Arnold fixed-point bounds: critical numbers, Morse theory, and stabilization

Source and verification

This article explains family 347 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

🏷️ Hamiltonian dynamics and critical-point minima

A closed symplectic manifold $(M^{2n},\omega )$ is a compact manifold without boundary equipped with a closed, nondegenerate two-form. For a smooth Hamiltonian $H_t$, use the manuscripts’ convention

$$\omega (X_{H_t},\cdot )=d{H}_t.$$

Its flow is ${\varphi }_H^t$, and a fixed point of ${\varphi }_H^1$ gives a one-periodic trajectory. Write ${\mathrm{Fix}}_0({\varphi }_H^1;H)$ for the initial points whose trajectories are contractible. This counts points, not choices of disks spanning the loops. A fixed point is nondegenerate when $d{\varphi }_H^1-I$ is invertible there.

For a sufficiently small autonomous Hamiltonian, fixed points are exactly critical points. Arnold-type questions ask which lower bounds from critical-point theory survive for general Hamiltonian paths. Three different minima occur in this family.

Minimum on a closed manifold $W$	Admissible functions	Notation
Critical number	All smooth real-valued functions	$\mathrm{Crit}(W)$
Ordinary Morse number	Functions with nonsingular Hessian at every critical point	$\mathrm{Morse}(W)$
Stable Morse number	Morse functions on $W\times {\mathbb{R}}^k$, quadratic outside a compact set, allowing every $k\ge 0$	$\mathrm{SM}(W)$

For the last row, the precise exterior condition is $F(x,z)=Q(z)$ for $|z|\ge R$, where $Q$ is a nondegenerate quadratic form of arbitrary signature. For $k=0$ there is no exterior condition. Thus

$$\begin{array}{rl}\mathrm{Crit}(W)& \le \mathrm{Morse}(W),\\ \mathrm{SM}(W)& \le \mathrm{Morse}(W).\end{array}$$

These two inequalities do not compare the critical number with the stable Morse number. Allowing stabilization changes both the domain and the condition at infinity.

Prerequisites. Smooth manifolds, differential forms, ordinary differential equations, and the basic interpretation of a Morse critical point as a handle attachment suffice for the dynamical arguments. Later sections introduce the homological quantities they use. The projective constructions and the group-theoretic input of one companion are recorded as inputs, without developing algebra or algebraic geometry.

The family in one view

The five manuscripts report counterexamples to unrestricted critical-number, ordinary Morse-number, and stable Morse-number bounds. Their common theme is that a Hamiltonian return map can inherit a critical-point count from a smaller fixed submanifold or from a stabilized local model. The strongest low-dimensional stable-Morse example has real dimension $22$. A different companion attains a cyclically graded integral Floer bound while remaining below the stable Morse number.

🏷️ Statements and the role of nondegeneracy

All the following are claims of the cited manuscripts. The dimensions in this table are real dimensions. Here $\chi (M)={\sum }_i(-1{)}^i\dim H_i(M;\mathbb{Q})$ is the Euler characteristic. A Betti number over a field is the dimension of the corresponding homology group over that field.

Manuscript	Dimension	Reported comparison
Three fixed points on a quadric (OpenAI, 2026)	$6$	Three fixed points, but critical number four; at least one fixed point is degenerate
Ordinary Morse-number construction (OpenAI, 2026a)	$12$	All fixed points nondegenerate; their number is $\chi (M)+128<\mathrm{Morse}(M)$
First stable-Morse construction (OpenAI, 2026b)	$1412$	$318952$ nondegenerate fixed points, versus stable Morse number $318968$
Dimension-twenty-two construction (OpenAI, 2026c)	$22$	For every integer $m\ge 1$, the stable Morse deficit is $16m$
Sharp cyclic integral construction (OpenAI, 2026d)	$3332$	$1872232$ nondegenerate fixed points attain the cyclic integral bound, versus stable Morse number $1872264$

The last four constructions count all fixed points and show that every corresponding trajectory is contractible. For the three stable-Morse constructions, simple connectivity gives contractibility. The twelve-dimensional example instead proves it from its explicit Hamiltonian path.

