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\title{A counterexample to Haglund's monotonic-zeros conjecture for the Riemann
$\Xi$ approximants}
\author{Kunal Tyagi\thanks{%
\texttt{hello.jay.tyagi@gmail.com}. \emph{Use of AI.} The mathematics in this
paper --- the derivations, the computations, the verification suite, and the
text --- was produced by Claude (Anthropic) working under the author's
direction; the author set the objectives, made the technical and editorial
decisions, and is responsible for the content. Claude is a tool and is not an
author. Code and certificates:
\url{https://github.com/x67ai/riemann-rh-program}.}}
\date{October 1, 2026; revised October 3, 2026}

\begin{document}

\maketitle

\begin{abstract}
Haglund introduced the partial sums $\Xi_N = \sum_{n \le N} \Phi_n$ of
Riemann's expansion of the $\Xi$-function in incomplete gamma functions and
conjectured that for every $N$ the zeros of $\Xi_N$ in the closed first
quadrant, listed by increasing real part, have nondecreasing imaginary parts.
The conjecture implies the Riemann hypothesis; Haglund's computations supported a
weak form of it for $N \le 10$. We show that the conjecture is false for
$N = 27$: $\Xi_{27}$ has a real zero in $(3144.8946, 3144.8947)$ and a non-real
zero $z_0 = 3143.2206824215\ldots + 0.3152587993\ldots\, i$ in the first
quadrant, whose real part is smaller. The weak form fails as well. Both facts
are certified twice, independently, in interval arithmetic (Arb ball
arithmetic; outward-rounded intervals with proved truncation bounds), and $z_0$
is enclosed in a square of half-width $10^{-48}$. The same violation occurs at
$N = 24$ for the companion approximants $\xi_N = \Xi_N + c_N$. The failure has a
mechanism: on the critical line $\Xi_N < \Xi < \xi_N$, so the real zeros of
$\Xi_N$ lie in positive lobes of $\Xi$ that rise above a positive level, and a
low lobe followed by a higher one near the height $4(N+1)^2$, where the
approximant departs from $\Xi$, can leave a non-real zero below real zeros. The same
structure shows that the number of positive real zeros of $\Xi_N$, counted with
multiplicity, is odd, and that $x^2\Xi_N(x)$ tends to a negative limit on the
real axis for every $N$.
\end{abstract}

%% SECTION 1
\section{Introduction}
\label{sec:intro}

\subsection{Haglund's approximants and his conjecture}\label{sec:hag}

Write $s = \tfrac12 + iz$ and let
\begin{equation}\label{eq:Xi}
\Xi(z) = \tfrac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s) = \xi(s),
\end{equation}
Riemann's $\Xi$-function, which is even, entire, and real on the real axis; the
Riemann hypothesis is the statement that all its zeros are real. Let
$\Q = \{z : \re z \ge 0,\ \im z \ge 0\}$ be the closed first quadrant. Haglund
\cite{Hag} starts from Riemann's representation
$\Xi(z) = \int_0^\infty \cos(zt)\varphi(t)\,dt$ with
$\varphi = \sum_{n \ge 1}\varphi_n$,
$\varphi_n(t) = \exp(-n^2\pi e^{2t})(8\pi^2 n^4 e^{4.5t} - 12\pi n^2 e^{2.5t})$,
and asks what happens when $\varphi$ is replaced by the finite sum
$\sum_{n \le N}\varphi_n$. With the ``hyperbolic gamma function''
\begin{equation}\label{eq:G}
G(z;a,b) = 4\int_0^\infty \cos(2zu)\,e^{2bu - a e^{2u}}\,du
= \frac{\Gamma(b+iz,a)}{a^{b+iz}} + \frac{\Gamma(b-iz,a)}{a^{b-iz}}
\qquad (a > 0,\ b \in \R),
\end{equation}
where $\Gamma(w,a) = \int_a^\infty e^{-t}t^{w-1}\,dt$ is the upper incomplete gamma
function, this gives Riemann's series
$\Xi(z) = \sum_{n \ge 1}\Phi_n(z)$ and Haglund's \emph{$\Xi$-approximates}
\begin{equation}\label{eq:XiN}
\Xi_N(z) := \sum_{n=1}^{N}\Phi_n(z), \qquad
\Phi_n(z) := 2\pi^2 n^4\, G\bigl(\tfrac{z}{2}; n^2\pi, \tfrac94\bigr)
- 3\pi n^2\, G\bigl(\tfrac{z}{2}; n^2\pi, \tfrac54\bigr)
\end{equation}
(\cite[(1)--(3), (6), (10)--(14), pp.~1--3]{Hag}; page and equation numbers refer
to the arXiv version). Each $\Xi_N$ is even, entire, and real on the real axis.
Haglund defines: ``We say that a given function $F(z)$ has monotonic zeros in a
region $D$ of the complex plane if, when we list the zeros of $F$ in $D$ by
increasing real part, the imaginary parts of the zeros are monotone
nondecreasing'', with the side assumption ``(We assume $F$ has at most one zero
on the intersection of any vertical line with $D$.)'' \cite[p.~3]{Hag}. He then
states:

\begin{conjecture}[Haglund {\cite[p.~3]{Hag}}]\label{conj:H}
``For $N \in \N$, $\Xi_N(z)$ has monotonic zeros in $\Q$.''
\end{conjecture}

His Proposition~1 reads ``Conjecture 1 implies the Riemann Hypothesis''
\cite[pp.~3--4]{Hag}; the proof is a Hurwitz-type argument, using that
$\Xi_N \to \Xi$ uniformly on compacta and that $\Xi$ has infinitely many real
zeros. His Remark~1 records a weak form: ``A weaker form of Conjecture 1, which
still implies RH, is that there are no non-real zeros of $\Xi_N(z)$ in $\Q$ whose
real part is less than the largest real zero of $\Xi_N(z)$'', and reports that
``Computations along these lines indicate this weaker form of Conjecture 1 is
true at least for $N \le 10$'' \cite[p.~4]{Hag}, the computations being
floating-point ones for which ``no attempt has been made to establish rigorous
error bounds'' \cite[p.~3]{Hag}. The paper was published as \cite{HagJ}.

\subsection{The result}

\begin{theorem}\label{thm:main}
\begin{enumerate}
\item[(a)] $\Xi_{27}$ has a real zero $x_1 \in (3144.8946,\ 3144.8947)$, and a
second real zero $x_2 \in (3145.5998,\ 3145.5999)$.
\item[(b)] $\Xi_{27}$ has exactly one zero $z_0$, which is simple, in the open
square of half-width $3.7\cdot 10^{-11}$ about
$c = 3143.2206824215 + 0.3152587994\,i$ (here $w + [-r, r]^2$ is the closed square
$\{w + a + bi : |a|, |b| \le r\}$). More precisely (certificate B),
$z_0 \in z^* + [-10^{-48}, 10^{-48}]^2$ with
\begin{align*}
\re z^* &= 3143.220682421536585287281295989417800119\ldots,\\
\im z^* &= 0.315258799378214845382286373009271765\ldots,
\end{align*}
where the full explicit center $z^*$ is given in Section~\ref{sec:cert}.
\item[(c)] Consequently $z_0 \in \Q$ is non-real, with $\im z_0 > 0.3152$ and
$\re z_0 < 3143.2207 < x_1$. Listed by increasing real part, the zeros of
$\Xi_{27}$ in $\Q$ do not have nondecreasing imaginary parts: Conjecture~\ref{conj:H}
is false for $N = 27$. Since $z_0$ is a non-real zero in $\Q$ whose real part is
less than the real zero $x_1$, and hence less than the largest real zero, the
weak form of Remark~1 is false for $N = 27$ as well.
\end{enumerate}
\end{theorem}

The pair $(z_0, x_1)$ violates the conjecture under every enumeration of the
zeros by increasing real part, so Haglund's side assumption (at most one zero on
each vertical line) plays no role. Statements (a) and (b) are established by two
independent computer-assisted proofs in rigorous interval arithmetic, described
in Section~\ref{sec:cert}: one in Arb ball arithmetic \cite{Arb, FLINT}, one in
outward-rounded interval arithmetic \cite{mpmath} with every truncation bounded
by a proved remainder estimate. Each was first run on a ladder of known cases
(two zeros from Haglund's table and a non-real zero of $\Xi_1$ from his
appendix).

\subsection{Scope}

The paper does not decide Conjecture~\ref{conj:H} for any $N \ne 27$: the scan
that located the violation at $N = 27$ (Section~\ref{sec:departure}) is
numerical, covers a window near one height for each $N$, and is not a proof that
any other $N$ satisfies the conjecture.

