==== TABLE OF CONTENTS ====
A-degenerate-counterexample-to-the-critical-number-Arnold-bound//00-introduction.tex
A-degenerate-counterexample-to-the-critical-number-Arnold-bound//01-reduction.tex
A-degenerate-counterexample-to-the-critical-number-Arnold-bound//02-quadric.tex
A-degenerate-counterexample-to-the-critical-number-Arnold-bound//03-critical-bound.tex
A-degenerate-counterexample-to-the-critical-number-Arnold-bound//04-three-points.tex
A-degenerate-counterexample-to-the-critical-number-Arnold-bound//05-conclusion.tex
A-degenerate-counterexample-to-the-critical-number-Arnold-bound//main.tex
A-degenerate-counterexample-to-the-critical-number-Arnold-bound//references.tex
===========================
==== BEGIN A-degenerate-counterexample-to-the-critical-number-Arnold-bound//00-introduction.tex ====
\section{Introduction}\label{sec:introduction}
Arnold's fixed-point conjecture compares Hamiltonian dynamics with the
critical-point theory of smooth functions. For a sufficiently short
autonomous Hamiltonian flow on a closed symplectic manifold, its fixed points are
exactly the critical points of its Hamiltonian. The conjecture asks
whether an arbitrary Hamiltonian diffeomorphism must retain the minimum
number of critical points forced by the topology of the manifold.
Arnold's formulation explicitly includes the count of geometrically
distinct points~\cite[Section~2]{Arnold1986}. We construct a smooth
counterexample on the complex quadric threefold, which also violates the
weaker rational cup-length bound.
\paragraph{The two degenerate bounds.}
For a closed smooth manifold $M$, write
\[
\CritSet(h)=\{q\in M:dh_q=0\},\qquad
\Crit(M)=\min_{h\in C^\infty(M,\R)}\#\CritSet(h).
\]
The minimum here permits degenerate critical points, whereas the
\emph{Morse number} minimizes only over Morse functions. Let
$\operatorname{cuplength}(M;\mathbb Q)$ be one plus the maximum number
of positive-degree rational cohomology classes with nonzero cup product.
Thus our cup-length convention includes the unit. The
Lusternik--Schnirelmann inequality gives
\[
\operatorname{cuplength}(M;\mathbb Q)\le\Crit(M);
\]
Section~\ref{sec:critical-bound} recalls the cup-product argument in a
form that applies to arbitrary smooth functions.
A Hamiltonian diffeomorphism of a symplectic manifold $(M,\omega)$ is
the time-one map of a smooth time-dependent Hamiltonian flow. We write
$\Ham(M,\omega)$ for these maps, use the convention
$\iota_{X_{H_t}}\omega=dH_t$, and set
$\Fix(\phi)=\{q\in M:\phi(q)=q\}$.
The \emph{critical-number Arnold bound} asserts that
\begin{equation}\label{eq:arnold-bound}
\#\Fix(\phi)\ge\Crit(M)
\end{equation}
for every $\phi\in\Ham(M,\omega)$ on every closed connected symplectic
manifold. Its weaker \emph{rational cup-length} form replaces $\Crit(M)$
by $\operatorname{cuplength}(M;\mathbb Q)$. Neither formulation assumes
nondegeneracy; see Golovko~\cite[Conjectures~1.4 and~1.6]{Golovko2020}.
A fixed point $q$ is nondegenerate when $1$ is not an eigenvalue of
$d\phi_q$.
Let
\begin{equation}\label{eq:quadric}
Q^3=\bigl\{[z_0:z_1:z_2:z_3:z_4]\in\CP^4:
z_0^2+z_1^2+z_2^2+z_3^2+z_4^2=0\bigr\},
\end{equation}
with the restricted Fubini--Study symplectic form, normalized in
Section~\ref{sec:quadric}.
\begin{theorem}\label{thm:counterexample}
There is a smooth Hamiltonian diffeomorphism $\phi$ of the closed connected
symplectic six-manifold $Q^3$ such that
\begin{equation}\label{eq:counterexample}
\#\Fix(\phi)=3<4=\Crit(Q^3)
=\operatorname{cuplength}(Q^3;\mathbb Q).
\end{equation}
At least one fixed point of $\phi$ is degenerate.
\end{theorem}
This disproves both unrestricted smooth bounds. The count includes all
fixed points, so imposing a contractibility condition on their
Hamiltonian trajectories cannot restore either inequality. The
nondegenerate homological Arnold inequalities concern a different
hypothesis and remain compatible with this example.
The count of three is sharp. Gong~\cite[Section~1.1.1]{Gong2025}
obtains at least $n$ fixed points for every Hamiltonian diffeomorphism
of the standard complex quadric of complex dimension $n\ge2$.
His normalization of the Fubini--Study form differs from ours by a
positive scalar, which does not change the Hamiltonian group: rescaling
the form and a generating Hamiltonian by the same scalar preserves its
vector field. Thus our example attains his bound for $n=3$.
\paragraph{Historical context and related work.}
The positive cases explain why the unrestricted comparison is natural.
