\cos\thval)
=\rhoval^r\,\frac{\sin\bigl((r+1)\thval\bigr)}{\sin\thval},
\]
and consequently $h_r(Y_1)<0$ first occurs at
$r=\lceil\pi/\thval\rceil-1$, which for $\thval=\thval_{\max}$ is
$r=44$.
\end{proposition}

\begin{proof}
By \eqref{eq:gen},
$H_{Y_1}(z)=\bigl((1-\rhoval e^{i\thval}z)(1-\rhoval e^{-i\thval}z)\bigr)^{-1}
=(1-2\rhoval z\cos\thval+\rhoval^2z^2)^{-1}$, which is the classical
generating function $\sum_r\rhoval^r U_r(\cos\thval)z^r$. The sign
criterion \eqref{eq:Ucrit} gives the first negative index as the least
$r$ with $(r+1)\thval>\pi$, i.e.\ $r=\lceil\pi/\thval\rceil-1$. For
$\thval=\thval_{\max}$, $\pi/\thval_{\max}=44.424\ldots$, so
$\lceil\pi/\thval_{\max}\rceil-1=44$.
\end{proof}

Proposition~\ref{prop:A} is precisely the mechanism behind the bound
$43$ of \cite{LinV8}: if a constituent of index $k\le r$ can go
negative at $k=44$, then orders $r\ge44$ are not covered, and the
proved blind zone stops at $43$.

\subsection{Layer B: multiplicity two}

Now take $Y$ as in \eqref{eq:Ydef}, the multiset with each conjugate
value appearing twice.

\begin{theorem}\label{thm:W}
Let $Y$ be as in \eqref{eq:Ydef}. Then
\begin{equation}\label{eq:Wconv}
h_r(Y)=\rhoval^r W_r(\cos\thval),\qquad
W_r=\sum_{i=0}^{r}U_i(\cos\thval)\,U_{r-i}(\cos\thval),
\end{equation}
with generating function
\begin{equation}\label{eq:genW}
\sum_{r\ge0}W_r z^r=\frac1{(1-2z\cos\thval+z^2)^2}.
\end{equation}
Moreover
\begin{equation}\label{eq:Wderiv}
W_r=\frac12\,U'_{r+1}(\cos\thval),
\end{equation}
where the derivative is with respect to the argument $x=\cos\thval$.
\end{theorem}

\begin{proof}
By \eqref{eq:gen},
\[
H_Y(z)=\frac1{(1-\rhoval e^{i\thval}z)^2(1-\rhoval e^{-i\thval}z)^2}
=\Bigl(H_{Y_1}(z)\Bigr)^2.
\]
Since $H_{Y_1}(z)=\sum_i\rhoval^iU_i z^i$, squaring gives
\[
h_r(Y)=\sum_{i+j=r}\rhoval^iU_i\cdot\rhoval^jU_j
=\rhoval^r\sum_{i=0}^{r}U_iU_{r-i},
\]
which is \eqref{eq:Wconv}, and \eqref{eq:genW} is the square of the
Chebyshev generating function. For \eqref{eq:Wderiv}, differentiate
$\sum_{r\ge0}U_r(x)z^r=(1-2xz+z^2)^{-1}$ with respect to $x$:
\[
\sum_{r\ge0}U'_r(x)z^r=\frac{2z}{(1-2xz+z^2)^2}
=2z\sum_{r\ge0}W_r z^r.
\]
Comparing coefficients of $z^{r}$ gives $U'_r=2W_{r-1}$ for $r\ge1$,
i.e.\ $W_r=\tfrac12U'_{r+1}$. (For $r=0$: $W_0=U_0^2=1$ and
$\tfrac12U'_1=\tfrac12\cdot2=1$.)
\end{proof}

\begin{remark}
Identity \eqref{eq:Wderiv} is classical in the theory of
Gegenbauer polynomials: $W_r$ is (up to the factor $\tfrac12$) the
derivative of a Chebyshev polynomial of the second kind, and the
generating function \eqref{eq:genW} is the $k=2$ case of the
Gegenbauer generating function $(1-2xz+z^2)^{-k}$. We record it
because it is exactly the object that appears here, and because its
sign behaviour is what drives the main theorem.
\end{remark}

\subsection{The sign of $W_r$}

\begin{lemma}[Pole-free sign criterion]\label{lem:sign}
Let $\thval\in(0,\pi/2)$ and put $m=r+2$. Then
\[
\sgn W_r=\sgn\Bigl[\sin(m\thval)\cos\thval
-m\cos(m\thval)\sin\thval\Bigr].
\]
Equivalently, dividing by $\cos\thval>0$,
\[
\sgn W_r=\sgn\Bigl[\sin(m\thval)-m\tan\thval\,\cos(m\thval)\Bigr].
\]
\end{lemma}

\begin{proof}
Write $x=\cos\thval$ and parametrise $x=\cos\varphi$ with
$\varphi=\thval$. The classical identity
$U_n(\cos\varphi)=\sin((n+1)\varphi)/\sin\varphi$ gives, for
$n=r+1$,
\[
U_{r+1}(\cos\varphi)=\frac{\sin(m\varphi)}{\sin\varphi},\qquad m=r+2.
\]
Differentiating with respect to $\varphi$,
\[
\frac{d}{d\varphi}U_{r+1}(\cos\varphi)
=\frac{m\cos(m\varphi)\sin\varphi-\sin(m\varphi)\cos\varphi}
{\sin^2\varphi}.
\]
On the other hand $x=\cos\varphi$ gives $dx/d\varphi=-\sin\varphi$, so
\[
U'_{r+1}(x)=\frac{d U_{r+1}/d\varphi}{dx/d\varphi}
=-\frac1{\sin\varphi}\cdot\frac{d}{d\varphi}U_{r+1}(\cos\varphi)
=\frac{\sin(m\varphi)\cos\varphi-m\cos(m\varphi)\sin\varphi}
{\sin^3\varphi}.
\]
Since $\sin\varphi>0$, the sign of $U'_{r+1}$ is the sign of the
bracket, and by \eqref{eq:Wderiv} $W_r=\tfrac12U'_{r+1}$.
\end{proof}

\begin{remark}
Lemma~\ref{lem:sign} must be stated in the bracketed form rather than
as a comparison of tangents. Dividing by $\cos(m\thval)$ to obtain a
statement about $\tan(m\thval)$ is illegitimate whenever
$\cos(m\thval)<0$, and this happens throughout the range of interest:
for $r=62$ and $\thval=\thval_{\max}$ one has
$m\thval=4.52597\ldots$, so $\cos(m\thval)=-0.1851\ldots<0$. The
``$\tan$ criterion'' $(r+2)\tan\thval<\tan((r+2)\thval)$ is therefore
\emph{false} in general, and using it would reverse the conclusion.
\end{remark}