\subsection{The mechanism}\label{sec:mech}

The failure is not an accident of $N = 27$. On the critical line the approximants
sit below $\Xi$ by a positive amount: for every $N \ge 1$ and real $t$,
\[
\Xi_N(t) = \Xi(t) - Q_N(t), \qquad Q_N(t) > 0
\]
(Theorem~\ref{thm:sandwich}). Hence $\Xi_N < 0$ wherever $\Xi \le 0$, and the real
zeros of $\Xi_N$ lie in the \emph{positive lobes} of $\Xi$ (the intervals between
consecutive zeros on which $\Xi > 0$): an even number in each positive lobe
$(\gamma_j, \gamma_{j+1})$ and an odd number in the central lobe $(0, \gamma_1)$,
where $0 < \gamma_1 < \gamma_2 < \cdots$ are the positive real zeros of $\Xi$. A
positive lobe carries real zeros of $\Xi_N$ only if $\Xi$ rises above the level $Q_N$ there. Near the height $4(N+1)^2$ the
level $Q_N$ becomes comparable to the size of $\Xi$, and the heights of
consecutive lobes fluctuate; a lobe that stays below the level, followed by a
lobe that rises above it, is expected to leave a pair of non-real zeros below real
zeros (it does in both cases computed below). At
$N = 27$ the positive lobe of $\Xi$ over $(3142.98, 3143.57)$ reaches only
$0.411$ of the level $Q_{27}$ and the next one, over $(3144.49, 3146.28)$,
reaches $1.325$ of it: the pair $z_0, \bar z_0$ sits over the first, and the real
zeros $x_1, x_2$ lie in the second (Section~\ref{sec:departure}; these lobe
figures are floating-point computations, not part of the proof).

The same sandwich gives two facts about the real zeros and the real-axis tail
of $\Xi_N$.
\begin{enumerate}
\item[(i)] The number of real zeros of $\Xi_N$ on $(0,\infty)$, counted with
multiplicity, is odd for every $N$ (Corollary~\ref{cor:odd}).
\item[(ii)] On the real axis $x^2\Xi_N(x)$ tends to a negative limit for every
$N \ge 1$ (Proposition~\ref{prop:tail}).
\end{enumerate}
Both agree with the copy of Haglund's paper on the author's web page
\cite{HagW}, dated February 9, 2011: its table lists $1$, $7$, $15$, $31$,
$53$, $79$, $113$, $155$, $207$, $263$ real zeros for $N = 1, \ldots, 10$
\cite[p.~4]{HagW}, and it states that the coefficient of $1/x^2$ in $\Xi_N(x)$
``approaches zero from below as $N \to \infty$'' \cite[p.~10]{HagW}. The arXiv
version \cite{Hag} and the journal version \cite{HagJ} have $32$ at $N = 4$
and, for the coefficient, ``positive for $k \ge 3$ and negative for
$1 \le k < 3$'' \cite[p.~10]{Hag} (his $k$ is $N$); \cite{HagW} corrects
both.

\subsection{The companion approximants, and why an ordering property}

Haglund's $\Xi_N$ differs by a constant from the more obvious truncation of
Riemann's formula,
\[
\xi_N(s) := \tfrac12 + \tfrac12 s(s-1)\sum_{n \le N} g_n(s), \qquad
\Xi_N(z) = \xi_N(\tfrac12 + iz) - c_N, \qquad
c_N = \sum_{n > N}(4\pi n^2 - 1)e^{-\pi n^2} > 0
\]
(Section~\ref{sec:riemann}). On the critical line $\xi_N$ lies \emph{above} $\Xi$
(Theorem~\ref{thm:polya}), so its real zeros lie in the negative lobes of $\Xi$,
and the same ordering property fails for it at $N = 24$
(Theorem~\ref{thm:xi24}). Neither family is real-rooted: no member of the
family
$\xi_w = \tfrac12 + \tfrac12 s(s-1)\sum_n w_n g_n$ with $w \not\equiv 0$ and either
$0 \le w_n \le 1$, $w \not\equiv 1$, or $w$ finitely supported, has all its zeros on the
critical line; it has finitely many zeros there and infinitely many off it
(Theorem~\ref{thm:A}), and the same holds for every $\Xi_N$. With polynomially bounded weights, having all zeros on
the line forces $w \equiv 1$ (Theorem~\ref{thm:B}). This is why a conjecture
about approximants of this kind has to be an ordering statement such as
Conjecture~\ref{conj:H}, and not real-rootedness; and any approach through
real-rootedness in a window that grows with $N$ is a height filtration of the
Riemann hypothesis, with nothing to prove beyond the Riemann hypothesis on the
new window (Proposition~\ref{prop:F}).

\subsection{Prior work}

Haglund's conjecture was stated in 2009 \cite{Hag, HagJ} and checked by him, in
its weak form, for $N \le 10$. Ahn's thesis \cite{Ahn}, written under Haglund's
supervision, computes zeros of the component functions of the approximants
(incomplete gamma functions, hyperbolic gamma functions, and small linear
combinations of the $\Phi_n$) and restates the check: ``Computations indicate
this weaker form of Conjecture 1.6 [sic: his Conjecture 1.5] is true at least for $N \le 10$''
\cite[pp.~10--11]{Ahn}; it tests no $\Xi_N$ beyond $N = 10$. Haglund's Conjecture~4
concerns the zeros of the pencil $\Xi_k + t\Phi_{k+1}$, $0 \le t \le 1$
\cite[p.~11]{Hag}; its first case $k = 1$ is the subject of a 2026 preprint of
Baccaro \cite{Bac}, whose record states that ``The result covers only the first
interpolation.'' We found no test of Conjecture~\ref{conj:H} beyond $N = 10$, and
no refutation of it, in the literature we could reach; a null search is not a
proof of absence. Lagarias and Montague study a related Fourier-integral family
whose members have only finitely many zeros on the critical line; they ask
whether monotone increase of the imaginary parts of the zeros in the first
quadrant would be a sufficient condition for the zeros of its derivative $\xi$ to
lie on the critical line, and point to Haglund's paper as a related situation ``involving the $\xi$-function,
where such a monotonicity implies the RH'' \cite[p.~24]{LM}. Truncations that keep
all but finitely many zeros \emph{on} the line are known in another setting: Ki
proved that for $y \ge 1$ all but finitely many complex zeros of any partial sum
of the Chowla--Selberg formula are simple and on the line \cite[Corollary~1]{KiP}, and
observed, restating it, that the Riemann hypothesis would follow if the words ``but finitely many'' could be
removed for infinitely many $N$ in the case $z = i$ \cite[p.~198]{Ki}. The
approximants here are the opposite case (Theorem~\ref{thm:A}). Finally,
Liu, Matiyasevich, Oesterl\'e and Zaharescu prove, for a different family of
approximants to $\xi$ built from truncated Euler products, that ``The Bounded
Riemann Hypothesis is equivalent to the Riemann Hypothesis plus the Simplicity
Conjecture'' \cite[Theorem~1.7]{LMOZ}; Proposition~\ref{prop:F} is the analogue
for the approximants of this paper.

\subsection{Organization}

Section~\ref{sec:riemann} recalls Riemann's identity and relates the two
families. Section~\ref{sec:positivity} proves the positivity and sandwich
theorems and their consequences for the real zeros and the tail. Section~\ref{sec:structure} gives the
structural theorems. Section~\ref{sec:departure} describes the departure height
and the lobe scan. Section~\ref{sec:cert} is the certificate for
Theorem~\ref{thm:main}, and Section~\ref{sec:xi24} the companion violation at
$N = 24$.

\section{Riemann's identity and the two families}
\label{sec:riemann}

Let $\psi(x) = \sum_{n \ge 1} e^{-\pi n^2 x}$. Splitting
$\int_0^\infty \psi(x)x^{s/2-1}\,dx$ at $x = 1$ and using
$\psi(1/x) = \sqrt{x}\,\psi(x) + (\sqrt{x}-1)/2$ gives Riemann's identity
\cite{Rie, Edw}, valid for all $s \in \Cx \setminus \{0, 1\}$:
\begin{equation}\label{eq:riemann}
\Lambda(s) := \pi^{-s/2}\Gamma(s/2)\zeta(s) = -\frac1s - \frac{1}{1-s}
+ \sum_{n \ge 1} g_n(s),
\end{equation}
where
\[
g_n(s) = \frac{\Gamma(s/2, X)}{X^{s/2}} + \frac{\Gamma((1-s)/2, X)}{X^{(1-s)/2}},
\qquad X = \pi n^2 .
\]
Hence $\xi(s) = \tfrac12 s(s-1)\Lambda(s) = \tfrac12 + \tfrac12 s(s-1)\sum_{n\ge1}
g_n(s)$. For weights $w\colon \N \to \R$ with $|w_n| \le Cn^A$ for some constants $C, A$
(so that the series converges absolutely and locally uniformly) put
\begin{equation}\label{eq:xiw}
\xi_w(s) := \tfrac12 + \tfrac12 s(s-1)\sum_{n \ge 1} w_n\, g_n(s),
\end{equation}
and write $\xi_N$ for $w = \mathbf{1}_{[1,N]}$. Every $\xi_w$ satisfies
$\xi_w(1-s) = \xi_w(s)$ and $\xi_w(\bar s) = \overline{\xi_w(s)}$, so
$\xi_w(\tfrac12 + it)$ is real for real $t$; and
$|g_n(s)| \le g_n(\re s)$, since $|\Gamma(a, X)| \le \Gamma(\re a, X)$.
Throughout, $s = \tfrac12 + iz$; on the critical line $z = t$ is real, and we
write $\Xi(t) = \xi(\tfrac12 + it)$ and
$P_w(t) := \xi_w(\tfrac12 + it) - \Xi(t)$.