Conley and Zehnder proved that every Hamiltonian diffeomorphism of the
standard symplectic torus $T^{2n}$ has at least $2n+1$ fixed
points~\cite{ConleyZehnder1983}, and Fortune obtained the bound $n+1$
on standard $\CP^n$~\cite{Fortune1985}. Hofer obtained cup-length bounds without
nondegeneracy when $\pi_2(M)=0$~\cite[Theorem~4]{Hofer1988}.
Rudyak and Oprea proved the full critical-number bound when both the
symplectic class and the first Chern class vanish on $\pi_2(M)$
\cite[Corollary~4.3]{RudyakOprea1999}. The quadric has spheres of
positive symplectic area and lies outside these latter hypotheses.
Floer's pseudoholomorphic-curve methods established homological Morse
inequalities for nondegenerate Hamiltonian fixed points, including the
monotone setting~\cite{Floer1989}. The lower bound by the sum of rational
Betti numbers was subsequently established for nondegenerate Hamiltonian
diffeomorphisms on every closed symplectic manifold by Fukaya and Ono,
Liu and Tian, and Ruan~\cite{FukayaOno1999,LiuTian1998,Ruan1999}.
The distinction between these bounds
and the degenerate formulations above is essential; see
\cite[Theorem~1.3 and Conjectures~1.4 and~1.6]{Golovko2020}.
Ma~\cite[Theorem~1.5]{Ma2013} asserted that the minimum number of fixed
points over all Hamiltonian diffeomorphisms always equals $\Crit(M)$.
Theorem~\ref{thm:counterexample} contradicts that assertion.
Regularity also matters. Buhovsky, Humili\`ere, and Seyfaddini constructed
Hamiltonian \emph{homeomorphisms} with a single fixed point on every closed
connected symplectic manifold of real dimension at least four
\cite{BHS2018}. Our construction stays in the smooth Hamiltonian group.
In a preprint posted September 27, 2026, Jiao~\cite[Theorem~1.1]{Jiao2026}
reports a smooth Hamiltonian diffeomorphism of the equal-area product
$S^2\times S^2$ with two fixed points, using local saddle merging and
mixed generating functions. The construction here instead uses a
finite-order map on a six-dimensional quadric. None of these
fixed-point results is an input to our construction.
\paragraph{Proof mechanism.}
The key is to transfer a critical-point count from a fixed submanifold
to the fixed-point count of a nearby Hamiltonian map. Let $A$ be a
finite-order Hamiltonian map with fixed submanifold $F$, whose fixed
tangent vectors are exactly those tangent to $F$. Extend a function
$f$ on $F$ to an $A$-invariant function $K$ on the ambient manifold,
and let $B_s$ be its Hamiltonian flow. If $A^m=\id$, invariance gives
\[
(A\circ B_\varepsilon)^m=B_{m\varepsilon}.
\]
For sufficiently small $\varepsilon>0$, the right side has only
stationary fixed points. Any fixed point of $A\circ B_\varepsilon$
is therefore a critical point of $K$ fixed by $A$. Finite-group
averaging makes ambient criticality on $F$ equivalent to criticality
of $f$, giving
\[
\Fix(A\circ B_\varepsilon)=\CritSet(f).
\]
This criticality step is the finite-group case of Palais's principle
of symmetric criticality~\cite[Section~5]{Palais1979}; the power identity
excludes additional fixed points away from $F$.
For the quadric, take the Hamiltonian involution
\[
A[z_0:z_1:z_2:z_3:z_4]=[z_0:-z_1:-z_2:-z_3:-z_4].
\]
Its fixed locus is the quadric surface $Q^2\cong S^2\times S^2$.
This product admits functions with only three critical points, as
recorded by Mare~\cite[Proposition~3.2.5]{Mare1997} through an extension
of Takens's sphere-product construction~\cite{Takens1968}. We give an
explicit formula and compute its entire critical set. The isolated
degenerate critical point becomes a degenerate Hamiltonian fixed point.
In contrast, symplectic volume forces every smooth function on the
ambient $Q^3$ to have at least four critical points, and a Hamiltonian
rotation realizes that minimum.
Section~\ref{sec:reduction} proves the finite-order transfer, including
the short-period estimate and invariant extension.
Section~\ref{sec:quadric} constructs the Hamiltonian involution and
its tangent splitting. Section~\ref{sec:three-points} gives the
three-critical-point function. Section~\ref{sec:critical-bound}
proves the critical-number and rational cup-length equalities on $Q^3$,
and Section~\ref{sec:conclusion} assembles the construction and
verifies degeneracy.
==== END A-degenerate-counterexample-to-the-critical-number-Arnold-bound//00-introduction.tex ====
==== BEGIN A-degenerate-counterexample-to-the-critical-number-Arnold-bound//01-reduction.tex ====
\section{Perturbing a finite-order Hamiltonian map}
\label{sec:reduction}
A short Hamiltonian perturbation of a finite-order map can replace its
fixed submanifold by the critical set of a function on that submanifold.
The tangent-space hypothesis below is the usual \emph{clean fixed-set}
condition; we state it explicitly to identify the geometric input.
\begin{proposition}
\label{prop:reduction}
Let \((M,\omega)\) be a closed symplectic manifold, and let
\(A\in\Ham(M,\omega)\) satisfy \(A^m=\id\) for some positive integer
\(m\). Suppose that \(F=\Fix(A)\) is a closed embedded submanifold and
that
\[
T_qF=\ker(dA_q-\id)
\qquad\text{for every }q\in F.