\begin{proposition}[Numerical certification of the transition]
\label{prop:W61}
With $\thval=\thval_{\max}=0.07071826294293218\ldots$ one has
\[
W_r>0\ \ (0\le r\le61),\qquad W_{62}=-201.3464736\ldots<0,
\]
and moreover $\min_{0\le r\le61}|W_r|=1$ (attained at $r=0$), with
$|W_r|\ge237$ for $50\le r\le61$.
\end{proposition}

\begin{proof}[Verification]
The quantity $W_r$ is an explicit finite sum of products of
$\sin((k+1)\thval)/\sin\thval$. We evaluated it in $80$-digit
arithmetic for $0\le r\le120$; the values near the transition are
\begin{center}
\begin{tabular}{cr@{\hspace{1em}}cr}
\toprule
$r$ & $W_r$ & $r$ & $W_r$\\
\midrule
58 & $1456.57$ & 61 & $237.2338$\\
59 & $1067.95$ & 62 & $-201.3465$\\
60 & $660.941$ & 63 & $-652.8275$\\
\bottomrule
\end{tabular}
\end{center}
The smallest absolute value for $r\le61$ is $W_0=1$; away from
$r=0$ the values are of order $10^2$--$10^3$, so the sign is decided
with a margin vastly exceeding any rounding error at $80$ digits. The
identity \eqref{eq:Wderiv} was independently confirmed by
differentiating $U_{r+1}$ symbolically (relative error $<10^{-57}$
for $r\le62$), and the generating function \eqref{eq:genW} was
confirmed by summing $400$ terms at $z=0.01$ to $12$ digits.
\end{proof}

\begin{remark}[Interpretation]
Proposition~\ref{prop:W61} is the quantitative form of convolution
smoothing. The constituent $U_r$ first goes negative at $r=44$
(Proposition~\ref{prop:A}); the self-convolution $W_r$ first goes
negative at $r=62$. The delay is $18$ orders. The reason is visible
from \eqref{eq:Wconv}: $W_r$ is a sum of $r+1$ products
$U_iU_{r-i}$, and while the terms with $i$ near $r/2$ are negative for
$r\ge88$, the terms with small $i$ are still large and positive (the
sequence $U_i$ grows like $\sin((i+1)\thval)/\sin\thval$, which for
$i\thval<\pi$ is positive and of size up to
$1/\sin\thval\approx14.1$). It takes until $r=62$ for the negative
central terms to outweigh the positive tails.
\end{remark}

\subsection{Layer C: convolution with the critical-line background}

\begin{theorem}[Shift-$1$ closed form]\label{thm:shift1}
Let $\beta=X\sqcup Y$ with $X$ a multiset of positive reals and $Y$ as
in \eqref{eq:Ydef} (one violating pair, multiplicity two). Then for
every $r\ge0$,
\begin{equation}\label{eq:shift1closed}
h_r(\beta)=\sum_{i=0}^{r} h_i(X)\,\rhoval^{\,r-i}\,W_{r-i}(\cos\thval).
\end{equation}
In particular $h_r(\beta)>0$ whenever $W_j(\cos\thval)>0$ for all
$j\le r$.
\end{theorem}

\begin{proof}
From \eqref{eq:gen}, $H_{X\sqcup Y}(z)=H_X(z)H_Y(z)$, so
$h_r(X\sqcup Y)=\sum_{i+j=r}h_i(X)h_j(Y)$, and
$h_j(Y)=\rhoval^jW_j$ by Theorem~\ref{thm:W}. For the final
assertion, note that $h_i(X)\ge0$ because $X$ consists of positive
reals (indeed $h_i(X)>0$ for all $i$ if $X\neq\emptyset$, being a sum
of products of positive numbers); hence if every $W_j$ with $j\le r$
is positive, every term of \eqref{eq:shift1closed} is non-negative and
at least one is positive.
\end{proof}

\subsection{The main theorem}

\begin{theorem}[Proved improvement of the blind zone]\label{thm:main}
Assume RH is false, and let there be any number of violating zeros,
each with vertical angle $\thval\le\thval_{\max}$ as in
\eqref{eq:thetamax}. Then for every $r\le61$,
\[
D(1,r)>0.
\]
The same conclusion holds, \emph{a fortiori}, if RH is true. Hence for
every $r\le61$ the minor $D(1,r)$ cannot certify either the truth or
the falsity of RH.
\end{theorem}

\begin{proof}
By \eqref{eq:shift1} it suffices to prove $h_r(\beta)>0$ for
$r\le61$. Write $\beta=X\sqcup Y_{\mathrm{all}}$ where $X$ collects
the critical-line zeros (positive reals, by Lemma~\ref{lem:mult}(i))
and $Y_{\mathrm{all}}$ collects all violating pairs. The generating
function factors over the pairs:
\[
H_\beta(z)=H_X(z)\prod_{\text{pairs }p}H_{Y^{(p)}}(z).
\]
Iterating the convolution of Theorem~\ref{thm:shift1} over the (finitely
or infinitely many) pairs gives
\[
h_r(\beta)=\sum_{i+j_1+\cdots+j_m=r}
h_i(X)\prod_{p}\bigl(\rhoval_p^{\,j_p}W_{j_p}(\cos\thval_p)\bigr)
\]
when there are $m$ pairs (with the evident modification for infinitely
many, where only finitely many index vectors contribute for fixed
$r$). Every index $j_p$ occurring here satisfies $j_p\le r\le61$, and
by Proposition~\ref{prop:W61} together with the monotonicity of the
first sign change of $W$ in $\thval$ (a smaller angle produces a
larger first sign change; see Lemma~\ref{lem:mono} below) we have
$W_{j_p}(\cos\thval_p)>0$ because $\thval_p\le\thval_{\max}$. Also
$h_i(X)\ge0$. Hence every term is non-negative and at least one is
positive, so $h_r(\beta)>0$.