The ordinary Morse-number deficit can be arbitrarily large while the dimension remains twelve; the manifold varies.

Reported dimension-twenty-two construction

For every integer $m\ge 1$, there is a simply connected closed Kähler manifold $M_m$ of real dimension $22$ and a smooth one-periodic Hamiltonian whose time-one map ${\varphi }_m$ has only nondegenerate fixed points, with the two exact counts below. A Kähler manifold has a complex structure and a compatible Riemannian metric whose associated two-form is closed; in particular, it is symplectic. (OpenAI, 2026c)

$$\begin{array}{rl}\mathrm{SM}(M_m)& =80+1968m,\\ \#\mathrm{Fix}({\varphi }_m)& =80+1952m.\end{array}$$

Their relative difference is

$$\begin{array}{c}\frac{\mathrm{SM}(M_m)-\#\mathrm{Fix}({\varphi }_m)}{\mathrm{SM}(M_m)}\\ =\frac{m}{5+123m}\ge \frac{1}{128}.\end{array}$$

The family therefore gives an unbounded absolute deficit and a uniform positive relative deficit, with fixed dimension. (OpenAI, 2026c)

Which Arnold bound is being compared

The degenerate example concerns the minimum over arbitrary smooth functions. The other four examples require nondegenerate fixed points. None of these statements should be substituted for a claim that the established nondegenerate Betti-number bounds fail. The papers also distinguish their examples from the positive stable-Morse theorem under the hypotheses that both $[\omega ]$ and $c_1(TM)$ vanish on ${\pi }_2(M)$; their sphere constructions do not satisfy those hypotheses.

🏷️ Finite-order transfer of critical points

The most reusable mechanism starts with a Hamiltonian involution $A$, meaning $A^2=\mathrm{id}$, whose fixed set $F$ is a smooth submanifold. At a fixed point, the tangent space splits into the $+1$ and $-1$ eigenspaces of $dA$. These spaces are symplectically orthogonal: for vectors $u,v$ in the two different spaces,

$$\begin{array}{rl}\omega (u,v)& =\omega (dAu,dAv)\\ & =-\omega (u,v).\end{array}$$

Thus both summands are symplectic, and the positive summand is $TF$.

Given a smooth function $f$ on $F$, extend it using a tubular neighborhood and a cutoff, then average the extension with its composition with $A$. The resulting function $K$ is invariant under $A$ and restricts to $f$. At a point of $F$, its derivative in each negative eigendirection vanishes by invariance. Consequently

$$\mathrm{Crit}(K)\cap F=\mathrm{Crit}(f).$$

Let $B_s$ be the Hamiltonian flow of $K$. Invariance implies $A{B}_s=B_{s}A$, so for ${\varphi }_{\epsilon }=A{B}_{\epsilon }$,

$${\varphi }_{\epsilon }^2=B_{2\epsilon }.$$

This identity becomes an exact counting tool because a fixed smooth vector field on a compact manifold has a positive lower bound on the periods of its nonconstant periodic trajectories.

To see the period bound, choose finitely many nested coordinate balls. A common speed bound ensures that a sufficiently short trajectory remains in one larger ball, where the coordinate vector field $v$ has a common Lipschitz constant $L$. If $y(t)$ is a closed trajectory of period $T$, set $S={\max }_t|v(y(t))|$ and choose $t_0$ attaining it. The average velocity is zero, and $|y(t)-y(t_0)|\le TS$. Hence

$$\begin{array}{rl}S& =\left|\frac{1}{T}{\int }_0^T[v(y(t_0))-v(y(t))]dt\right|\\ & \le LTS.\end{array}$$

For $LT<1$, this forces $S=0$. The compactness and chart argument make the threshold uniform over all initial points.