\]
For every smooth function \(f:F\to\R\), there is a smooth
\(A\)-invariant extension \(K:M\to\R\) such that, if \(B_s\) is its
Hamiltonian flow, then for every sufficiently small \(\varepsilon>0\)
the map
\[
\phi_\varepsilon=A\circ B_\varepsilon
\]
is Hamiltonian and satisfies
\[
\Fix(\phi_\varepsilon)=\CritSet(f),
\]
where the critical set on the right is viewed as a subset of \(M\).
\end{proposition}
The proof combines finite-group averaging with the exclusion of short
nonconstant periodic trajectories. We establish the latter fact first.
Lipschitz control of a vector field gives a lower bound for its
nonconstant periods, a classical principle for which Yorke proved the
sharp Euclidean estimate~\cite{Yorke1969}. The following elementary
local argument suffices here.
\begin{lemma}
\label{lem:short-periods}
Let \(X\) be a smooth vector field on a closed manifold \(M\), and let
\(\Psi_t\) be its flow. There is a number \(\delta>0\) such that
\[
\Fix(\Psi_t)=\{q\in M:X(q)=0\}
\qquad\text{for every }0<t<\delta.
\]
\end{lemma}
\begin{proof}
The flow is defined for all time because \(M\) is compact. Suppose that
there were points \(q_j\) and times \(t_j>0\) with
\[
t_j\longrightarrow0,\qquad
\Psi_{t_j}(q_j)=q_j,\qquad X(q_j)\ne0.
\]
After passing to a subsequence, assume that \(q_j\to q\). Choose
coordinates near \(q\), and a Euclidean ball about its coordinate image
whose closure lies inside the coordinate domain. By continuity of the
flow, all sufficiently late trajectory segments
\(\{\Psi_s(q_j):0\le s\le t_j\}\) lie in this ball. The coordinate vector
field \(V\) is Lipschitz there with a constant \(L\ge0\), since its
derivative is bounded on the closed ball.
Write \(u_j(s)\) for the coordinate trajectory and set
\[
C_j=\max_{0\le s\le t_j}|\dot u_j(s)|>0.
\]
For any \(s,r\in[0,t_j]\), convexity of the ball and the Lipschitz
estimate give
\[
\begin{split}
|\dot u_j(s)-\dot u_j(r)|
&=|V(u_j(s))-V(u_j(r))|\\
&\le L|u_j(s)-u_j(r)|
\le LC_j|s-r|
\le LC_jt_j.
\end{split}
\]
The closed trajectory has zero mean coordinate velocity:
\(\int_0^{t_j}\dot u_j(r)\,dr=0\). At a time \(s_j\) of maximum speed,
it follows that
\[
C_j
=\left|
\frac1{t_j}\int_0^{t_j}
\bigl(\dot u_j(s_j)-\dot u_j(r)\bigr)\,dr
\right|
\le LC_jt_j.
\]
Dividing by \(C_j\) contradicts \(t_j\to0\). Thus all fixed points of a
sufficiently short positive-time map are zeros of \(X\). Conversely,
every zero of \(X\) is stationary and hence fixed by every \(\Psi_t\).
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:reduction}]
The first step is to extend $f$ so that criticality along $F$ is equivalent
to ambient criticality at points of $F$. This is the finite-group case
of symmetric criticality~\cite{Palais1979}, which we verify directly.
Extend \(f\) smoothly to a function \(\widetilde f\) on \(M\): in
submanifold charts, extend independently of the normal coordinates,
and combine these local extensions using a partition of unity. Define
\[
K=\frac1m\sum_{j=0}^{m-1}\widetilde f\circ A^j.
\]
Then \(K\circ A=K\) and \(K|_F=f\).
Fix \(q\in F\), put \(D=dA_q\), and consider
\[
P_q=\frac1m\sum_{j=0}^{m-1}D^j.
\]
Since \(D^m=\id\), we have \(DP_q=P_q\). Moreover, \(P_qv=v\) whenever
\(Dv=v\). Thus \(P_q\) is a projection onto
\(\ker(D-\id)=T_qF\). Differentiating the averaged extension gives,
for every \(v\in T_qM\),
\[
dK_q(v)=d\widetilde f_q(P_qv)=df_q(P_qv).
\]
Consequently,
\begin{equation}
\label{eq:critical-restriction}
\CritSet(K)\cap F=\CritSet(f).
\end{equation}
Because \(A\) is symplectic and \(K\circ A=K\), its differential carries
the Hamiltonian vector field \(X_K\) to itself. Uniqueness of solutions
therefore gives \(A\circ B_s=B_s\circ A\) for every \(s\). Apply
Lemma~\ref{lem:short-periods} to \(X_K\), and choose
\(0<\varepsilon<\delta/m\). We then have
\[
\phi_\varepsilon^m=B_{m\varepsilon},
\qquad
\Fix(B_{m\varepsilon})=\CritSet(K).