\begin{lemma}\label{lem:relation}
For every $n \ge 1$, $\Phi_n(z) = \tfrac12 s(s-1)g_n(s) + (4\pi n^2 - 1)e^{-\pi n^2}$.
Consequently
\[
\Xi_N(z) = \xi_N(s) - c_N, \qquad
c_N := \sum_{n > N}(4\pi n^2 - 1)e^{-\pi n^2} > 0 .
\]
\end{lemma}

\begin{proof}
Put $h(w) = X^{-w}\Gamma(w, X)$ with $X = \pi n^2$. Since $iz = s - \tfrac12$,
one has $\tfrac94 + \tfrac{iz}{2} = 2 + \tfrac s2$ and
$\tfrac94 - \tfrac{iz}{2} = 2 + \tfrac{1-s}{2}$, and the same with $1$ in place
of $2$ for $b = \tfrac54$; so definition \eqref{eq:XiN} reads
$\Phi_n = 2X^2[h(\tfrac s2 + 2) + h(\tfrac{1-s}2 + 2)]
- 3X[h(\tfrac s2 + 1) + h(\tfrac{1-s}2 + 1)]$.
The recurrence $\Gamma(w+1, X) = w\Gamma(w, X) + X^w e^{-X}$ gives
$h(w+1) = (w\,h(w) + e^{-X})/X$, and applying it twice,
$2X^2 h(a+2) - 3X h(a+1) = (2a^2 - a)h(a) + (2a - 1 + 2X)e^{-X}$. For
$a = s/2$ and for $a = (1-s)/2$ one has $2a^2 - a = \tfrac12 s(s-1)$, while
$2a - 1$ equals $s - 1$ and $-s$ respectively; adding the two gives the first
claim. Summing over $n \le N$ and using
$\sum_{n \ge 1}(4\pi n^2 - 1)e^{-\pi n^2} = \tfrac12$ (differentiate
$\theta(1/x) = \sqrt{x}\,\theta(x)$, $\theta = 1 + 2\psi$, at $x = 1$) gives the
second.
\end{proof}

Numerically $c_1 = 1.7180566258547875025\cdot 10^{-4}$,
$c_2 = 5.891258854396012012\cdot 10^{-11}$,
$c_3 = 2.9589851955455663928\cdot 10^{-20}$,
$c_4 = 2.4342009049056017\cdot 10^{-32}$ and
$c_5 = 3.443519624082649\cdot 10^{-47}$. So Haglund's zero sets are the
level sets $\{\xi_N = c_N\}$ of the truncations $\xi_N$, not their zero sets; the
two families have different zeros, and both are studied below.

\section{Positivity on the critical line}
\label{sec:positivity}

\begin{theorem}[positivity]\label{thm:polya}
For every $n \ge 1$ and real $t$, $g_n(\tfrac12 + it) > 0$. Consequently, for
weights $0 \le w_n \le 1$,
\[
P_w(t) = \tfrac12\bigl(\tfrac14 + t^2\bigr)\sum_{n \ge 1}(1 - w_n)\,g_n(\tfrac12 + it)
\ \ge\ 0,
\]
and $P_w(t) > 0$ for every real $t$ unless
$w \equiv 1$. In particular
$\xi_1(\tfrac12+it) > \xi_2(\tfrac12+it) > \cdots > \Xi(t)$ for every real $t$.
\end{theorem}

\begin{proof}
With $x = e^{2u}$,
\[
g_n(\tfrac12+it) = \int_1^\infty e^{-Xx}\,2x^{1/4}\cos\bigl(\tfrac t2\log x\bigr)
\frac{dx}{x} = 4\int_0^\infty k(u)\cos(tu)\,du, \qquad
k(u) = \exp\bigl(u/2 - Xe^{2u}\bigr).
\]
With $y = Xe^{2u} \ge X \ge \pi$ one has $k' = k(\tfrac12 - 2y) < 0$ and
$k'' = k(4y^2 - 6y + \tfrac14) > 0$, since the larger root of
$4y^2 - 6y + \tfrac14$ is $(3 + 2\sqrt2)/4 = 1.457\ldots < \pi$. So $k$ is
positive, strictly decreasing and strictly convex on $[0,\infty)$, and tends to
$0$. For $t \ne 0$,
$\int_0^\infty k(u)\cos(tu)\,du = t^{-1}\int_0^\infty(-k'(u))\sin(tu)\,du$; since
$-k'$ is positive and strictly decreasing, the pieces of the sine integral over
consecutive half-periods alternate in sign with strictly decreasing sizes, and
the first is positive, so the integral is positive (this is the argument behind
P\'olya's criterion). At $t = 0$ the integral is $\int_0^\infty k > 0$. The
formula for $P_w$ follows by subtracting
$\xi = \tfrac12 + \tfrac12 s(s-1)\sum_n g_n$ from \eqref{eq:xiw}, with
$\tfrac12 s(s-1) = -\tfrac12(\tfrac14 + t^2)$ on the line.
\end{proof}

\begin{corollary}[lobe law for $\xi_w$]\label{cor:lobe}
Let $0 \le w \le 1$, $w \not\equiv 1$. Every zero of $\xi_w$ on the critical line
lies in an open negative lobe of $\Xi$ (an interval between consecutive zeros of
$\Xi$ on which $\Xi < 0$), and each negative lobe contains an even number of
them, counted with multiplicity.
\end{corollary}

\begin{proof}
At the ends of a lobe $\xi_w = P_w > 0$, and on positive lobes
$\xi_w \ge \Xi > 0$.
\end{proof}

The real-zero counts of $\xi_N$ observed numerically are $2, 6, 16, 32, 52, 80$
for $N = 1, \ldots, 6$, all even. Haglund's approximants lie on the other side of
$\Xi$. For $n \ge 1$ put $k_n(u) = \exp(u/2 - \pi n^2 e^{2u})$ and
\[
\tphi_n := k_n'' - \tfrac14 k_n = 2y(2y-3)\,e^{u/2 - y}, \qquad y = \pi n^2 e^{2u},
\]
which equals $(4\pi^2 n^4 e^{9u/2} - 6\pi n^2 e^{5u/2})\exp(-\pi n^2 e^{2u})$,
one half of the term $\varphi_n(u)$ of Riemann's kernel
(Section~\ref{sec:hag}).

\begin{theorem}[sandwich]\label{thm:sandwich}
For every $N \ge 1$ and real $t$,
\[
\Xi_N(t) = \Xi(t) - Q_N(t) < \Xi(t) < \Xi(t) + P_N(t) = \xi_N(\tfrac12 + it),
\]
where
\[
Q_N(t) = c_N - P_N(t) = 2\int_0^\infty \sum_{n > N}\tphi_n(u)\cos(tu)\,du > 0 .
\]
\end{theorem}

\begin{proof}
Integrating by parts twice,
\[
\tfrac12\bigl(\tfrac14 + t^2\bigr)g_n(\tfrac12+it)
= 2\bigl(\tfrac14 + t^2\bigr)\int_0^\infty k_n\cos(tu)\,du
= -2k_n'(0) - 2\int_0^\infty\bigl(k_n'' - \tfrac14 k_n\bigr)\cos(tu)\,du,
\]
and $-2k_n'(0) = (4\pi n^2 - 1)e^{-\pi n^2}$; summing over $n > N$ gives
$P_N = c_N - Q_N$, and $\Xi_N = \xi_N - c_N = \Xi + P_N - c_N = \Xi - Q_N$ by
Lemma~\ref{lem:relation}. For $n \ge 2$ one has $y \ge 4\pi$, and then
$\tphi_n > 0$,
$\tphi_n' = e^{u/2-y}\, y\,(-8y^2 + 30y - 15) < 0$ and
$\tphi_n'' = e^{u/2-y}\, y\,(16y^3 - 112y^2 + 165y - 37.5) > 0$ (because
$16y^2(y - 7) > 0$ and $165y > 37.5$). So $\sum_{n > N}\tphi_n$ is positive,
strictly decreasing and strictly convex on $[0, \infty)$ for $N \ge 1$, and the
argument of Theorem~\ref{thm:polya} gives $Q_N(t) > 0$.
\end{proof}

For $n = 1$ the kernel is not monotone at $u = 0$: the factor
$-8y^2 + 30y - 15$ is $+0.291$ at $y = \pi$, so
$\tphi_1'(0) = e^{-\pi}\cdot\pi\cdot 0.291 = +0.0395 > 0$. This is why the theorem
needs $N \ge 1$.

\begin{proposition}[the $1/t^2$ tail]\label{prop:tail}
For every $N \ge 1$, with $X = \pi n^2$,
\[
\lim_{t \to \infty} t^2\,\Xi_N(t)
= 2\sum_{n > N}\tphi_n'(0)
= -2\sum_{n > N}\bigl(8X^3 - 30X^2 + 15X\bigr)e^{-X} \ <\ 0 .
\]
\end{proposition}

\begin{proof}
$\Xi(t)$ decays exponentially, and two integrations by parts in the integral
for $Q_N(t)$ in Theorem~\ref{thm:sandwich} give
$t^2 Q_N(t) \to -2\sum_{n > N}\tphi_n'(0)$. At $u = 0$,
$\tphi_n'(0) = -(8X^3 - 30X^2 + 15X)e^{-X}$, and each summand
$8X^3 - 30X^2 + 15X$ is positive for $n \ge 2$.
\end{proof}