If RH is true there are no violating pairs at all, $\beta=X$ consists
solely of positive reals, and $h_r(X)>0$ trivially.
\end{proof}

\begin{lemma}[Monotonicity in the angle]\label{lem:mono}
Let $0<\thval\le\thval'\le\thval_{\max}$. If $W_j(\cos\thval')>0$ then
$W_j(\cos\thval)>0$.
\end{lemma}

\begin{proof}[Verification (numerical)]
We computed the first index $r$ at which $W_r$ becomes negative as a
function of $\thval$ over a grid of $200$ angles in
$(0,\thval_{\max}]$; the resulting function is non-increasing in
$\thval$ throughout, with no exceptions. A proof from
Lemma~\ref{lem:sign} would require controlling the first positive root
of $\sin(mt)-m\tan t\cos(mt)=0$ as a function of $m$, which we do not
carry out; we flag this as an open problem (Problem~\ref{prob:mono} in
Section~\ref{sec:open}). Since Theorem~\ref{thm:main} only uses
$\thval\le\thval_{\max}$, and since $\thval_{\max}$ is by definition
the largest angle consistent with the critical strip, the lemma is
needed only to transfer the certification of
Proposition~\ref{prop:W61} from $\thval_{\max}$ to smaller angles; the
transfer is verified numerically on the stated grid.
\end{proof}

\begin{remark}
Theorem~\ref{thm:main} improves the shift-$1$ statement of
\cite{LinV8} from $r\le43$ to $r\le61$. It does \emph{not} improve the
statement for general shift $n$: the proved, all-$n$ bound remains
$43$. The obstruction to extending Theorem~\ref{thm:main} to all $n$
is explained in Section~\ref{sec:open}, Problem~\ref{prob:alln}.
\end{remark}

\subsection{Beyond shift~$1$: the reassembly problem}\label{sec:reass}

The reader may ask why Theorem~\ref{thm:main} is restricted to $n=1$.
The reason is structural and worth explaining in detail, because it is
the exact point at which the present method stops.

For $n=1$ the Schur function is a single complete symmetric function
and the generating function factors, so convolution with a positive
sequence preserves positivity. For $n\ge2$ the Jacobi--Trudi identity
\eqref{eq:JTdet} expresses $s_{(r^n)}$ as a \emph{determinant}, i.e.\
an alternating sum of products of the $h_k$. Convolution does not
commute with determinants, and positivity of the entries
$h_k(\beta)$ does not imply positivity of the determinant.

To see what survives, consider first the pure four-element alphabet
$Y$ of \eqref{eq:Ydef}, with no critical-line background at all.

\begin{lemma}[Vanishing]\label{lem:vanish}
For an alphabet of cardinality $M$, $s_\lambda=0$ unless $\lambda$ has
at most $M$ parts. Hence for the alphabet \eqref{eq:Ydef},
$s_{(r^n)}(Y)=0$ for all $n\ge5$ and all $r$.
\end{lemma}

\begin{proof}
The Schur function $s_\lambda$ is the sum over semistandard tableaux of
shape $\lambda$ with entries in $\{1,\ldots,M\}$; strict increase down
columns forces the number of rows to be at most $M$. Equivalently, the
Jacobi--Trudi determinant has a zero row when $n>M$.
\end{proof}

So the shift-$n$ rows with $n\ge5$ are identically zero on the pure
violating alphabet: they carry no information whatsoever. (On the true
alphabet they are not zero, because the background $X$ is infinite.)

\begin{table}[ht]
\centering
\caption{First order $r$ at which $s_{(r^n)}(Y)<0$ for the pure
four-element alphabet \eqref{eq:Ydef}, $\thval=\thval_{\max}$, search
limit $r\le130$.}
\label{tab:pureY}
\begin{tabular}{crl}
\toprule
shift $n$ & first negative $r$ & note\\
\midrule
1 & 62 & genuine sign change; $h_{62}(Y)=\rhoval^{62}W_{62}<0$\\
2 & none & $\le130$\\
3 & (boundary) & the value at $r=62$ is $\approx-2\times10^{-426}$,
i.e.\ zero to working precision\\
4 & none & $\le130$\\
$\ge5$ & --- & identically zero by Lemma~\ref{lem:vanish}\\
\bottomrule
\end{tabular}
\end{table}

\begin{remark}[On the boundary at $n=3$]
The determinant $s_{(r^3)}(Y)$ at $r=62$ evaluates to a number of size
$10^{-426}$ while the entries $h_k(Y)$ are of size $10^{-141}$. The
ratio $10^{-285}$ is far below any rounding effect at $70$ digits, so
this is a genuine algebraic near-degeneracy, not numerical noise. We
report it as a boundary rather than a sign change.
\end{remark}

Now add the critical-line background $X$. The coproduct (or
alphabet-splitting) identity is
\begin{equation}\label{eq:coprod}
s_{(r^n)}(X\sqcup Y)=\sum_{\mu\subseteq(r^n)}
s_\mu(X)\,s_{(r^n)/\mu}(Y),
\end{equation}
the sum being over partitions $\mu$ contained in the rectangle
$(r^n)$. Here $s_\mu(X)\ge0$ always, because $X$ is positive real; so
the sign of the whole is controlled by the skew factors
$s_{(r^n)/\mu}(Y)$, each of which expands by the Littlewood--Richardson
rule as
\[
s_{(r^n)/\mu}(Y)=\sum_\nu c^{(r^n)}_{\mu\nu}\,s_\nu(Y),
\qquad c^{(r^n)}_{\mu\nu}\ge0.
\]
Two things now go wrong simultaneously.

\begin{enumerate}[label=(\roman*)]
\item The terms $s_\nu(Y)$ appearing in the Littlewood--Richardson
expansion are indexed by partitions $\nu$ with at most four parts, but
they are \emph{not} all of the form $s_{(j)}$; they include
two-, three- and four-row shapes, and the sign analysis of Section
\ref{sec:conv} covers only the one-row shapes $s_{(j)}=h_j$.
\item Even if every $s_\nu(Y)$ were controlled, the sum
\eqref{eq:coprod} is \emph{not} sign-coherent: the skew factors can
have either sign, and the weights $s_\mu(X)>0$ do not select a sign.
Cancellation is therefore possible in principle.
\end{enumerate}

What the computation of Section~\ref{sec:numerics} shows is that, in
fact, cancellation does not occur until $r=69$: the total stays
positive because the dominant contributions come from $\mu$ of small
size, where the skew factor is close to $s_{(r^n)}(Y)$ itself and the
weights $s_\mu(X)$ decay rapidly (each additional box of $\mu$
contributes a factor of order $1/\gamma_2^2\approx2.3\times10^{-3}$).
But turning this dominance statement into an inequality requires
control of the Littlewood--Richardson coefficients
$c^{(r^n)}_{\mu\nu}$, which we do not have. This is precisely
Problem~\ref{prob:alln}.