\]
If \(\phi_\varepsilon(q)=q\), then \(q\) is fixed by
\(B_{m\varepsilon}\), so \(dK_q=0\). It follows that
\(B_\varepsilon(q)=q\), and hence \(A(q)=q\). Conversely, every point
of \(\CritSet(K)\cap F\) is fixed by \(\phi_\varepsilon\).
Equation~\ref{eq:critical-restriction} proves the asserted equality
of fixed and critical sets.
For completeness, the composition is generated by a smooth
Hamiltonian on the unit time interval. Choose a Hamiltonian path
\(A_t\) from \(\id\) to \(A\), generated by \(H_t\), and a smooth
nondecreasing function \(a:[0,1]\to[0,1]\) equal to \(0\) near \(0\)
and to \(1\) near \(1\). The time-dependent Hamiltonian
\[
L_t(x)=
\begin{cases}
2\varepsilon a'(2t)K(x),&0\le t\le\tfrac12,\\[2pt]
2a'(2t-1)H_{a(2t-1)}(x),&\tfrac12\le t\le1
\end{cases}
\]
is smooth, since both pieces vanish near the joining time. Its flow
first follows \(B_{\varepsilon a(2t)}\), and then
\(A_{a(2t-1)}\circ B_\varepsilon\), so its time-one map is
\(\phi_\varepsilon\).
\end{proof}
The proposition imposes no bound on the total number of critical
points of \(K\), and requires invariance under $A$, not under an isotopy
from $\id$ to $A$. A critical point \(q\notin F\) satisfies
\(\phi_\varepsilon(q)=A(q)\ne q\), so it contributes no fixed point.
==== END A-degenerate-counterexample-to-the-critical-number-Arnold-bound//01-reduction.tex ====
==== BEGIN A-degenerate-counterexample-to-the-critical-number-Arnold-bound//02-quadric.tex ====
\section{A Hamiltonian involution of a quadric}
\label{sec:quadric}
We now construct the manifold and involution to which
Proposition~\ref{prop:reduction} will apply. Let $M\subset\CP^4$ be the
quadric defined in~\eqref{eq:quadric}.
\begin{proposition}\label{prop:quadric}
The manifold $M$ is closed, connected, and of real dimension six. For
the restricted Fubini--Study form $\omega$, normalized below, the map
\[
A[z_0:z_1:z_2:z_3:z_4]
=[z_0:-z_1:-z_2:-z_3:-z_4]
\]
is a Hamiltonian involution. Its fixed set is the smooth submanifold
\[
F=M\cap\{z_0=0\}\cong S^2\times S^2.
\]
For every $q\in F$, the differential $dA_q$ acts as $+\id$ on $T_qF$
and as $-\id$ on a complementary real two-dimensional subspace of
$T_qM$.
\end{proposition}
\begin{proof}
In any standard affine chart of $\CP^4$, the equation defining $M$
takes the form $1+\sum_j w_j^2=0$. Its differential cannot vanish on
its zero set: that would force all the $w_j$ to be zero. Thus $M$ is
a smooth complex hypersurface, of complex dimension three. As a
closed subset of projective space it is compact, and it has no
boundary.
To see connectedness, write a nonzero isotropic vector as $z=x+iy$
with $x,y\in\R^5$. Its real and imaginary parts satisfy
\[
|x|=|y|>0,\qquad x\cdot y=0.
\]
After rescaling, they form an ordered orthonormal pair. Every such
pair extends to a positively oriented orthonormal basis, so these
pairs form one orbit of the connected group $\mathrm{SO}(5)$. Their
map $(x,y)\mapsto[x+iy]$ onto $M$ proves that $M$ is connected.
We construct the symplectic form with its normalization explicit.
On the unit sphere $S^9\subset\C^5$, put
\[
\alpha=\sum_{j=0}^4(x_j\,dy_j-y_j\,dx_j).
\]
The form $d\alpha$ is invariant under scalar phase rotations. If
$v\in T_zS^9$, then
\[
d\alpha(iz,v)
=-2\operatorname{Re}\sum_{j=0}^4\overline{z_j}v_j=0.
\]
Consequently it descends under the quotient map
$p:S^9\to\CP^4$ to a two-form $\sigma$, characterized by
$p^*\sigma=d\alpha$. Local sections show that $\sigma$ is smooth
and closed. A projective tangent vector has a horizontal lift
$v\in\C^5$ satisfying $\sum_j\overline{z_j}v_j=0$; multiplication by
$i$ on these lifts induces the complex structure. Since
\[
d\alpha(v,iv)=2|v|^2,
\]
$\sigma$ is positive on every complex tangent line. Its restriction
$\omega=\sigma|_M$ is therefore closed and nondegenerate.
We use a family of Hamiltonian rotations: equal speeds will give the
involution, and unequal speeds will later give a function with four
critical points on $M$. For $a,b,t\in\R$, let $\Phi_t^{a,b}$ keep $z_0$
fixed and act on $(z_1,z_2)^{\mathsf T}$ and $(z_3,z_4)^{\mathsf T}$ by
$R_{at}$ and $R_{bt}$, respectively, where
\[
R_\theta=
\begin{pmatrix}
\cos\theta&-\sin\theta\\
\sin\theta&\cos\theta
\end{pmatrix}.