Since the full kernel $\sum_{n \ge 1}\tphi_n$ is even (Haglund states it
\cite[p.~2]{Hag}; with $K(u) = e^{u/2}\psi(e^{2u})$ the theta relation gives
$K(u) - K(-u) = -\sinh(u/2)$, so $K'' - K/4$ is even),
$\sum_{n\ge1}\tphi_n'(0) = 0$, and the limit is also the sum over $n \le N$ of
the coefficients of $1/t^2$ in the individual $\Phi_n$, namely
$-2\tphi_n'(0)$; this is how Haglund computes it. The limits are $-0.0789975$,
$-1.65306\cdot10^{-7}$, $-2.78345\cdot10^{-16}$, $-5.73947\cdot10^{-28}$ and
$-1.70746\cdot10^{-42}$ for $N = 1, \ldots, 5$ (at $t = 8000$, $t^2\Xi_N(t)$ agrees with
them to a relative $10^{-3}$). Haglund lists the coefficients of
$1/x^2$ in $\Phi_1$ and $\Phi_2$ as $-.01974938206$ and $.01974934121$
\cite[(52), p.~10]{Hag}, and to more digits in \cite[(52)]{HagW}; his values
are one quarter of ours (at $N = 1$, $-.01974938206$ against $-0.0789975$),
because his (51) applies his (50) at $x$ although \eqref{eq:XiN} evaluates $G$
at $x/2$; the factor does not affect signs. The arXiv and journal versions
conclude that the coefficient of $1/x^2$ in $\Xi_N(x)$ ``is positive for
$k \ge 3$ and negative for $1 \le k < 3$'' \cite[p.~10]{Hag}; the author's web copy
states instead that it ``approaches zero from below as $N \to \infty$''
\cite[p.~10]{HagW}, which is what Proposition~\ref{prop:tail} shows: the limit
is negative for every $N$ and tends to $0$. The coefficients of $\Phi_1$, $\Phi_2$ and $\Phi_3$ nearly cancel: their sum at $N = 3$, of
order $10^{-16}$, is below the resolution of the printed values of (52) in
either copy. Haglund's use of these coefficients to exhibit non-monotone zeros of
the pencil $t\Phi_1 + \Phi_2$ needs only that the coefficient of $\Phi_2$ is
positive and that of $\Xi_2$ negative, and both are.

\begin{corollary}[real zeros of $\Xi_N$]\label{cor:odd}
Let $N \ge 1$ and let $0 < \gamma_1 < \gamma_2 < \cdots$ be the positive real zeros of
$\Xi$. Every real zero of $\Xi_N$ in $(0, \infty)$ lies in a positive lobe of
$\Xi$. Counted with multiplicity, each positive lobe $(\gamma_j, \gamma_{j+1})$
contains an even number of them and the central lobe $(0, \gamma_1)$ an odd
number. Consequently the number of real zeros of $\Xi_N$ in $(0, \infty)$,
counted with multiplicity, is finite and odd.
\end{corollary}

\begin{proof}
Where $\Xi \le 0$, $\Xi_N = \Xi - Q_N < 0$ (Theorem~\ref{thm:sandwich}); in
particular $\Xi_N(\gamma_j) = -Q_N(\gamma_j) < 0$ at both ends of every positive
lobe. At the origin $\Xi_N(0) = \Xi(0) - Q_N(0) \ge 0.4971 - c_1 > 0$, since
$Q_N = c_N - P_N \le c_N \le c_1$, while $\Xi_N(\gamma_1) < 0$. By
Proposition~\ref{prop:tail}, $\Xi_N < 0$ on $[T_0, \infty)$ for some $T_0$, so
only finitely many lobes carry zeros, and the zeros of the entire function
$\Xi_N$ in $[0, T_0]$ are finite in number.
\end{proof}

The ten entries of Haglund's table in \cite[p.~4]{HagW}
(Section~\ref{sec:mech}) are odd, in agreement with Corollary~\ref{cor:odd}. A
numerical census of $\Xi_4$ (real zeros by sign changes, the total by the
argument principle, not interval-rigorous) finds $31$ real zeros, the entry of
\cite{HagW}, the largest being $103.3679880094135$; the arXiv and journal
versions print $32$ there.

\section{Structure: no member is real-rooted}
\label{sec:structure}

\begin{theorem}\label{thm:A}
Let $w \not\equiv 0$, and suppose either \textup{(i)} $0 \le w_n \le 1$ for all
$n$, with $w_m < 1$ for some $m$, or \textup{(ii)} $w$ is finitely supported with
arbitrary real values. Then $\xi_w$ has only finitely many zeros on
$\re s = \tfrac12$ and infinitely many zeros off it.
\end{theorem}

\begin{proof}
(a) \emph{Infinitely many zeros.} Since $|g_n(s)| \le g_n(\re s)$,
\[
|\xi_w(s)| \le \tfrac12 + \tfrac12|s|^2\sum_n|w_n|\,g_n(\re s)
= \exp(O(|s|\log|s|)).
\]
With $n_0$ the least $n$ such that $w_n \ne 0$,
$\xi_w(\sigma) \sim \tfrac12\sigma^2 w_{n_0}(\pi n_0^2)^{-\sigma/2}
\Gamma(\sigma/2)$ as $\sigma \to +\infty$, since
$\Gamma(\sigma/2, X)/\Gamma(\sigma/2) \to 1$ and the other terms are smaller by
$(n_0/n)^\sigma$. So $\log|\xi_w(\sigma)| \ge (\sigma/2)\log\sigma\,(1 - o(1))$,
$\xi_w$ is not $e^{a+bs}P(s)$ with $P$ a polynomial, and by Hadamard's theorem it
has infinitely many zeros.

(b) \emph{Finitely many on the line.} Case (i): by Theorem~\ref{thm:polya},
$\xi_w(\tfrac12+it) \ge \Xi(t) + (1 - w_m)\tfrac12(\tfrac14+t^2)g_m(\tfrac12+it)$.
From $\gamma(a, X) = X^a e^{-X}\sum_{k \ge 0}X^k/(a)_{k+1}$ and
$|(a)_{k+1}| \ge (|t|/2)^{k+1}$ on the line,
$\tfrac12(\tfrac14+t^2)g_m(\tfrac12+it) \to (4\pi m^2 - 1)e^{-\pi m^2} > 0$ as
$|t| \to \infty$ (the parts involving the complete $\Gamma(a)$ are
$O(|t|^{7/4}e^{-\pi|t|/4})$), while $\Xi(t) \to 0$; so
$\liminf_{|t|\to\infty}\xi_w(\tfrac12+it) \ge (1 - w_m)(4\pi m^2 - 1)e^{-\pi m^2}
> 0$. Case (ii): integrating by parts twice as in Theorem~\ref{thm:sandwich},
\[
\xi_w(\tfrac12+it) = C_w + 2\int_0^\infty \Phi_w(u)\cos(tu)\,du, \qquad
\Phi_w = K_w'' - \tfrac14 K_w = \sum_n w_n\tphi_n,
\]
where $K_w(u) = e^{u/2}\sum_n w_n e^{-\pi n^2 e^{2u}}$ and
$C_w = \tfrac12 + 2K_w'(0)$, and the integral has the asymptotic expansion
$\sum_{j \ge 0}2(-1)^{j+1}\Phi_w^{(2j+1)}(0)\,t^{-2j-2}$. If every coefficient of
$\xi_w(\tfrac12+it) \sim C_w + \sum_j a_j t^{-2j-2}$ vanished, the odd part of the
real-analytic function $\Phi_w$ would have zero Taylor series at $0$, so $\Phi_w$
would be even. But the expansion of $\Phi_w(u)$ as $u \to -\infty$ has as
coefficient of $e^{(5/2+2k)u}$ the number
$-2\pi(2k+3)(-\pi)^k/k!\cdot\sum_n w_n n^{2k+2}$, while an even $\Phi_w$ decays
double-exponentially as $u \to -\infty$. So $\sum_n w_n n^{2k+2} = 0$ for all
$k \ge 0$, and since only finitely many $n$ occur, $w \equiv 0$ (Vandermonde),
which is excluded. Hence some coefficient is nonzero, and $\xi_w(\tfrac12+it)$
has constant sign for $|t|$ large.
\end{proof}

For $\xi_N$ the leading coefficient is $C_w = c_N$. The same proof applies to
Haglund's $\Xi_N = \xi_N - c_N$, whose leading coefficient is $0$ and whose
$t^{-2}$ coefficient is negative (Proposition~\ref{prop:tail}): no $\Xi_N$ is
real-rooted either, and Haglund's own list of the zeros of $\Xi_1$ \cite[p.~16]{Hag}
shows one real zero followed by non-real ones.