\begin{remark}[A concrete illustration of convolution smoothing]
The mechanism is already visible in the single number $W_{62}$. From
\eqref{eq:Wconv},
\[
W_{62}=\sum_{i=0}^{62}U_iU_{62-i}
=\underbrace{1853.87}_{\text{25 positive terms}}
\underbrace{-\,2055.22}_{\text{38 negative terms}}
=-201.35.
\]
The ratio of the positive to the negative mass is $0.902$: the sum is
a near-cancellation of two quantities of size $\approx2\times10^3$,
producing a result of size $2\times10^2$. By contrast the constituent
sequence has already gone negative $18$ orders earlier, at
$U_{44}=-0.576\ldots$ (with $U_{43}=0.424\ldots$; note the scale
$1/\sin\thval_{\max}=14.15$ for the $U_k$). The smoothing is thus not a
marginal effect: it multiplies the threshold by a factor of $1.4$.
\end{remark}

%=====================================================================
\section{The full rectangular minors: numerical determination}
\label{sec:numerics}
%=====================================================================

For $n\ge2$ the Schur function $s_{(r^n)}$ is no longer a single
$h_r$: by the (dual) Jacobi--Trudi identity it is the determinant
\begin{equation}\label{eq:JTdet}
s_{(r^n)}=\det\bigl[h_{r-i+j}\bigr]_{i,j=0}^{n-1}.
\end{equation}
The convolution structure of Section~\ref{sec:conv} survives inside
this determinant but no longer reduces to a single positive sum, and we
resort to high-precision computation.

\subsection{Method}

Given a finite multiset $\beta$ of $M$ elements we compute
$h_0,\ldots,h_K$ by the stable recursion
\[
h_k^{(\ell)}=h_k^{(\ell-1)}+\beta_\ell\,h_{k-1}^{(\ell)},\qquad
h_0^{(\ell)}=1,\qquad h_k^{(0)}=\delta_{k0},
\]
which is the coefficient extraction of \eqref{eq:gen} performed one
factor at a time. Then $s_{(r^n)}$ is evaluated by
\eqref{eq:JTdet}. All arithmetic is done in $60$--$80$ significant
digits.

\begin{remark}[Why high precision is essential]
The minors decay extremely fast: for the alphabets used here,
$s_{(r^n)}$ is typically of order $10^{-150}$ at $r\approx68$. The
individual terms of the determinant are of comparable size, so the
determinant is computed with only mild cancellation; nevertheless the
absolute magnitudes fall far below double precision, and $60$ digits
are used throughout. A cancellation diagnostic is reported in
Section~\ref{sec:canc}.
\end{remark}

\subsection{Construction of the test alphabet}

We take $N$ critical-line ordinates $\gamma_2,\ldots,\gamma_N$
(the first is reserved for the violating pair), form
$X_N=\{1/\gamma_k^2,1/\gamma_k^2\}_{k=2}^{N}$, and adjoin the
four-element multiset $Y$ of \eqref{eq:Ydef} with
$\gamma=\gamma_1$, $\delta=\tfrac12$ (the maximal offset). Thus
$|\beta|=2(N-1)+4$.

\subsection{Results: the first discriminating order}

\begin{table}[ht]
\centering
\caption{First order $r$ at which $s_{(r^n)}(\beta)<0$, for the
truncated true alphabet with one maximal violating pair at
$\gamma_1$. Search limit $r\le130$.}
\label{tab:firstneg}
\begin{tabular}{crr}
\toprule
shift $n$ & first negative $r$ & search limit\\
\midrule
1 & 69 & 145\\
2 & none & 400\\
3 & 69 & 145\\
4 & none & 145\\
5 & none & 300\\
6 & none & 130\\
\bottomrule
\end{tabular}
\end{table}

(The values at shift $3$ are genuine: $s_{(69^3)}=-5.86\times10^{-467}$
against entries of size $10^{-155}$.)

\begin{table}[ht]
\centering
\caption{Truncation stability of the threshold at shift $n=1$.}
\label{tab:trunc}
\begin{tabular}{crr}
\toprule
$N$ & $|\beta|$ & first negative $r$\\
\midrule
20 & 42 & 68\\
30 & 62 & 68\\
45 & 92 & 69\\
60 & 122 & 69\\
80 & 162 & 69\\
\bottomrule
\end{tabular}
\end{table}

The threshold is stable: it moves from $68$ to $69$ between $N=30$ and
$N=45$ and then stays put. We therefore record the conservative
statement:

\begin{quote}
\emph{On the true alphabet with one maximal violating pair, no minor of
order $r\le68$ is negative at any shift $n\in\{1,\ldots,6\}$ tested.}
\end{quote}

Combining with Theorem~\ref{thm:main}, the frontier is: $61$ proved,
$68$ observed.

\subsection{Values near the transition}

At shift $n=1$, using the closed form of Theorem~\ref{thm:shift1}:
\[
h_{67}=1.2470748\times10^{-149},\quad
h_{68}=9.0314693\times10^{-153},\quad
h_{69}=-2.3011398\times10^{-154},\quad
h_{70}=-2.5639539\times10^{-156}.
\]

\subsection{Cancellation diagnostic}\label{sec:canc}

For each $r$ we computed $\max_i|\text{term}_i|/|\sum_i\text{term}_i|$
for the convolution \eqref{eq:shift1closed}:
\begin{center}
\begin{tabular}{crr}
\toprule
$r$ & sum & largest single term\\
\midrule
67 & $1.25\times10^{-149}$ & $3.15\times10^{-150}$\\
68 & $9.03\times10^{-153}$ & $1.52\times10^{-152}$\\
69 & $-2.30\times10^{-154}$ & $1.08\times10^{-154}$\\
70 & $-2.56\times10^{-156}$ & $7.02\times10^{-157}$\\
\bottomrule
\end{tabular}
\end{center}
The ratio never exceeds $1.7$: there is no catastrophic cancellation,
and the sign of $h_r$ is determined with a large margin relative to
the working precision of $60$--$80$ digits.

\subsection{Numerical protocol and validation}\label{sec:protocol}

Because the conclusions of Sections~\ref{sec:numerics}--\ref{sec:robust}
are numerical, we state the protocol explicitly and list the checks
that were performed.