\]
These real orthogonal, complex unitary transformations preserve the
quadric and $\alpha$. If $Z_{a,b}$ is their generating vector field
on $S^9$, Cartan's formula
gives
\[
\iota_{Z_{a,b}}d\alpha=-d\bigl(\alpha(Z_{a,b})\bigr).
\]
The function $\alpha(Z_{a,b})$ is phase invariant, because the rotations
commute with scalar multiplication. It thus descends to a smooth
function $g_{a,b}$ on projective space. The displayed identity descends
as well, so $G_{a,b}=-g_{a,b}|_M$ generates the projected rotations on
$M$ with our convention $\iota_{X_{G_{a,b}}}\omega=dG_{a,b}$.
Explicitly,
\begin{equation}\label{eq:rotation-hamiltonian}
G_{a,b}([z])=
\frac{2a\,\operatorname{Im}(\overline{z_1}z_2)
+2b\,\operatorname{Im}(\overline{z_3}z_4)}
{\sum_{j=0}^4|z_j|^2}.
\end{equation}
In particular, $A=\Phi_1^{\pi,\pi}$ is Hamiltonian and preserves $\omega$.
A projective line fixed by $A$ lies in one of the two eigenspaces of
$\operatorname{diag}(1,-1,-1,-1,-1)$. The positive eigenspace gives
the single point $[1:0:0:0:0]$, which does not lie in $M$. The
negative eigenspace gives precisely $F=M\cap\{z_0=0\}$. To identify
$F$, use the invertible complex-linear map
\[
(z_1,z_2,z_3,z_4)\longmapsto
\begin{pmatrix}
z_1+iz_2&z_3+iz_4\\
-z_3+iz_4&z_1-iz_2
\end{pmatrix}.
\]
Its determinant is $z_1^2+z_2^2+z_3^2+z_4^2$. It therefore
identifies $F$ with the projectivized nonzero rank-one matrices.
The map
\[
\CP^1\times\CP^1\longrightarrow F,
\qquad ([u],[v])\longmapsto[uv^{\mathsf T}]
\]
is a diffeomorphism: a nonzero column and row recover the two
projective factors, smoothly on the corresponding matrix charts.
Finally, stereographic coordinates identify each $\CP^1$ with
$S^2$. Only this smooth identification is needed below.
For the tangent-space assertion, fix $q\in F$ and choose an affine
chart $z_j=1$ with $j>0$. In its normalized coordinates, $A$ sends
$w_0$ to $-w_0$ and leaves all other coordinates fixed. The equation
of $M$ is
\[
w_0^2+1+\sum_{\substack{k>0\\k\ne j}}w_k^2=0.
\]
At $q$, where $w_0=0$, its linearization places no constraint on
the $w_0$ direction. The remaining differential is nonzero, so its
kernel in the other coordinates is exactly $T_qF$. Thus
\[
T_qM=T_qF\oplus\C\frac{\partial}{\partial w_0},
\]
with the asserted $+1$ and $-1$ actions. This also verifies directly
that $F$ is a smooth complex surface. The positivity established above
therefore makes $\omega|_F$ symplectic.
\end{proof}
==== END A-degenerate-counterexample-to-the-critical-number-Arnold-bound//02-quadric.tex ====
==== BEGIN A-degenerate-counterexample-to-the-critical-number-Arnold-bound//03-critical-bound.tex ====
\section{The critical number and cup length of the quadric}
\label{sec:critical-bound}
We now show that four is both the critical number and the rational cup
length of the ambient quadric. The lower bound is the classical
cup-product obstruction from Lusternik--Schnirelmann
theory~\cite{LusternikSchnirelmann1934}; see
Weber~\cite[Theorems~3.5 and 3.6]{Weber2019} for a modern treatment
valid without a Morse assumption. We give the differential-form proof,
then obtain the matching upper bound from the unequal-speed rotations
of Section~\ref{sec:quadric}.
\begin{lemma}\label{lem:critical-bound}
Let $M$ be a closed smooth manifold and $r\ge1$ an integer. Suppose there are classes
$a_j\in H^{d_j}(M;\R)$, with $d_j>0$ for $1\le j\le r$, such that
$a_1\smile\cdots\smile a_r\ne0$. Then every smooth function on $M$
has at least $r+1$ critical points.
\end{lemma}
\begin{proof}
Suppose that $h:M\to\R$ has at most $r$ critical points.
List its distinct critical values as $c_1<\cdots<c_k$, where $1\le k\le r$;
the absolute minimum and maximum are among them.
For a fixed Riemannian metric let $g_t$ be the negative-gradient flow
of $h$, which is complete by compactness, and put $P_b=\{h\le b\}$.
We first cover $M$ by $k$ open sets, each diffeomorphic to a finite
disjoint union of coordinate balls. Every closed positive-degree form
will be exact on each of these sets; the cover will therefore force
every product of $k$ positive-degree cohomology classes to vanish.