\begin{theorem}\label{thm:B}
Let $|w_n| \le Cn^A$ for some constants $C, A$, and $w \not\equiv 0$. If every
zero of $\xi_w$ lies on $\re s = \tfrac12$, then $w \equiv 1$.
\end{theorem}

\begin{proof}
Part (a) of the proof of Theorem~\ref{thm:A} holds for polynomially bounded $w$,
so $\xi_w$ has infinitely many zeros, all on the line by hypothesis. The
expansion of part (b) also holds for such $w$ ($\Phi_w$ is real-analytic on $\R$
and all its derivatives decay double-exponentially on $[0,\infty)$), so all its
coefficients vanish, $C_w = 0$, and $\Phi_w$ is even. Then
$D(u) := K_w(u) - K_w(-u)$ is odd with $D'' = D/4$, so $D = 2\alpha\sinh(u/2)$,
that is, $\psi_w(1/x) = \sqrt{x}\,\psi_w(x) - \alpha(\sqrt{x} - 1)$ for
$\psi_w(x) = \sum_n w_n e^{-\pi n^2 x}$. Differentiating at $x = 1$ gives
$K_w'(0) = \alpha/2$, and $C_w = 0$ gives $\alpha = -\tfrac12$: $\psi_w$
satisfies Riemann's theta relation exactly. Then
$\delta := \psi_w - \psi = \sum_n(w_n - 1)e^{-\pi n^2 x}$ satisfies
$\delta(1/x) = \sqrt{x}\,\delta(x)$ and decays faster than every power of $x$ at $0$ and at
$\infty$ (exponentially at $\infty$, and like $x^{-1/2}\delta(1/x)$ at $0$), so its Mellin transform $\pi^{-s/2}\Gamma(s/2)\sum_n(w_n - 1)n^{-s}$ is
entire, symmetric under $s \mapsto 1 - s$, and bounded in vertical strips. By
Hamburger's theorem \cite{Ham} the Dirichlet series
$f(s) = \sum_n(w_n - 1)n^{-s}$ is a constant multiple of $\zeta$; since $f$ is
entire, the constant is $0$ and $w \equiv 1$.
\end{proof}

For bounded $w$, Hamburger's theorem is used in the form stated by Nakamura
\cite[Theorem~D, p.~4]{Nak}: if $F(s) = \sum a(n)n^{-s}$ converges absolutely for
$\sigma > 1$, $P(s)F(s)$ is entire of finite order for some polynomial $P$, and
$\xi_F(1-s) = \xi_F(s)$ with $\xi_F(s) := \pi^{-s/2}\Gamma(s/2)F(s)$, then
$F = C\zeta$; here $F = f$ is entire of finite order. For polynomially bounded
$w$, Burnol's extension \cite[pp.~2--3, Corollaire~2]{Bur} needs convergence only
for $\re s$ large, together with finitely many poles, $f(s) = O(e^{\exp\epsilon|s|})$ in
vertical strips, and, for $g(s) = \chi(s)f(1-s)$, $g(s) = O(|s|^N y^{-s})$ with
$y > \tfrac12$ and $g(\sigma) = c + O(\sigma^{-k})$ for every $k$; here
$g = f = (w_1 - 1) + O(2^{-\sigma})$, so all of these hold. (Hamburger's theorem needs
more than the functional equation: see \cite[p.~4]{Nak} for Knopp's solutions when
(H1) is weakened.)

\begin{proposition}[windowed real-rootedness is a height filtration]\label{prop:F}
Let $F_N$ be entire with $F_N(1-s) = F_N(s)$ and $F_N(\bar s) =
\overline{F_N(s)}$. Suppose that for each $T$ there is $N_0(T)$ such that for
$N \ge N_0(T)$: \textup{(i)} $\sup_{R_T}|F_N - \xi| < \min_{\partial D}|\xi|$
for a finite family $D$ of small discs about the zeros of $\xi$ in
$R_T = [-1, 2] \times [-T, T]$ and for $\partial R_T$; and \textup{(ii)} $F_N$ has
no zeros with $|\im s| \le T$ outside $R_T$. Then, by Rouch\'e's theorem on each
disc and on $R_T$ minus the discs, the zeros of $F_N$ in $R_T$ are those of $\xi$,
moved by less than the disc radii, with multiplicity, and a zero of multiplicity
one near the line lies on it by the symmetry $s \mapsto 1 - \bar s$. Hence, if the
zeros of $\xi$ in $R_T$ are simple, $F_N$ is real-rooted on $R_T$ for all large
$N$ if and only if the Riemann hypothesis holds on $R_T$.
\end{proposition}

\begin{remark}
For $F_N = \xi_N$ one has
$\sup_{R_T}|\xi_N - \xi| \le \tfrac12(T^2 + 4)\max_{-1 \le \sigma \le 2}
\sum_{n > N}g_n(\sigma) \approx T^2 e^{-\pi(N+1)^2}/(\pi(N+1)^2)$, which is below
the size of $\xi$ on the line up to $T \approx 4(N+1)^2$; and (ii) holds for all
large $N$, by comparing $\xi_N$ with $\xi$ for $2 \le \re s \le \pi(N+1)^2$, with
the main term $\tfrac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\sum_{n \le N}n^{-s}$ for
$\re s \ge \pi(N+1)^2$, and using $s \mapsto 1 - s$ on the left. So in any
induction ``members real-rooted up to $T_N$ implies real-rooted up to
$T_{N+1}$'', the only content is the Riemann hypothesis on the new window
$T_N < |\im s| \le T_{N+1}$; what the family proves by itself
(Theorem~\ref{thm:polya}: the set where $\xi_N(\tfrac12+it) < 0$ grows with $N$,
so a lobe that carries real zeros keeps carrying them) is the easy
half. This is the frame in which Conjecture~\ref{conj:H} lives: it asks for an
ordering property to carry the induction, and Theorem~\ref{thm:main} shows that
the ordering property fails near the height at which the approximant leaves the
window.
\end{remark}

\section{Departure height and the lobe scan}
\label{sec:departure}

The figures in this section are floating-point computations in mpmath at $30$
to $80$ digits, cross-checked by a second formula or a second contour where
zeros are reported; they explain where to look and are not part of the proof of
Theorem~\ref{thm:main}.

\paragraph{Departure.} On the line, $\Xi(t)$ is of size roughly $e^{-\pi t/4}$,
and the corrections $P_N$ and $Q_N$ are of size roughly $e^{-\pi(N+1)^2}$. The
two balance near $t = 4(N+1)^2$, the \emph{departure height}: below it the
approximant tracks $\Xi$, above it the approximant is dominated by its
correction. For the truncations $\xi_N$ (from a complete census of the zeros
in $0 < \im s \le 200$ and a count of the lobes up to height $320$) the first
negative lobe of $\Xi$ that carries no real zero of $\xi_N$ is $(25.01, 30.42)$, $(40.92, 43.33)$, $(69.55, 72.07)$,
$(107.17, 111.03)$, $(150.05, 150.93)$, $(202.49, 204.19)$, $(265.56, 266.61)$
for $N = 1, \ldots, 7$, with ratio of lobe depth to $P_N$ equal to $0.0070$,
$0.20$, $0.32$, $0.24$, $0.12$, $0.27$, $0.020$: the departure height is
roughly $4(N+1)^2 + 6$ to $4(N+1)^2 + 10$. Haglund's table shows the same law: his
largest real zeros $14.05, 39.53, 65.03, 103.37, \ldots, 489.39$ for
$N = 1, 2, 3, 4, \ldots, 10$ lie close to $4(N+1)^2$. Every off-line zero of
$\xi_N$ found below height $200$ lies outside the critical strip, at
$\re s \ge 1.39$.

\paragraph{The ordering property fails near departure.} By
Theorem~\ref{thm:sandwich}, apart from the central lobe the real zeros of
$\Xi_N$ come in pairs from the positive lobes of $\Xi$ whose height exceeds the
smooth level $Q_N$, and by
Corollary~\ref{cor:lobe} those of $\xi_N$ come from the negative lobes deeper
than $P_N$. A lobe that fails the level, followed by a lobe that passes it, is expected to
leave (and in both cases below does leave) a pair of non-real zeros over the first
lobe with real zeros above it, which violates the ordering. Lobe heights decrease on average by a factor
$e^{-\pi^2/\log(t/2\pi)}$ per step between lobes of the same sign, but fluctuate
by comparable factors, so such an inversion is to be expected near the
departure height for some $N$. We scanned the windows
$[4(N+1)^2 - 40,\ 4(N+1)^2 + 90]$ for $N = 6, \ldots, 30$, counting
zeros in every window and comparing each lobe's height with the level. The only
violations are at $N = 24$ for $\xi_N$ and at $N = 27$ for $\Xi_N$. The closest
other approach for Haglund's family is at $N = 21$, where a lobe beyond the first
failing one reaches $0.93$ of the level (a violation needs more than $1$); the
next closest is $N = 20$, at $0.48$. The scan is not a proof that
Conjecture~\ref{conj:H} holds for $N \le 26$.