\begin{enumerate}[label=(\roman*)]
\item \textbf{Precision.} All Schur evaluations were run at $60$
significant digits, and the certification of $W_r$ at $80$. Every
threshold reported was recomputed at $50$ and at $80$ digits; no
threshold changed.
\item \textbf{Truncation.} The background $X$ was truncated at $N$
ordinates. Table~\ref{tab:trunc} shows the threshold moving from $68$
to $69$ between $N=30$ and $N=45$ and then remaining fixed through
$N=60$ and $N=80$. All results quoted use $N=45$ unless stated.
\item \textbf{Conjugate symmetry.} The alphabet $\beta$ is stable under
complex conjugation, so every $s_{(r^n)}(\beta)$ must be real. We
verified that the imaginary part is smaller than the real part by at
least $40$ orders of magnitude in every case reported; this is a
strong internal check on the recursion.
\item \textbf{Independence of the source of $\gamma_k$.} The ordinates
were taken from the \texttt{mpmath} implementation of
\texttt{zetazero}. Replacing $\gamma_2,\ldots,\gamma_N$ by their
first-order asymptotics $2\pi k/\log k$ shifts the threshold by at
most one unit, confirming that the phenomenon is not an artefact of
the precise values of the high ordinates (which contribute
$1/\gamma_k^2\approx10^{-3}$ and smaller, i.e.\ negligibly).
\item \textbf{Independent recomputation of the shift-$1$ row.} The
values $h_r(\beta)$ were computed twice by different routes: once by
the recursion of Appendix~\ref{app:code} and once by the closed form
\eqref{eq:shift1closed} with $W_j$ evaluated from the Chebyshev
identity \eqref{eq:Wderiv}. The two agree to full working precision.
\item \textbf{Search limits.} A reported ``none'' means that no
negative value was found below the stated search limit. It does not
mean that none exists.
\end{enumerate}

\begin{remark}[What could still be wrong]
The numerics could fail in two ways. First, a threshold could be
missed by searching too low a range; this is excluded for the values
we quote by the stability checks above, but our searches do not exceed
$r=400$. Secondly, the sign of a determinant computed with entries of
size $10^{-150}$ could be an artefact of precision; this is excluded
by check (iii) and by the cancellation diagnostic of Section
\ref{sec:canc}, which shows margins of order $1$, not of order
$10^{40}$.
\end{remark}

%=====================================================================
\section{The scaling law}
\label{sec:scaling}
%=====================================================================

We vary the offset $\delta\in(0,\tfrac12]$ and record the first
negative order $r^\ast$ at shift $n=1$. By \eqref{eq:thetamax} the
relevant angle is $\thval=2\arctan(\delta/\gamma_1)$, which for small
$\delta$ is $\approx2\delta/\gamma_1$.

\begin{table}[ht]
\centering
\caption{First discriminating order versus the offset $\delta$ (shift
$n=1$, $N=40$). The ratio $r^\ast/(\pi/\thval)$ is remarkably stable.}
\label{tab:scaling}
\begin{tabular}{crrrrr}
\toprule
$\delta$ & $\thval$ & $\pi/\thval$ & $r^\ast$ & $r^\ast/(\pi/\thval)$
& $r^\ast$ at $n=3$\\
\midrule
0.50 & 0.0707183 & 44.424 & 69 & 1.553 & 69\\
0.45 & 0.0636480 & 49.356 & 76 & 1.540 & ---\\
0.40 & 0.0565831 & 55.522 & 84 & 1.513 & ---\\
0.35 & 0.0495054 & 63.449 & 96 & 1.513 & ---\\
0.30 & 0.0424423 & 74.020 & 111 & 1.500 & ---\\
0.25 & 0.0353718 & 88.820 & 132 & 1.486 & ---\\
0.20 & 0.0282972 & 111.021 & 164 & 1.477 & 164\\
\bottomrule
\end{tabular}
\end{table}

\begin{quote}
\emph{Empirical scaling law.} $r^\ast\approx\kappa\,\pi/\thval$ with
$\kappa$ decreasing slowly from $1.553$ at $\delta=0.5$ to $1.477$ at
$\delta=0.2$.
\end{quote}

Compare with the two naive bounds: the constituent bound of
\cite{LinV8} gives blindness up to $\pi/\thval-1$; the multiplicity-two
convolution of Theorem~\ref{thm:W} gives (at shift $1$, without the
critical-line background) a threshold around $1.39\,\pi/\thval$
($62/44.42=1.395$). Adding the background raises the constant to about
$1.5$. Thus the three effects---constituent truncation, multiplicity,
background---each contribute a multiplicative factor of roughly $1.0$,
$1.4$, and $1.08$ respectively.

\begin{remark}
For $\delta\le0.1$ the threshold exceeds $300$ and our search did not
reach it; the scaling law predicts $r^\ast\approx330$ at
$\delta=0.1$ and $r^\ast\approx660$ at $\delta=0.05$. Violations of RH
at small offset are correspondingly harder to detect by this route.
\end{remark}

%=====================================================================
\section{Robustness}
\label{sec:robust}
%=====================================================================

\subsection{Position of the violating zero}

\begin{table}[ht]
\centering
\caption{First discriminating order at shift $n=1$ as a function of
which ordinate carries the violation ($\delta=0.5$, $N=40$, search
limit $200$).}
\label{tab:position}
\begin{tabular}{crrr}
\toprule
violation at & $\gamma_k$ & $\thval$ & first negative $r$\\
\midrule
$\gamma_1$ & 14.134725 & 0.0707183 & 69\\
$\gamma_2$ & 21.022040 & 0.0475602 & none ($>200$)\\
$\gamma_3$ & 25.010858 & 0.0399773 & none ($>200$)\\
$\gamma_5$ & 32.935062 & 0.0303604 & none ($>200$)\\
$\gamma_{10}$ & 49.773832 & 0.0200902 & none ($>200$)\\
\bottomrule
\end{tabular}
\end{table}

The lowest ordinate gives the largest angle and hence the \emph{least}
blind configuration. Since even that configuration is blind to order
$68$, the value $68$ is a conservative global bound over all single-pair
violations.

\subsection{Several violating pairs}

\begin{table}[ht]
\centering
\caption{First discriminating order at shift $n=1$ as a function of
the number of violating pairs (all at $\delta=0.5$, attached to
$\gamma_1,\gamma_2,\gamma_3$, $N=40$).}
\label{tab:pairs}
\begin{tabular}{cr}
\toprule
number of pairs & first negative $r$\\
\midrule
1 & 69\\
2 & 70\\
3 & 71\\
\bottomrule
\end{tabular}
\end{table}

Adding violating pairs makes the hierarchy \emph{more} blind, by
approximately one order per pair. This is consistent with, and
quantifies, the ``arbitrarily many pairs'' clause of the blind zone
theorem of \cite{LinV8}: additional pairs add further convolution
factors, each of which smooths the sign.

\subsection{The shift-$2$ anomaly}

Table~\ref{tab:firstneg} shows that at shift $n=2$ no negative minor
was found up to $r=400$, whereas at $n=1$ and $n=3$ the threshold is
$69$. Two explanations suggest themselves.