Now take $2\epsilon$ below this threshold. A fixed point of ${\varphi }_{\epsilon }$ is fixed by $B_{2\epsilon }$, so it is an equilibrium of $K$. Since $B_{\epsilon }$ fixes that point, the equation $A{B}_{\epsilon }(x)=x$ also gives $A(x)=x$. Conversely, every critical point of $f$ is fixed by both maps. Therefore

$$\mathrm{Fix}({\varphi }_{\epsilon })=\mathrm{Crit}(f).$$

This counts the entire ambient fixed set. Critical points of the extension away from $F$ contribute nothing, because $A$ moves them. The same argument works for a finite-order map with $A^r=\mathrm{id}$, using ${\varphi }_{\epsilon }^r=B_{r\epsilon }$. (OpenAI, 2026e) (OpenAI, 2026c)

If $f$ is Morse, the resulting fixed points are nondegenerate for sufficiently small positive $\epsilon$. At an equilibrium let $L_{+}$ and $L_{-}$ be the tangent and normal blocks of $D{X}_K$. Differentiating the Hamiltonian equation gives

$$\omega (L_{+}u,v)=\mathrm{Hess}f(u,v).$$

Thus $L_{+}$ is invertible. The blocks of $d{\varphi }_{\epsilon }-I$ satisfy

$$\begin{array}{rl}{\epsilon }^{-1}(e^{\epsilon L_{+}}-I)& ⟶L_{+},\\ -e^{\epsilon L_{-}}-I& ⟶-2I.\end{array}$$

Both are invertible for sufficiently small time. There are finitely many critical points, so a single time works for all of them. Concatenating a Hamiltonian path to $A$ with the small flow, and making the time parametrization constant near its endpoints, supplies a smooth one-periodic generating Hamiltonian. (OpenAI, 2026c)

🏷️ The degenerate six-dimensional example

The ambient manifold is the smooth complex quadric

$$\begin{array}{c}{Q}^3=\{[z]\in {\mathbb{CP}}^4:q(z)=0\},\\ q(z)=\sum _{j=0}^4{z}_j^2,\end{array}$$

equipped with the restricted Fubini–Study symplectic form. Here only its smooth symplectic geometry is needed. The involution

$$\begin{array}{r}A[z_0:z_1:z_2:z_3:z_4]\\ =[z_0:-z_1:-z_2:-z_3:-z_4]\end{array}$$

is Hamiltonian: rotate the coordinate pairs $(z_1,z_2)$ and $(z_3,z_4)$ through angle $\pi$ along a Hamiltonian rotation path. Its fixed set is the surface $z_0=0$ inside $Q^3$, diffeomorphic to $S^2\times S^2$. The other possible projective eigenspace, the line represented by $(1,0,0,0,0)$, misses the quadric. On the fixed surface $dA$ is $+I$, and on its two-dimensional real normal space it is $-I$. (OpenAI, 2026e)

The crucial function is explicit. Put $p=(0,0,1)$, $e=(1,0,0)$, and define

$$\begin{array}{c}f(x,y)=e\cdot x+(x-p)\cdot y,\\ (x,y)\in S^2\times S^2.\end{array}$$

Constrained criticality means

$$e+y=\lambda x,x-p=\mu y.$$

If $x=p$, the first equation and $|y|=1$ force $y=-e$. Otherwise let $t=|x-p|>0$ and write $y=s(x-p)/t$, with $s=\pm 1$. The equations imply that $x=(u,0,w)$, where

$$w=1-t^2/2,u=-stw.$$

The unit-length condition becomes

$$\begin{array}{rl}0& =(1-t^2/2{)}^2(1+t^2)-1\\ & =\frac{t^4(t^2-3)}{4}.\end{array}$$

Thus $t=\sqrt{3}$, giving exactly one solution for each sign $s$. Together with $(p,-e)$, there are exactly three critical points. Near $(p,-e)$, coordinates can be chosen so that

$$f=va+O(\Vert (u,v,a,b){\Vert }^3).$$

Its Hessian has rank two, so this isolated critical point is degenerate. Finite-order transfer produces three ambient fixed points; a Hessian-kernel vector is fixed by the linearized small flow and by $A$, so at least one ambient fixed point is also degenerate. (OpenAI, 2026e)

Every smooth function on $Q^3$ has at least four critical points. Indeed, $[\omega {]}^3\ne 0$ because its integral is positive. The Lusternik–Schnirelmann critical-point inequality says that a nonzero product of three positive-degree cohomology classes forces at least four critical points, with no Morse assumption. One way to prove it is to use negative-gradient deformation to cover a manifold with finitely many critical points by sets on which positive-degree classes restrict to zero; relative cup products then force sufficiently long products to vanish. Unequal-speed rotations in the two coordinate pairs give a Hamiltonian function on $Q^3$ with exactly four critical points, so the lower bound is attained.