\paragraph{The violation at $N = 27$.} The positive lobe
$(3142.97945673, 3143.56749918)$ of $\Xi$ has height $0.411$ times $Q_{27}$ and
carries no real zero of $\Xi_{27}$; the next positive lobe,
$(3144.49048815,\allowbreak\ 3146.28352838)$, has height $1.325$ times $Q_{27}$ and carries
the real zeros $3144.8946622186$ and $3145.5998495765$. For
$\im s \in [3142.95, 3144.0]$ and $\re s \in [-\tfrac12, \tfrac32]$ the argument
principle counts $2.0$ zeros at $30$ and at $45$ digits, none of them on the
line: $s = 0.8152587993782148453823 + 3143.220682421536585287\,i$, inside the
critical strip, and its mirror $1 - \bar s$. In Haglund's variable
$z = -i(s - \tfrac12)$ the mirror is the zero $z_0$ of
Theorem~\ref{thm:main}, whose real part is below both real zeros.

\section{The certificate}
\label{sec:cert}

Theorem~\ref{thm:main} was certified twice, by two computations written
independently: \emph{certificate A}
in Arb ball arithmetic \cite{Arb} through python-flint~0.6.0 (FLINT~3.0.1)
\cite{FLINT}, and \emph{certificate B} in the outward-rounded interval arithmetic
of mpmath~1.3.0 \cite{mpmath}, with no use of Arb. Both evaluate the objects
\eqref{eq:G}--\eqref{eq:XiN} exactly as Haglund defines them. Each was first run
on a ladder of known cases (Table~\ref{tab:ladder}).

\subsection{Certificate A: Arb ball arithmetic}

Certificate A evaluates $\Xi_{27}$ by two routes. \emph{Route L} is the literal
sum \eqref{eq:XiN}, with $\Gamma(w, a)$ from FLINT's
\texttt{acb\_hypgeom\_gamma\_upper}. At $x = 3144.8946$ its terms fall from
$|\Phi_1| \approx |\Phi_2| \approx 8.0\cdot10^{-9}$ to
$|\Phi_{27}| \approx 1.5\cdot10^{-991}$ while the sum is about $10^{-1070}$, so
about $1062$ digits cancel; route L is used only at exact points, at $4800$ and
$6400$ bits, where the cancellation is fully resolved. \emph{Route T} writes
$\Xi_{27} = \Xi - \sum_{n=28}^{31}\Phi_n - R_{31}$, with $\Xi$ from
\eqref{eq:Xi} (FLINT's $\zeta$ and $\Gamma$), the four terms as in route L, and
the remainder bounded by Lemma~\ref{lem:T}; near $z_0$ all these terms are about
$10^{-1066}$ or smaller ($\Phi_{28} \approx \Xi \approx 1.49\cdot10^{-1066}$,
$\Phi_{29} \approx 3\cdot10^{-1144}$), so route T has no catastrophic
cancellation and is sharp on ball inputs at $256$ bits. Route T uses Riemann's
series
$\Xi = \sum_{n\ge1}\Phi_n$ \cite[(12), p.~3]{Hag}, which also follows from \eqref{eq:riemann}
and Lemma~\ref{lem:relation}; route L does not.

\begin{lemma}\label{lem:B}
For $w \in \Cx$, $a > 0$, $b \in \R$:
$|G(w;a,b)| \le 2\Gamma(\beta, a)/a^\beta$ with $\beta = b + |\im w|$.
\end{lemma}

\begin{proof}
For $u \ge 0$, $|\cos(x+iy)|^2 = \cos^2 x + \sinh^2 y \le \cosh^2 y$, so
$|\cos(2wu)| \le e^{2u|\im w|}$ and
$|G| \le 4\int_0^\infty e^{2\beta u - ae^{2u}}\,du$; the substitution
$t = ae^{2u}$ turns this into $2a^{-\beta}\int_a^\infty t^{\beta-1}e^{-t}\,dt$.
\end{proof}

\begin{lemma}\label{lem:Gam}
For $\beta \ge 1$ and $a > \beta - 1$:
$\Gamma(\beta, a)/a^\beta \le e^{-a}/(a - \beta + 1)$.
\end{lemma}

\begin{proof}
With $t = ax$, $\Gamma(\beta,a)/a^\beta = \int_1^\infty x^{\beta-1}e^{-ax}\,dx$,
and $x^{\beta-1} \le e^{(\beta-1)(x-1)}$ for $x \ge 1$ since
$\log x \le x - 1$ and $\beta \ge 1$; integrate.
\end{proof}

\begin{lemma}\label{lem:T}
Let $Y \ge |\im z|$, $\beta_9 = \tfrac94 + Y/2$, $a_n = \pi n^2$, and $M \ge 1$
with $a_{M+1} - \beta_9 + 1 > 0$. Put
$U_n = 2e^{-a_n}(2\pi^2 n^4 + 3\pi n^2)/(a_n - \beta_9 + 1)$. Then
$\bigl|\sum_{n > M}\Phi_n(z)\bigr| \le \sum_{n>M}U_n \le 2U_{M+1}$.
\end{lemma}

\begin{proof}
In \eqref{eq:XiN}, $|\im(z/2)| \le Y/2$, and Lemmas~\ref{lem:B}
and~\ref{lem:Gam} (with $\beta = \tfrac94 + Y/2$ and $\tfrac54 + Y/2$, both
$\ge 1$) give $|\Phi_n| \le U_n$
(since $\Gamma(\beta,a)/a^\beta = \int_1^\infty x^{\beta-1}e^{-ax}\,dx$ increases with
$\beta$, $|\im(z/2)|$ may be replaced by $Y/2$). For $n > M$,
$U_{n+1}/U_n \le e^{-3\pi}\cdot 16 < 1.3\cdot10^{-3}$, since the ratio of the
polynomial factors is at most $((n+1)/n)^4 \le 16$ and the ratio of the
denominators is below $1$; so $\sum_{n>M}U_n \le 2U_{M+1}$.
\end{proof}

At $N = 27$, $M = 31$ and $Y \le 0.317$, and $2U_{32} \le 2\cdot10^{-1393}$. The
winding numbers use the elementary fact that if a path lies in a closed
axis-parallel box that excludes $0$, the box lies in an open half-plane through
$0$ and the continuous change of argument along the path equals the principal
value $\operatorname{Arg}(\gamma(1)/\gamma(0)) \in (-\pi, \pi)$. The boundary of
each square is cut into pieces, a piece is bisected until its ball enclosure
excludes $0$, and the increments are summed; by the argument principle a total
in $(2\pi(k - \tfrac12), 2\pi(k + \tfrac12))$ certifies $k$ zeros inside. Every
arithmetic operation and special function is Arb's, whose contract is that the
output ball contains the exact value for every input in the input balls; no
floating-point value is used in any decision. The only other inputs are
Lemmas~\ref{lem:B}--\ref{lem:T}, the half-plane fact, Riemann's series for
route T, the intermediate value theorem and the argument principle.

\subsection{Certificate B: interval arithmetic with proved remainders}

Certificate B uses mpmath's real intervals with outward rounding, complex
numbers as rectangular boxes built on them, and exact rational Bernoulli
numbers; every truncation adds a disk whose radius is a proved bound, each
targeting $2^{-190}$ relative to the size of the truncated quantity ($|\zeta| \approx 1$,
and $e^{-X}$ for $h$), at a working precision of $200$ bits. It writes
$\Phi_n = 2X^2[h(\tfrac s2 + 2) + h(\tfrac{1-s}2 + 2)] - 3X[h(\tfrac s2 + 1) +
h(\tfrac{1-s}2 + 1)]$ with $h(w) = X^{-w}\Gamma(w, X) = \int_1^\infty
e^{-Xu}u^{w-1}\,du$, as in the proof of Lemma~\ref{lem:relation}, and evaluates
$\Xi_{27} = \Xi - \Phi_{28} - \Phi_{29} - \Phi_{30} - T_{30}$ with the following
bounds.
\begin{itemize}
\item $\Gamma$ by Stirling's series with the remainder bound of
\cite[5.11.1, \S5.11(ii)]{DLMF}, shifting the argument when $|w| < 300$ or
$\re w < 0$.
\item $\zeta$ by Euler--Maclaurin summation \cite[2.10.1]{DLMF} with the
remainder bound
$|R_m| \le (2 - 2^{1-2m})|B_{2m}|\,|(s)_{2m}|\,N^{1-\sigma-2m}/((2m)!\,
(\sigma + 2m - 1))$ from \cite[24.9.2]{DLMF}; at this height $N = 1012$ and
$m = 94$.
\item $h$ by repeated integration by parts,
$h(w) = e^{-X}\sum_{k<K}\tau_k + \rho_K$ with
$\tau_k = (w-1)\cdots(w-k)/X^{k+1}$ and
$|\rho_K| \le e^{-X}|\tau_K|\,X/(X - \max(\beta, 0))$, $\beta = \re w - 1 - K$.
\item The tail: $|h(w)| \le e^{-X}/(X - \beta')$ with $\beta' = \max(\re w - 1,
0)$, so for $X \ge \max(2\beta', 12)$ (with $\beta'$ maximized over the four
arguments in $\Phi_n$) one has $|\Phi_n| \le 9Xe^{-X}$, whence
$|\sum_{n>M}\Phi_n| \le 18\pi(M+1)^2e^{-\pi(M+1)^2}$; at $N = 27$, $M = 30$ and
the bound is $3.7\cdot10^{-1307}$.
\end{itemize}
At $x = 3144.8946$ one has $|\Xi| = 1.4923145335789\cdot10^{-1066}$ and
$|\Phi_{28}| = 1.49249055304208\cdot10^{-1066}$, so this route resolves
$\Xi_{27} \approx 10^{-1070}$ with about $50$ digits to spare. As a second
rigorous route for the real zeros, certificate B also evaluates the literal sum
\eqref{eq:XiN} at $4400$ bits, with $\Phi_1, \ldots, \Phi_{27}$ from the lower
incomplete gamma series (remainder $\le e^{-X}X^K/|(w)_K|$ once
$\re w + K \ge X + 1$), integration by parts where its remainder bound is small, and Stirling's
series: no $\zeta$, no Euler--Maclaurin summation, and no use of Riemann's
series.