First, for a two-element alphabet, $s_{(r,r)}=\rhoval^{2r}>0$
identically (put $a=b=r$ in \eqref{eq:two}), so the $n=2$ row is
trivially positive in the constituent analysis; this positivity may be
inherited by the reassembled alphabet. Secondly, the Jacobi--Trudi
determinant
\eqref{eq:JTdet} at $n=2$ is
$s_{(r,r)}=h_r^2-h_{r-1}h_{r+1}$, the classical Tur\'an-type
determinant of the sequence $(h_r)$ itself; the log-concavity of
$(h_r)$ for the alphabets in question would force positivity for all
$r$, which is exactly what is observed.

We state this as a conjecture.

\begin{conjecture}\label{conj:n2}
For every alphabet of the form $X\sqcup Y_{\mathrm{all}}$ arising from
$\Xif$ as in Section~\ref{sec:setup}, the sequence
$\bigl(h_r(\beta)\bigr)_{r\ge0}$ is log-concave, and hence
$s_{(r,r)}(\beta)=h_r^2-h_{r-1}h_{r+1}\ge0$ for all $r$.
\end{conjecture}

\begin{remark}
Conjecture~\ref{conj:n2}, if true, would mean that the entire
shift-$2$ row of the Tur\'an hierarchy is blind at \emph{every} order:
not a finite threshold but an infinite one. This would be a genuinely
new phenomenon, and it is the most interesting open direction suggested
by the present computations. Note that log-concavity of $(h_r)$ is
\emph{not} implied by total positivity of $(c_k)$ in any direct way we
are aware of; it appears to be an independent structural feature.
\end{remark}

%=====================================================================
\section{Status: what is proved and what is not}
\label{sec:status}
%=====================================================================

We give an explicit ledger. This section is deliberately pedantic; the
distinction between the two columns is the difference between a theorem
and a computation.

\begin{center}
\begin{longtable}{p{0.42\textwidth}p{0.42\textwidth}}
\toprule
\textbf{Proved} & \textbf{Numerical (not proved)}\\
\midrule
\endfirsthead
\toprule
\textbf{Proved} & \textbf{Numerical (not proved)}\\
\midrule
\endhead
\bottomrule
\endlastfoot
Lemma~\ref{lem:mult}: the multiplicity structure of $\beta$.
& Table~\ref{tab:firstneg}: $r^\ast=69$ at shift $1$.\\
Proposition~\ref{prop:A}: $h_r(Y_1)=\rhoval^rU_r$ and the threshold
$44$ for the two-element alphabet.
& Table~\ref{tab:trunc}: stability of $69$ under truncation.\\
Theorem~\ref{thm:W}: $h_r(Y)=\rhoval^rW_r$, the generating function
\eqref{eq:genW}, and $W_r=\tfrac12U'_{r+1}$.
& Table~\ref{tab:scaling}: the law $r^\ast\approx1.5\pi/\thval$.\\
Lemma~\ref{lem:sign}: the pole-free sign criterion.
& Tables~\ref{tab:position}, \ref{tab:pairs}: robustness in position
and in the number of pairs.\\
Proposition~\ref{prop:W61}: $W_r>0$ for $r\le61$, $W_{62}<0$
(certified finite computation).
& The absence of negative minors at shift $2$ up to $r=400$.\\
Theorem~\ref{thm:shift1}: the convolution formula
\eqref{eq:shift1closed}.
& Lemma~\ref{lem:mono}: monotonicity of the threshold in $\thval$.\\
\textbf{Theorem~\ref{thm:main}: $D(1,r)>0$ for all $r\le61$,
unconditionally.}
& \\
\bottomrule
\end{longtable}
\end{center}

\begin{remark}[On Proposition~\ref{prop:W61}]
The statement ``$W_r>0$ for $r\le61$'' is a finite computation, but
it is a computation with an enormous margin: the values in question
are of order $10^2$--$10^3$ and were computed to $80$ digits. We
regard it as certified in the same sense that a finite check of a
finite identity is certified. A reader who insists on a symbolic proof
may observe that the claim is equivalent, by
Lemma~\ref{lem:sign}, to $61$ explicit trigonometric inequalities.
\end{remark}

\begin{remark}[What is \emph{not} claimed]
No claim of a proof of RH is made, and none of the results here
constitute one. Theorem~\ref{thm:main} is a statement about the
\emph{insensitivity} of certain minors, i.e.\ an obstruction result.
It says that a certain class of certificates cannot exist, not that RH
is true or false.
\end{remark}

%=====================================================================
\section{Open problems}
\label{sec:open}
%=====================================================================

\begin{enumerate}[label=\textbf{P\arabic*.},ref=P\arabic*]
\item\label{prob:alln} \textbf{Extend Theorem~\ref{thm:main} to all
shifts.} The obstruction is that for $n\ge2$ the Jacobi--Trudi
determinant \eqref{eq:JTdet} involves products
$h_{r-i+j}$ with indices that are \emph{not} bounded by $r$ in a way
compatible with the positivity argument: the determinant is an
alternating sum, and the convolution smoothing that works for a single
$h_r$ does not obviously control the determinant. A resolution would
upgrade the proved bound from $61$ (shift $1$) to $61$ (all shifts).
\item\label{prob:mono} \textbf{Prove Lemma~\ref{lem:mono}.} Show that
the first sign change of $W_r(\cos\thval)$ is monotone in $\thval$ on
$(0,\thval_{\max}]$. By Lemma~\ref{lem:sign} this amounts to
controlling the first positive root of
$f_m(t)=\sin(mt)-m\tan t\,\cos(mt)$ as a function of $m$.
\item\label{prob:kappa} \textbf{Explain the constant $\kappa\approx1.5$
in Table~\ref{tab:scaling}.} Three effects are known to contribute
(constituent truncation, multiplicity $2$, critical-line background),
with approximate factors $1.0$, $1.4$, $1.08$. Is $\kappa$ a
universal constant of this hierarchy, and does it tend to a limit as
$\thval\to0$? The data suggest a slow drift, possibly towards
$\kappa\approx1.47$.
\item\label{prob:n2} \textbf{Prove Conjecture~\ref{conj:n2}
(log-concavity of $h_r$).} This is the most valuable item on the list:
it would establish an \emph{infinite} blind row in the hierarchy. A
natural first target is the log-concavity of $h_r(Y)$ for the pure
four-element alphabet \eqref{eq:Ydef}, which is a question about the
sequence $\rhoval^rW_r$ and hence about the Chebyshev derivative
sequence $\tfrac12U'_{r+1}(\cos\thval)$.
\item\label{prob:partial} \textbf{Counting versus angle.} The blind
zone is governed by an angle, not by a count: a single violating pair
at the maximal offset produces blindness to order $68$, while
arbitrarily many pairs at small offset produce blindness well beyond
$300$. Can one formulate a \emph{counting-free} certificate, i.e.\ a
minor whose positivity is sensitive to the \emph{existence} rather than
the \emph{geometry} of a violation?
\end{enumerate}

%=====================================================================
\section{Implications for certificate design}
\label{sec:impl}
%=====================================================================

We collect the practical consequences of the thresholds, and comment on
what a certificate based on this hierarchy would have to look like.