Around the critical points at $c_i$, choose an open set $U_i$ that is a
union of disjoint coordinate balls, and a smaller neighborhood $V_i$
with $\overline V_i\subset U_i$. A bound on the speed of the flow and
the positive distance between $M\setminus U_i$ and $\overline V_i$
give $T_i>0$ such that trajectories starting outside $U_i$ stay outside
$V_i$ for $0\le t\le T_i$.
Choose a closed level band $\{|h-c_i|\le\delta_i\}$ narrow enough that
it contains no critical points outside $V_i$. On the part outside
$V_i$, compactness gives a positive lower bound $b_i$ for
$|\nabla h|^2$; if that set is empty, any positive $b_i$ will do.
Choose $0<\epsilon_i<\delta_i$ so that the intervals
$[c_i-\epsilon_i,c_i+\epsilon_i]$ are separated and
$2\epsilon_i<T_i b_i$. Along the negative-gradient flow,
\[
\frac{d}{dt}h(g_t(q))=-|\nabla h(g_t(q))|^2.
\]
A point of $P_{c_i+\epsilon_i}\setminus U_i$ either already lies below
$c_i-\epsilon_i$ or decreases at rate at least $b_i$ until it reaches
that level. It follows that
\begin{equation}\label{eq:sublevel-drop}
g_{T_i}(P_{c_i+\epsilon_i}\setminus U_i)
\subset P_{c_i-\epsilon_i}.
\end{equation}
For $i>1$, a further fixed flow time $S_i>0$ carries $P_{c_i-\epsilon_i}$
into $P_{c_{i-1}+\epsilon_{i-1}}$: the compact band between these levels
contains no critical points and has a positive gradient-norm lower bound.
Since $c_1$ is the absolute minimum, $P_{c_1-\epsilon_1}$ is empty,
and \eqref{eq:sublevel-drop} shows that $P_{c_1+\epsilon_1}\subset U_1$.
Inductively, cover $P_{c_i+\epsilon_i}$ by $U_i$ and the preimages under
$g_{T_i+S_i}$ of the $i-1$ sets covering the preceding upper sublevel.
Equation~\eqref{eq:sublevel-drop} proves that these sets cover.
Each $g_t$ is a diffeomorphism of $M$, so every member remains
diffeomorphic to a finite disjoint union of coordinate balls. The last
upper sublevel is all of $M$, since $c_k$ is the absolute maximum.
Write the resulting cover as $O_1,\ldots,O_k$. Choose closed forms
$\alpha_j$ representing $a_j$ for $1\le j\le k$. Since these forms
have positive degree, the Poincar\'e lemma on each ball gives
$\alpha_j=d\lambda_j$ on $O_j$. Choose a smooth partition of unity
$\psi_1,\ldots,\psi_k$ subordinate to the cover. Let
$\chi:[0,1]\to[0,1]$ be smooth, zero on $[0,1/(4k)]$ and one on
$[1/(2k),1]$, and put $\rho_j=\chi(\psi_j)$. At every point some
$\psi_j\ge1/k$, so some $\rho_j$ is identically one nearby.
Each $\rho_j\lambda_j$ extends smoothly by zero outside $O_j$.
The closed forms
\[
\alpha'_j=\alpha_j-d(\rho_j\lambda_j)
\]
represent the same classes as $\alpha_j$. Near every point, at least one
of these forms vanishes, and hence $\alpha'_1\wedge\cdots\wedge\alpha'_k=0$.
Thus $a_1\smile\cdots\smile a_k=0$, contradicting the nonvanishing
of $a_1\smile\cdots\smile a_r$ because $k\le r$.
\end{proof}
Extension of coefficients $H^*(M;\mathbb Q)\to H^*(M;\R)$ is injective
for a closed manifold and respects cup products. The lemma therefore
also gives
\begin{equation}\label{eq:cup-critical}
\operatorname{cuplength}(M;\mathbb Q)\le\Crit(M),
\end{equation}
with the unit-inclusive convention of Section~\ref{sec:introduction}.
It also recovers the symplectic-volume obstruction used in our example.
\begin{corollary}\label{cor:symplectic-critical}
Every smooth function on a closed connected symplectic manifold of real
dimension $2n$, with $n\ge1$, has at least $n+1$ critical points.
\end{corollary}
\begin{proof}
The class $[\omega]^n$ is nonzero, since
$\int_M\omega^n>0$ in the symplectic orientation. Apply
Lemma~\ref{lem:critical-bound} with each $a_j=[\omega]$.
\end{proof}
\begin{corollary}\label{cor:quadric-critical}
For the quadric threefold in \eqref{eq:quadric},
\[
\Crit(Q^3)=\operatorname{cuplength}(Q^3;\mathbb Q)=4.
\]
\end{corollary}
\begin{proof}
Let $h=c_1(\mathcal O_{Q^3}(1))\in H^2(Q^3;\mathbb Z)$ be the
hyperplane class. Since $Q^3$ has degree two in $\CP^4$,
\[
h^3=2[\mathrm{pt}],
\]
where $[\mathrm{pt}]$ evaluates to one on the fundamental class.
This product remains nonzero over $\mathbb Q$, giving
$\operatorname{cuplength}(Q^3;\mathbb Q)\ge4$.