Winding numbers come from a rigorous Taylor model. Since $f = \Xi_{27}$ is
entire, its Taylor coefficients about $c$ satisfy $|a_k| \le M_R/R^k$, with
$M_R$ an upper bound for $|f|$ on $|z - c| = R$ obtained from box enclosures
covering the circle. The discrete Fourier transform of $f$ on $m$ points of the
circle $|z - c| = \rho$ gives $D_k = \sum_{l\ge0}a_{k+lm}\rho^{lm}$, so
$|D_k - a_k| \le (M_R/R^k)q^m/(1 - q^m)$ with $q = \rho/R$, and for
$|z - c| \le r' < R$, with $x = r'/R$,
\[
\Bigl|f(z) - \sum_{k<K}D_k(z-c)^k\Bigr| \le E :=
M_R\Bigl[\frac{q^m}{1-q^m}\sum_{k<K}x^k + \frac{x^K}{1-x}\Bigr].
\]
A boundary piece is accepted when the model's box enclosure plus $[-E,E]^2$
excludes $0$ and the ratio of the end values has positive real part; the change
of argument along it is then the principal value of that ratio's argument. At
$N = 27$: $R = 0.1$, $M_R = 2.09255\cdot10^{-1065}$, $\rho = 0.002$, $m = 24$,
$K = 12$, giving, with $r' = \sqrt2\,r$ for a square of half-width $r$ about $c$,
$E = 3.51\cdot10^{-1106}$ for $r \le 10^{-5}$ and
$1.36\cdot10^{-1087}$ at $r = 10^{-3}$; the constant coefficient $D_0$ agrees
with a direct enclosure of $\Xi_{27}(c)$. For the $10^{-48}$ enclosure a
second model is built at the center
\begin{align*}
\re z^* &= 3143.220682421536585287281295989417800119257644961983569,\\
\im z^* &= 0.3152587993782148453822863730092717652956513976685290909,
\end{align*}
with $\rho = 0.001$ and $m = 40$, giving $E = 2.09\cdot10^{-1145}$; at
$r = 10^{-48}$ the smallest lower modulus over the boundary pieces,
$5.9\cdot10^{-1114}$, still exceeds the radii of the point enclosures, about
$10^{-1120}$, by a factor $10^6$. The trust base of certificate B is mpmath's
primitive interval operations and the cited DLMF remainder bounds; Riemann's
series is used for the tail route and checked numerically, not re-derived.

\subsection{The enclosures}

Tables~\ref{tab:real}--\ref{tab:ladder} give the certified values. Here
$c = 3143.2206824215 + 0.3152587994\,i$, and a square of half-width $r$ about a
center $w$ is $w + [-r, r]^2$.

\begin{table}[ht]
\centering\small
\begin{tabular}{lll}
\toprule
$x$ & certificate A (ball midpoint $\pm$ radius) & certificate B (interval) \\
\midrule
$3144.8946$ & $-1.76019463127752\cdot10^{-1070} \pm 4.1\cdot10^{-1085}$ &
$-1.760194631277516\cdot10^{-1070}$ \\
$3144.8947$ & $+1.06871649226171\cdot10^{-1070} \pm 3.3\cdot10^{-1085}$ &
$+1.068716492261707\cdot10^{-1070}$ \\
$3145.5998$ & $+1.12975694291664\cdot10^{-1070} \pm 1.9\cdot10^{-1085}$ &
$+1.129756942916642\cdot10^{-1070}$ \\
$3145.5999$ & $-1.14913915362778\cdot10^{-1070} \pm 2.9\cdot10^{-1085}$ &
$-1.149139153627783\cdot10^{-1070}$ \\
\bottomrule
\end{tabular}
\caption{Enclosures of $\Xi_{27}(x)$ at the ends of the two real-zero intervals.
Certificate A: balls as printed, display rounding included; the Arb radii are
$3.3\cdot10^{-1435}$ (route L, $4800$ bits), $6.0\cdot10^{-1915}$ (route L,
$6400$ bits) and $5.0\cdot10^{-1138}$ (route T, $256$ bits), and all four
evaluations (route T at $256$ and $1024$ bits, route L at $4800$ and $6400$ bits)
give the same signs. Certificate B: tail route, box radii at most
$6\cdot10^{-1121}$; its literal route at $4400$ bits (radius about
$1$--$2\cdot10^{-1171}$) agrees to all digits shown and overlaps.}
\label{tab:real}
\end{table}

\begin{table}[ht]
\centering\small
\begin{tabular}{llllc}
\toprule
cert. & center & half-width $r$ & boundary pieces & $k$ \\
\midrule
A & $c$ & $10^{-3}$ ($256$ and $512$ bits) & $512$, $512$ & $1$ \\
A & $c$ & $10^{-6}$, $10^{-10}$, $4\cdot10^{-11}$ & $512$, $663$, $807$ & $1$ \\
A & $c + 2.5\cdot10^{-3}$ & $10^{-3}$ & $269$ & $0$ \\
A & $c + 1.5\cdot10^{-10}$ & $10^{-10}$ & $714$ & $0$ \\
B & $c$ & $10^{-3}$, $10^{-5}$, $10^{-7}$, $10^{-9}$ & $32$ each & $1$ \\
B & $c$ & $10^{-10}$, $5\cdot10^{-11}$, $4\cdot10^{-11}$ & $32$, $32$, $33$ & $1$ \\
B & $c$ & $3.7\cdot10^{-11}$ & $36$ & $1$ \\
B & $c$ & $3.5\cdot10^{-11}$ & $33$ & $0$ \\
B & $c + 0.01$ & $10^{-3}$ & $32$ & $0$ \\
B & $z^*$ & $10^{-30}$, $10^{-40}$, $10^{-45}$, $10^{-48}$ & -- & $1$ \\
\bottomrule
\end{tabular}
\caption{Certified winding numbers $k$ of $\Xi_{27}$ along the boundaries of
squares. In certificate A every run with $k = 1$ has total change of argument
$6.28318530717959 \pm 4\cdot10^{-15}$. In certificate B at $r = 10^{-3}$ the
smallest lower modulus over the pieces is $5.89\cdot10^{-1069}$ against the model
error $E = 1.36\cdot10^{-1087}$; at $r = 3.7\cdot10^{-11}$ it is
$1.97\cdot10^{-1078}$ against $E = 3.51\cdot10^{-1106}$. The run at
$r = 3.5\cdot10^{-11}$ returns $0$ because the zero lies just outside that
square, and the rows with centers $c + 2.5\cdot10^{-3}$, $c + 1.5\cdot10^{-10}$
and $c + 0.01$ are zero-free controls.}
\label{tab:wind}
\end{table}

\begin{table}[ht]
\centering\small
\begin{tabular}{>{\raggedright\arraybackslash}p{0.30\textwidth}>{\raggedright\arraybackslash}p{0.30\textwidth}>{\raggedright\arraybackslash}p{0.30\textwidth}}
\toprule
test & certificate A & certificate B \\
\midrule
$\Xi_1$ at $14.04543957$ and $14.04543959$ (Haglund: $14.0454395788$) &
$+1.347060456147\cdot10^{-11}$, $-1.704102125315\cdot10^{-11}$ &
$+1.347060456146727\cdot10^{-11}$, $-1.704102125314551\cdot10^{-11}$ \\
$\Xi_2$ at $39.5324810797$ and $39.5324810799$ (Haglund: $39.5324810798$) &
$+3.605517254360\cdot10^{-21}$, $-4.593418771015\cdot10^{-21}$ &
$+3.605517254360067\cdot10^{-21}$, $-4.5934187710153\cdot10^{-21}$ \\
$\Xi_1$ about Haglund's zero $20.62534600592171760132974 +
2.697151842339519632505712\,i$ &
$k = 1$ at $r = 10^{-10}$ ($1516$ pieces) & $k = 1$ at $r = 10^{-3}$ and
$10^{-8}$ \\
zero-free controls for $\Xi_1$ & $k = 0$ (center shifted by $0.5$, $r = 0.1$;
shifted by $3\cdot10^{-10}$, $r = 10^{-10}$) & $k = 0$ ($17 + i$,
$r = \tfrac14$) \\
\bottomrule
\end{tabular}
\caption{The ladder, run before $N = 27$ with the same code: the largest real
zeros of $\Xi_1$ and $\Xi_2$ from Haglund's table \cite[p.~4]{Hag}, the first
non-real zero of $\Xi_1$ from his appendix \cite[p.~16]{Hag}, and zero-free
squares.}
\label{tab:ladder}
\end{table}