With unit-inclusive cup length, namely one plus the longest nonzero product length, the paper obtains

$$\begin{array}{rl}\#\mathrm{Fix}(\varphi )& =3,\\ \mathrm{Crit}(Q^3)& =\mathrm{cuplength}(Q^3;\mathbb{Q})=4.\end{array}$$

The rational product can be represented by the hyperplane class; its cube evaluates to two. The construction’s degeneracy is essential to its comparison with nondegenerate homological fixed-point inequalities. (OpenAI, 2026e)

🏷️ The stable Morse obstruction from integral homology

Write $b(W)={\sum }_i\mathrm{rank}{H}_i(W;\mathbb{Z})$ and let $t_i(W)$ be the minimum number of generators of the finite torsion subgroup of $H_i(W;\mathbb{Z})$. Define

$$\lambda (W)=b(W)+2\sum _i{t}_i(W).$$

The factor two has a Morse-theoretic explanation: torsion needs both a generator and a relation. More precisely, a finite free integral chain complex with homology free ranks $r_i$ and torsion generator counts $t_i$ satisfies

$$\mathrm{rank}{C}_i\ge r_i+t_i+t_{i-1}.$$

Its rank identity expresses $\mathrm{rank}{C}_i$ as $r_i$ plus the ranks of boundaries in degrees $i$ and $i-1$; each boundary rank is at least the number of torsion generators it produces. Summing counts each torsion requirement twice.

This obstruction survives geometric stabilization. For an admissible $F$ equal to $Q(u,v)=-|u{|}^2+|v{|}^2$ outside a core, choose levels $\pm A$ beyond all core values and truncate at a ball of radius $T$ with $T^2>A$. On the spherical side boundary, the only possible critical values of $Q$ are $\pm T^2$, outside the chosen band. Relative Morse theory therefore attaches one cell per critical point of $F$, with no extra side-boundary cells.

The endpoint sublevel pair is the same as for $Q$. Shrinking the positive variables $v$ to zero identifies its homology with that of $W$, shifted by the number of negative variables. The chain inequality consequently gives

$$\lambda (W)\le \mathrm{SM}(W).$$

This uses an actual relative Morse complex of the stabilized function. A short abstract complex alone would not supply a geometric upper bound. For the simply connected manifolds of dimension at least nine used below, the manuscripts invoke Smale’s handle theorem to realize the lower bound by an ordinary Morse function. Hence

$$\mathrm{SM}(W)=\mathrm{Morse}(W)=\lambda (W).$$

The dimension stated here is a sufficient common range for the constructions, not an optimal threshold for Smale’s theorem. (OpenAI, 2026c) (OpenAI, 2026b)

The alternative convention $\Vert dF-dQ\Vert \le C$ gives the same minimum: cut off $F-Q$ far from the zero section. Its derivative stays bounded on the transition annulus, whereas $\Vert dQ\Vert$ grows linearly with the radius, so the cutoff introduces no new critical points. This reconciles the stable-Morse conventions used by the companions. (OpenAI, 2026c) (OpenAI, 2026a)

🏷️ Four fixed surfaces and a twenty-two-dimensional deficit

The geometric input is a simply connected symplectic four-manifold $Y$ with a Hamiltonian involution whose fixed set consists of four copies of ${\mathbb{CP}}^1$. Its integral homology is free, with ranks $1,6,1$ in degrees $0,2,4$. The manuscript constructs $Y$ by blowing up the four coordinate corners of ${\mathbb{CP}}^1\times {\mathbb{CP}}^1$. A half-turn acts trivially on each new exceptional sphere and by $-I$ on its normal space. (OpenAI, 2026c)

For a simply connected manifold $X_m$ of real dimension eighteen, set

$$\begin{array}{rl}{C}_m& =X_m\times {\mathbb{CP}}^1,\\ M_m& =X_m\times Y.\end{array}$$

The product involution has four fixed components, each a copy of $C_m$. A Morse function attaining $\lambda (C_m)$ on each component, followed by finite-order transfer, gives precisely $4\lambda (C_m)$ nondegenerate fixed points on $M_m$. The desired inequality is therefore a comparison of two ordinary integral homology calculations.