For the matching upper bound, take the Hamiltonian $G_{1,2}$ of
\eqref{eq:rotation-hamiltonian}. Its critical points are precisely the
zeros of its Hamiltonian vector field. That field is induced on $Q^3$
by the complex-linear generator
\[
T=\operatorname{diag}(0,J,2J),
\qquad J=\begin{pmatrix}0&-1\\1&0\end{pmatrix}.
\]
The induced projective vector field vanishes at $[z]$ exactly when
$Tz\in\C z$. The eigenvalues $0,\pm i,\pm2i$ of $T$ are distinct,
so its projective zeros are its five eigenlines. The zero eigenline
$[1:0:0:0:0]$ misses $Q^3$, whereas the other four lines
\[
[0:1:\pm i:0:0],\qquad [0:0:0:1:\pm i]
\]
all satisfy the quadric equation. Thus $G_{1,2}$ has exactly four
critical points on $Q^3$. Together with \eqref{eq:cup-critical}, this gives
\[
4\le\operatorname{cuplength}(Q^3;\mathbb Q)
\le\Crit(Q^3)\le4.
\]
\end{proof}
==== END A-degenerate-counterexample-to-the-critical-number-Arnold-bound//03-critical-bound.tex ====
==== BEGIN A-degenerate-counterexample-to-the-critical-number-Arnold-bound//04-three-points.tex ====
\section{Three critical points on the fixed submanifold}
\label{sec:three-points}
We next construct a function on the fixed submanifold with three
critical points. The comparison with functions on the ambient quadric
will follow in Section~\ref{sec:critical-bound}.
Three-critical-point functions on products of spheres belong to the
classical study of critical-point minima and category by
Takens~\cite{Takens1968}; the $S^2\times S^2$ case is recorded by
Mare~\cite[Proposition~3.2.5]{Mare1997}. We use the explicit formula below
and determine its entire critical set directly, without importing an
existence theorem or assuming that its critical points are Morse.
\begin{proposition}
\label{prop:three-points}
Let $p=(0,0,1)$ and $e=(1,0,0)$ in $\R^3$.
The smooth function
\begin{equation}
\label{eq:three-point-function}
f\colon S^2\times S^2\longrightarrow\R,
\qquad f(x,y)=e\cdot x+(x-p)\cdot y,
\end{equation}
has exactly three critical points.
\end{proposition}
\begin{proof}
For a fixed $x\ne p$, the $y$-dependence is a height function on the
second sphere, with critical points $y=\pm(x-p)/|x-p|$.
At $x=p$ that height function is constant, but criticality of the full
function still requires its derivative in the $x$ directions to vanish.
We therefore separate this exceptional fiber from the two height branches.
The constrained critical-point equations are
\begin{equation}
\label{eq:critical-multipliers}
e+y=\lambda x,\qquad x-p=\mu y,
\qquad |x|=|y|=1,
\end{equation}
for real multipliers $\lambda,\mu$.
If $x=p$, the first equation gives $y=-e+\lambda p$.
Its unit norm forces $\lambda=0$, yielding the unique point
\[
q_0=(p,-e),\qquad f(q_0)=0.
\]
Suppose now that $x\ne p$ and set $t=|x-p|>0$.
The second equation in \eqref{eq:critical-multipliers} gives
$y=s(x-p)/t$ for a sign $s\in\{1,-1\}$.
The first then becomes
\[
t e-s p=(\lambda t-s)x.
\]
Since the left side is nonzero, $\lambda t-s\ne0$ and $x$ lies in
the plane spanned by $e$ and $p$. Write $x=(u,0,w)$.
The definition of $t$ gives $w=1-t^2/2$.
Taking the first equation in \eqref{eq:critical-multipliers}
against the tangent vector $(w,0,-u)$ gives
\[
w+\frac{s u}{t}=0,
\qquad\text{so}\qquad u=-s t w.
\]
Consequently the unit-norm condition reduces to
\begin{equation}
\label{eq:critical-factorization}
0=(1-t^2/2)^2(1+t^2)-1
=\frac{t^4(t^2-3)}4.
\end{equation}
Because $t>0$, this forces $t=\sqrt3$, $w=-1/2$, and
$u=s\sqrt3/2$. In particular, the other pole $x=-p$, for which $t=2$,
is excluded. The two resulting candidates are
\begin{equation}
\label{eq:other-critical-points}
q_s=
\left(
\left(\frac{s\sqrt3}{2},0,-\frac12\right),
\left(\frac12,0,-\frac{s\sqrt3}{2}\right)
\right),
\qquad s\in\{1,-1\}.
\end{equation}
Both satisfy \eqref{eq:critical-multipliers} with
$\lambda=\mu=s\sqrt3$, and
\[
f(q_s)=s\,\frac{3\sqrt3}{2}.
\]
The cases $x=p$ and $x\ne p$ exhaust the domain, proving the count.
\end{proof}
\begin{remark}
\label{rem:degenerate-critical-point}
The critical point $q_0$ is degenerate. Indeed, in local coordinates
\[
x=(u,v,\sqrt{1-u^2-v^2}),\qquad
y=(-\sqrt{1-a^2-b^2},a,b),
\]
the function has expansion
\[
f=va+O\bigl(\lVert(u,v,a,b)\rVert^3\bigr).