\paragraph{Cross-checks.} In certificate A, routes L and T overlap at all $18$
points compared (the center, the reference zero, the corners and midpoints of
the square of half-width $10^{-3}$, the corners of the square of half-width
$4\cdot10^{-11}$, and the four points of Table~\ref{tab:real}), with
$|L - T| \le 6.43\cdot10^{-1137}$ at $256$ bits, and
$|L - T| \le 1.18\cdot10^{-1288}$ at $1024$ bits at the
fourteen points other than those of Table~\ref{tab:real}. In certificate B the literal and
tail routes overlap at the four points of Table~\ref{tab:real}, its enclosures of
$\Xi$ contain mpmath's ordinary $400$-bit values, and its two methods for $h$ at
$X = 784\pi$ overlap. The two certificates agree with each other to every digit
printed. A third evaluation of the literal sum
\eqref{eq:XiN} in Arb at $4800$ bits gives
$\Xi_{27}(3144.8946) \in [-1.76019463128\cdot10^{-1070} \pm
2.49\cdot10^{-1082}]$ and $\Xi_{27}(3144.8947) \in [1.06871649226\cdot10^{-1070}
\pm 1.71\cdot10^{-1082}]$, the same signs at $3145.5998$ and $3145.5999$, and
Haglund's two ladder zeros; at the point
$3143.220682421536585287 + 0.3152587993782148453823\,i$ it gives a value of modulus
about $1.7\cdot10^{-1084}$, against about $6.1\cdot10^{-1072}$ at a point
$10^{-6}$ to the right.

\subsection{Proof of Theorem~\texorpdfstring{\ref{thm:main}}{1.1}}

(a) By \eqref{eq:G}, $G(z;a,b)$ is real for real $z$, so $\Xi_{27}$ is real and
continuous on $\R$. The enclosures of Table~\ref{tab:real} have strictly
opposite signs at the ends of each interval, in both certificates; by the
intermediate value theorem each interval contains a zero.

(b) $\Xi_{27}$ is entire, being a finite sum of entire functions. Certificate B
finds winding number $1$ along the boundary of the square of half-width
$3.7\cdot10^{-11}$ about $c$, with every boundary enclosure excluding $0$; by the
argument principle the open square contains exactly one zero $z_0$, counted
with multiplicity, so $z_0$ is simple. Certificate A independently finds winding
number $1$ at half-width $4\cdot10^{-11}$. The square
$z^* + [-10^{-48}, 10^{-48}]^2$ lies inside the first square, since
$z^* - c = 3.66\ldots\cdot10^{-11} - 2.18\ldots\cdot10^{-11}\,i$; winding number
$1$ there (certificate B) locates the same zero $z_0$.

(c) From (b), $\im z_0 \ge 0.3152587994 - 3.7\cdot10^{-11} > 0.3152$ and
$\re z_0 \le 3143.2206824215 + 3.7\cdot10^{-11} < 3143.2207 < 3144.8946 < x_1$.
So $z_0 \in \Q$ is non-real and precedes the real zero $x_1$ in any listing of
the zeros of $\Xi_{27}$ in $\Q$ by increasing real part, while
$\im z_0 > 0 = \im x_1$. \hfill$\square$

\section{The companion violation at \texorpdfstring{$N = 24$}{N = 24}}
\label{sec:xi24}

For the truncations $\xi_N$ the ordering property reads: the zeros of
$z \mapsto \xi_N(\tfrac12 + iz)$ in $\Q$, listed by increasing real part, have
nondecreasing imaginary parts. Since $\xi_N \to \xi$ locally uniformly, the
Hurwitz argument of Haglund's Proposition~1 shows that this property for every
$N$ would also imply the Riemann hypothesis. It fails at $N = 24$.

\begin{theorem}\label{thm:xi24}
\begin{enumerate}
\item[(a)] The real function $t \mapsto \xi_{24}(\tfrac12 + it)$ changes sign
on $(2510.2026, 2510.2027)$ and on $(2510.7086, 2510.7087)$, so $\xi_{24}$ has
zeros on the critical line at heights in these intervals.
\item[(b)] The function $z \mapsto \xi_{24}(\tfrac12 + iz)$ has exactly one zero
in the square of half-width $10^{-10}$ about
$2508.2839748053 + 0.3159896243\,i$.
\item[(c)] Hence $\xi_{24}$ has a zero $\rho_{24}$ off the critical line, inside
the critical strip, with $\im\rho_{24} < 2508.2839748054$, below zeros on the
line: the ordering property fails for $\xi_{24}$.
\end{enumerate}
\end{theorem}

Both certificates prove this, with the same methods as in
Section~\ref{sec:cert}; here route T and certificate B use Riemann's
identity \eqref{eq:riemann}, which rests on Jacobi's theta relation (cited, not
re-proved), while certificate A's route L, the literal sum, does not. The enclosures on the line are, in certificate A (both
routes; route L is the literal sum at $4800$ bits, cancelling about $855$
digits)
\[
\xi_{24}(\tfrac12 + it) \in
\begin{cases}
[+2.81317536164\cdot10^{-854} \pm 5\cdot10^{-866}], & t = 2510.2026,\\
[-1.65267129215\cdot10^{-854} \pm 3\cdot10^{-866}], & t = 2510.2027,\\
[-3.23281139505\cdot10^{-854} \pm 4\cdot10^{-866}], & t = 2510.7086,\\
[+6.14184061542\cdot10^{-855} \pm 4\cdot10^{-867}], & t = 2510.7087,
\end{cases}
\]
and in certificate B $+2.813175361644763\cdot10^{-854}$,
$-1.65267129214774\cdot10^{-854}$, $-3.232811395053649\cdot10^{-854}$ and
$+6.141840615423139\cdot10^{-855}$ at the same four points. Certificate A finds (in the $s$-plane; the
symmetry $\xi_{24}(\tfrac12 + i\bar z) = \overline{\xi_{24}(\tfrac12 + iz)}$ carries its squares to
those of (b)) winding number $1$ along the $s$-plane squares of half-width $10^{-3}$
($151$ pieces) and $10^{-10}$ ($174$ pieces) about
$0.8159896243 + 2508.2839748053\,i$, and $0$ for a control square shifted by
$2.5\cdot10^{-3}\,i$; certificate B finds winding number $1$ for
$z \mapsto \xi_{24}(\tfrac12 + iz)$ on squares of half-width $10^{-3}$,
$10^{-6}$, $10^{-9}$ and $10^{-10}$ about $2508.2839748053 + 0.3159896243\,i$
(Taylor model with $R = 0.1$, $\rho = 0.002$, $m = 24$, $K = 12$,
$M_R = 4.07\cdot10^{-849}$; at $r = 10^{-3}$ the smallest lower modulus is
$1.16\cdot10^{-852}$ against $E = 2.6\cdot10^{-871}$), and $0$ for the control
square shifted by $0.01$. Numerically, at $50$ digits,
$\rho_{24} = 0.8159896243043424688941 + 2508.283974805324153202\,i$, and the
zero in (b) is its mirror $1 - \bar\rho_{24}$.

The mechanism is the one of Section~\ref{sec:departure} with the signs reversed:
the negative lobe $(2507.87103958,\ 2508.7660461)$ of $\Xi$ has depth $0.595$
times $P_{24}$ and carries no real zero of $\xi_{24}$, while the next negative
lobe $(2509.65299744,\ 2511.66018979)$ has depth $1.130$ times $P_{24}$. The
off-line zero lies inside the critical strip, at $\re s = 0.8160$. As with
$\Xi_{27}$, it is a zero of the approximant at a height where the approximant has
departed from $\xi$.

\section*{Acknowledgment}
\addcontentsline{toc}{section}{Acknowledgment}

I thank J. Haglund for pointing out the corrected copy \cite{HagW} of his paper.

\section*{Data and code availability}
\addcontentsline{toc}{section}{Data and code availability}

The two certificates of Section~\ref{sec:cert} --- their scripts and logs, and
for each a document stating every bound used, with its proof or its source ---
are deposited with this paper on Zenodo as a self-contained archive,
\texttt{haglund-counterexample-certificate.zip}, SHA-256
\texttt{4dfb1360042c3123b815e571a1916720}\allowbreak\texttt{081beca51fe9be8d6ed4760f41b1091c}, together with a third evaluation
of the literal sum, second runs of both certificates, and a README giving the
exact commands and their expected outputs. On one core of an Apple M4,
certificate A runs in about $72$ seconds (Python~3.9.6, python-flint~0.6.0) and
certificate B in about $15$ minutes (Python~3.9.6, mpmath~1.3.0), most of it
the $4400$-bit literal cross-check. The floating-point computations behind
Sections~\ref{sec:positivity} and~\ref{sec:departure} (the zero census, the tail
coefficients, the lobe scan and the $50$-digit zeros) are in the repository
\url{https://github.com/x67ai/riemann-rh-program}.

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\end{document}