\subsection{Summary of thresholds}

\begin{table}[ht]
\centering
\caption{The three thresholds, with their status.}
\label{tab:summary}
\begin{tabular}{lccc}
\toprule
Quantity & bound & status & shift\\
\midrule
constituent (ignores multiplicity and background) & $43$ & proved
\cite{LinV8} & all $n$\\
multiplicity two, no background & $61$ & proved, Theorem
\ref{thm:main} & $n=1$ only\\
true alphabet, empirically & $68$ & numerical, \S\ref{sec:numerics}
& $n=1,2,3,4$\\
\bottomrule
\end{tabular}
\end{table}

The gap between the second and third rows ($61$ versus $68$) measures
exactly what Theorem~\ref{thm:main} leaves on the table: the
contribution of the critical-line background, which is worth a further
factor of about $1.11$ in the order. The gap between the first and
second rows ($43$ versus $61$) measures the contribution of
multiplicity, worth a factor of about $1.42$.

\subsection{What a certificate would require}

Suppose one wished to use this hierarchy to \emph{disprove} RH: one
would exhibit integers $n,r$ with $D(n,r)<0$. The results here say
what such a pair must look like.

\begin{enumerate}[label=(\roman*)]
\item The order must satisfy $r\gtrsim1.5\,\pi/\thval$, where $\thval$
is the largest vertical angle among the violating zeros. Since
$\thval\le\thval_{\max}=0.0707\ldots$, one needs
$r\gtrsim69$ in the best case; for a violation at offset $\delta=0.1$
one needs $r\gtrsim330$; at $\delta=0.05$, $r\gtrsim660$. \emph{The
hierarchy is a detector of geometry, not of existence}: a violation
sitting close to the critical line is invisible to it.
\item The shift $n=2$ should be avoided entirely: on the evidence of
Table~\ref{tab:firstneg} and Conjecture~\ref{conj:n2}, that row may be
blind at every order.
\item Increasing the number of violating pairs does not help: each
additional pair raises the threshold by about one
(Table~\ref{tab:pairs}). A \emph{single} violation at maximal offset
is the most detectable configuration, and it is still invisible below
order $69$.
\end{enumerate}

\begin{remark}[A caution on the word ``invisible'']
``Invisible below order $r$'' means that the minors of order $\le r$
are positive. It does not mean that no computation can detect the
violation: direct evaluation of $\zeta$ on the critical line, or of
$\Xif$ off it, is of course unaffected. The statement is about this
particular family of certificates.
\end{remark}

\subsection{The asymmetric cost of the two directions}

There is an asymmetry worth recording. To \emph{confirm} RH by this
route one must verify $D(n,r)\ge0$ for all $n$ and all $r$---infinitely
many inequalities. To \emph{refute} it one needs a single negative
minor. The blind zone theorem says that the cheapest refutations are
not available: the first $68$ orders are closed in every direction at
once. This is the sense in which the result is an obstruction rather
than a partial verification: it removes the low-hanging fruit from
both sides.

\subsection{Relation to the cost of computation}

A companion note estimates the precision needed to resolve the sign of
$D(n,r)$ as approximately $3.2\,n\,r$ significant digits (measured for
$r\le30$). At the threshold found here, $(n,r)=(1,69)$, this is about
$220$ digits---negligible by modern standards. The obstacle is
therefore emphatically \emph{not} the cost of evaluating a determinant
of order $69$; it is the fact that the determinant is positive. What
makes the blind zone hard to penetrate is not computational expense
but the structure of the Schur expansion.

%=====================================================================
\section{Conclusion}
\label{sec:concl}
%=====================================================================

The Tur\'an hierarchy is blinder than previously established. The
mechanism is convolution: the true $\beta$-alphabet carries multiplicity
two at every zero (by evenness of $\Xif$), and it carries, in addition
to the violating pairs, an infinite positive real background (the
critical-line zeros). Each of these two features convolves the
Chebyshev sign sequence with a non-negative sequence, and each
convolution delays the sign change by a fixed multiplicative factor.
The constituent analysis of \cite{LinV8}, which bounds each
two-variable piece separately, sees only the innermost of the three
layers and therefore stops at $43$.

What is proved here is the improvement to $61$ on the shift-$1$ row
(Theorem~\ref{thm:main}), together with the exact algebraic structure
of that row (Theorem~\ref{thm:W}, Lemma~\ref{lem:sign}). What is
observed is that the true threshold is $69$, that it scales as
$1.5\,\pi/\thval$, that it is robust in the position and number of
violations, and that the shift-$2$ row may be blind at every order.
Turning the observations into theorems is the programme of Section
\ref{sec:open}.

\subsection{Where this sits in a longer programme}

The results of this paper are obstruction results, and obstruction
results have a specific role: they do not advance a proof of RH, they
advance the \emph{map} of where a proof cannot go cheaply. The value of
such a map is that it prevents the repeated rediscovery of dead ends.

Concretely, the sequence of results now available on this route is:

\begin{center}
\begin{tabular}{ll}
\toprule
Statement & status\\
\midrule
Level $2$ is not discriminating & proved \cite{Csordas}\\
Levels $\le43$ are not discriminating (all shifts) & proved
\cite{LinV8}\\
Level $\le61$ at shift $1$ is not discriminating & proved,
Theorem~\ref{thm:main}\\
Levels $\le68$ are not discriminating (observed) & numerical,
\S\ref{sec:numerics}\\
\bottomrule
\end{tabular}
\end{center}

and the natural continuation is Problem~\ref{prob:alln}: prove the
bound $61$, or something between $61$ and $68$, for all shifts. Until
that is done, the honest summary is that the proved frontier is $43$
for the whole hierarchy and $61$ on the shift-$1$ row, while the true
frontier appears to be $68$.

The most intriguing item remains Conjecture~\ref{conj:n2}. If the
shift-$2$ row is blind at every order, then the hierarchy contains an
infinite blind direction, and the search for a certificate should be
conducted entirely away from $n=2$. We regard this as the single most
promising structural question raised by the present work, and it is
amenable to attack: it reduces to a log-concavity statement about the
sequence $\rhoval^r W_r=\tfrac12\rhoval^rU'_{r+1}(\cos\thval)$, which
is a question about Chebyshev polynomials alone.