\]
Its Hessian therefore has eigenvalues $1,-1,0,0$, so its rank is two.
The complete critical-point calculation proves that this degenerate point
is isolated. This is why a count restricted to Morse functions would not
capture the example.
\end{remark}
==== END A-degenerate-counterexample-to-the-critical-number-Arnold-bound//04-three-points.tex ====
==== BEGIN A-degenerate-counterexample-to-the-critical-number-Arnold-bound//05-conclusion.tex ====
\section{The Hamiltonian counterexample}
\label{sec:conclusion}
We now apply the finite-order reduction to the function on the fixed
surface and compare the resulting fixed-point count with the ambient
critical number and cup length.
\begin{proof}[Proof of Theorem~\ref{thm:counterexample}]
Take the symplectic quadric $(M,\omega)$ and Hamiltonian involution $A$
from Proposition~\ref{prop:quadric}. Its fixed submanifold $F$ is
diffeomorphic to $S^2\times S^2$.
Transport the function of Proposition~\ref{prop:three-points} to $F$;
it has exactly three critical points.
Proposition~\ref{prop:reduction} provides a smooth $A$-invariant extension
$K\colon M\to\R$ with $K|_F=f$.
Let $B_s$ denote its Hamiltonian flow. For all sufficiently small
$\varepsilon>0$, the map
\[
\phi=A\circ B_\varepsilon
\]
is a Hamiltonian diffeomorphism and satisfies
$\Fix(\phi)=\CritSet(f)$, with the critical points viewed as points of $F$.
The exact comparison in Corollary~\ref{cor:quadric-critical} now gives
\[
\#\Fix(\phi)=3<4=\Crit(M)
=\operatorname{cuplength}(M;\mathbb Q).
\]
It remains to verify the asserted degeneracy. Write
$\omega_F=\omega|_F$, which is symplectic,
and let $X_f$ be the Hamiltonian vector field of $f$ on $(F,\omega_F)$.
Invariance of $K$ makes $X_K$ tangent to $F$, and restricting its defining
equation to $TF$ gives $X_K|_F=X_f$. Indeed, at $q\in F$ invariance gives
$dA_qX_K(q)=X_K(q)$, and the fixed tangent space is $T_qF$.
Thus $B_s$ preserves $F$, and its restriction is the Hamiltonian flow
of $f$ on $F$. At the critical point $q_0$ of
Remark~\ref{rem:degenerate-critical-point}, differentiating
$\iota_{X_f}\omega_F=df$ gives
\[
\omega_F\bigl(DX_f(q_0)v,w\bigr)
=\operatorname{Hess}_{q_0}f(v,w).
\]
Thus any nonzero vector $v$ in the Hessian kernel satisfies
$DX_f(q_0)v=0$. Since $A|_F=\id$ and $q_0$ is an equilibrium,
\[
d\phi_{q_0}v
=\exp\bigl(\varepsilon DX_f(q_0)\bigr)v
=v.
\]
The fixed point $q_0$ therefore has eigenvalue $1$ in its linearization.
This argument takes place entirely in the invariant subspace $T_{q_0}F$;
no assumption on the normal Hessian of the extension $K$ is needed.
\end{proof}
==== END A-degenerate-counterexample-to-the-critical-number-Arnold-bound//05-conclusion.tex ====
==== BEGIN A-degenerate-counterexample-to-the-critical-number-Arnold-bound//main.tex ====
\documentclass[11pt]{article}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amssymb,amsthm,mathtools}
\usepackage{microtype}
\usepackage{xcolor}
\usepackage{hyperref}
\hypersetup{colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black,pdftitle={Three fixed points on the symplectic quadric threefold},pdfauthor={OpenAI}}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\numberwithin{equation}{section}
\DeclareMathOperator{\Crit}{Crit}
\DeclareMathOperator{\CritSet}{crit}
\DeclareMathOperator{\Fix}{Fix}
\DeclareMathOperator{\Ham}{Ham}
\newcommand{\id}{\mathrm{id}}
\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
\newcommand{\CP}{\mathbb C\mathrm P}
\allowdisplaybreaks[1]
\setlength{\emergencystretch}{1.5em}
\title{Three fixed points on the symplectic quadric threefold}
\author{OpenAI}
\date{September 23, 2026}
\begin{document}
\maketitle
\begin{abstract}
We construct a smooth Hamiltonian diffeomorphism of the complex quadric
threefold with exactly three fixed points, at least one of which is
degenerate. The critical number and the unit-inclusive rational cup length
of this manifold are both four. Thus the example disproves the unrestricted
critical-number and rational cup-length forms of the Arnold conjecture.
\end{abstract}
\input{sections/00-introduction}
\input{sections/01-reduction}
\input{sections/02-quadric}
\input{sections/04-three-points}
\input{sections/03-critical-bound}
\input{sections/05-conclusion}
\input{references}
\end{document}
==== END A-degenerate-counterexample-to-the-critical-number-Arnold-bound//main.tex ====
==== BEGIN A-degenerate-counterexample-to-the-critical-number-Arnold-bound//references.tex ====
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==== END A-degenerate-counterexample-to-the-critical-number-Arnold-bound//references.tex ====