%=====================================================================
\appendix
\section{Algorithms}
\label{app:code}
%=====================================================================

\subsection{Complete symmetric functions}

\begin{verbatim}
h[0] = 1; h[k] = 0 for k > 0
for b in alphabet:
    new[0] = h[0]
    for k = 1..K:  new[k] = h[k] + b*new[k-1]
    h = new
\end{verbatim}

\subsection{Rectangular Schur function}

By \eqref{eq:JTdet}, $s_{(r^n)}=\det[h_{r-i+j}]_{i,j=0}^{n-1}$.

\subsection{Search for the first negative order}

\begin{verbatim}
def first_negative(beta, shift, rmax):
    K = rmax + shift + 3
    h = h_complete(beta, K)
    for r in 1..rmax:
        M[i][j] = h[r - i + j]        (i,j = 0..shift-1)
        if det(M).real < 0: return r
    return None
\end{verbatim}

\subsection{Reproducibility}

All numbers in this paper are reproduced by the accompanying scripts
(\texttt{verify\_v9.py}, \texttt{v9\_cheby.py}, \texttt{v9\_ident.py},
\texttt{v9\_tan2.py}, \texttt{v9\_curve.py}, \texttt{v9\_multi.py}),
which depend only on \texttt{mpmath}. Working precision is $60$
significant digits for the Schur computations and $80$ for the
certification of $W_r$; the imaginary parts of all computed
$s_{(r^n)}$ are smaller than the real parts by at least $40$ orders of
magnitude, as they must be for a conjugate-symmetric alphabet.

\subsection{Software independence}

The ordinates $\gamma_k$ were obtained from the \texttt{mpmath}
library's \texttt{zetazero}. Every threshold reported here is stable
under increasing the working precision from $50$ to $80$ digits and
under changing the truncation $N$ (Table~\ref{tab:trunc}).

%=====================================================================
\section{Full numerical tables}
\label{app:tables}
%=====================================================================

\subsection{Constants}

\[
\gamma_1=14.134725141734693\ldots,\quad
\thval_{\max}=0.07071826294293218\ldots,\quad
\frac{\pi}{\thval_{\max}}=44.42406420707728\ldots
\]
\[
\rhoval_{\max}=\frac1{\gamma_1^2+\tfrac14}=0.00499898883372314\ldots
\]

\subsection{The sequence $W_r$ near the transition}

\begin{center}
\begin{tabular}{cr|cr|cr}
\toprule
$r$ & $W_r$ & $r$ & $W_r$ & $r$ & $W_r$\\
\midrule
0 & 1 & 40 & 4420.87 & 58 & 1456.57\\
1 & 3.99000 & 43 & 4445.24 & 59 & 1067.95\\
2 & 9.94009 & 44 & 4420.87 & 60 & 660.941\\
5 & 54.7492 & 50 & 3568.1 & 61 & 237.234\\
20 & 1380.56 & 55 & 2601.4 & 62 & $-201.346$\\
\bottomrule
\end{tabular}
\end{center}

\subsection{Comparison of the three layers}

\begin{center}
\begin{tabular}{lcc}
\toprule
Layer & alphabet & first negative $r$ (shift $1$)\\
\midrule
A & $\{\rhoval e^{i\thval},\rhoval e^{-i\thval}\}$ & 44\\
B & A with multiplicity two & 62\\
C & B plus $N$ critical-line zeros & 69\\
\bottomrule
\end{tabular}
\end{center}

The three numbers $44$, $62$, $69$ are the backbone of this paper:
the first is the constituent bound recovered from \cite{LinV8}, the
second is proved here (Theorem~\ref{thm:W} and
Proposition~\ref{prop:W61}), and the third is observed (Section
\ref{sec:numerics}).

\subsection{Full scaling data}

Shift $n=1$, $N=40$ critical-line ordinates, one violating pair at
$\gamma_1$.

\begin{center}
\begin{tabular}{rrrrr}
\toprule
$\delta$ & $\thval$ & $\pi/\thval$ & $r^\ast$ &
$r^\ast/(\pi/\thval)$\\
\midrule
0.50 & 0.07071826 & 44.4241 & 69 & 1.5532\\
0.45 & 0.06364803 & 49.3562 & 76 & 1.5398\\
0.40 & 0.05658310 & 55.5218 & 84 & 1.5129\\
0.35 & 0.04950538 & 63.4495 & 96 & 1.5130\\
0.30 & 0.04244228 & 74.0204 & 111 & 1.4996\\
0.25 & 0.03537176 & 88.8204 & 132 & 1.4862\\
0.20 & 0.02829721 & 111.0213 & 164 & 1.4772\\
0.15 & 0.02122251 & 148.0300 & (not reached) & ---\\
0.10 & 0.01414931 & 222.0315 & (not reached) & ---\\
0.05 & 0.00707475 & 444.0573 & (not reached) & ---\\
\bottomrule
\end{tabular}
\end{center}

Extrapolating the ratio $\kappa\approx1.47$ to the unreached rows
predicts $r^\ast\approx218$ at $\delta=0.15$, $r^\ast\approx326$ at
$\delta=0.10$ and $r^\ast\approx653$ at $\delta=0.05$.

\subsection{Shift scan on the full alphabet}

$N=45$, one violating pair at $\gamma_1$ with $\delta=0.5$, search to
$r=145$.

\begin{center}
\begin{tabular}{crr}
\toprule
shift $n$ & first negative $r$ & count of negative $r\le145$\\
\midrule
1 & 69 & 45\\
2 & none & 0\\
3 & 69 & 46\\
4 & none & 0\\
\bottomrule
\end{tabular}
\end{center}

\subsection{Sample values at shift $n=1$}

Signed values of $h_r(\beta)$ near the transition
($N=45$, $\delta=0.5$):

\begin{center}
\begin{tabular}{cr}
\toprule
$r$ & $h_r(\beta)$\\
\midrule
65 & $1.3030821\times10^{-144}$\\
66 & $4.5484089\times10^{-147}$\\
67 & $1.2470748\times10^{-149}$\\
68 & $9.0314693\times10^{-153}$\\
69 & $-2.3011398\times10^{-154}$\\
70 & $-2.5639539\times10^{-156}$\\
\bottomrule
\end{tabular}
\end{center}

\subsection{Sensitivity of the threshold to $N$ and to precision}

\begin{center}
\begin{tabular}{ccr}
\toprule
$N$ & working digits & first negative $r$\\
\midrule
30 & 60 & 68\\
45 & 60 & 69\\
45 & 80 & 69\\
60 & 60 & 69\\
80 & 60 & 69\\
\bottomrule
\end{tabular}
\end{center}

%=====================================================================
\begin{thebibliography}{9}
%=====================================================================

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\end{document}