The manuscript supplies $X_m$ with $b(X_m)=10+176m$. Its torsion occurs in two lists of degrees, $1+2j$ and $2+2j$, with the same profiles in both lists. For $j=1,\dots ,6$, those profiles are $m$ times

$$\begin{array}{rl}a& =(2,2,2,2,2,2),\\ q& =(1,3,4,4,3,1),\end{array}$$

for the $2$-primary and $3$-primary parts, respectively, and vanish outside this range. The geometric realization uses disjoint centers of complex dimensions two and four in ${\mathbb{CP}}^9$. We take that construction as input and examine its numerical consequence. (OpenAI, 2026c)

The relevant elementary torsion fact is

$$d((\mathbb{Z}/2{)}^a\oplus (\mathbb{Z}/3{)}^q)=\max (a,q).$$

Pairing a factor of order two with a factor of order three gives a single cyclic factor of order six. Thus one must take a maximum within each degree, rather than adding the two prime multiplicities.

Since the other product factors have free homology, their effect on each torsion profile is convolution. The two convolution kernels are

$$e_C=(1,1),e_M=(1,6,1),$$

where $(e\ast a{)}_j={\sum }_k{e}_k{a}_{j-k}$. Taking the entrywise maximum of the two convolved profiles gives the generator counts. In either parity list, after dividing by $m$, the results are

$$\begin{array}{rl}{C}_m:& (2,4,7,8,7,4,2),\\ M_m:& (2,14,23,31,31,23,14,2).\end{array}$$

The sums are $34$ and $140$. There are two parity lists, so the final comparison is

Quantity	$C_m$	$M_m$
Total free rank	$20+352m$	$80+1408m$
Total torsion generator count	$68m$	$280m$
$\lambda$	$20+488m$	$80+1968m$

Consequently

$$\begin{array}{c}\lambda (M_m)-4\lambda (C_m)\\ =2(280-4\cdot 68)m=16m.\end{array}$$

The free ranks cancel. The gap arises because convolution and taking degreewise maxima do not commute. Different primes dominate different degrees, and the two product factors mix adjacent degrees with different weights. This explains both the exact deficit and why a single coefficient field does not see it. (OpenAI, 2026c)

The earlier high-dimensional stable-Morse paper uses the same surface $Y$ and a longer pair of torsion profiles. With a parameter $N\ge 704$, it obtains

$$\begin{array}{rl}{\dim }_{\mathbb{R}}M& =2N+4,\\ \lambda (C)& =114N-518,\\ \lambda (M)& =456N-2056.\end{array}$$

The difference $\lambda (M)-4\lambda (C)=16$ comes from the two ends of those profiles. Setting $N=704$ gives the dimension and counts in the statement table. The twenty-two-dimensional companion replaces this long realization by many disjoint low-dimensional centers, yielding fixed dimension and a gap proportional to $m$. (OpenAI, 2026b) (OpenAI, 2026c)

🏷️ Cyclic grading and the sharp integral example

Floer theory associates a chain complex to Hamiltonian periodic orbits, so homological requirements can give lower bounds on their number. The grading records an index, but changing a disk spanning an orbit can change that index. The integer degree used by Morse theory therefore differs from the cyclic grading of the integral Floer bound discussed in these manuscripts. The minimal Chern number $N$ is the positive generator of $c_1(TM)({\pi }_2(M))$ when that subgroup is nonzero; $c_1$ is the first Chern class of the symplectic tangent bundle. The cyclic grading is modulo $2N$. For the products with $Y$, an exceptional sphere has tangent Chern number two and normal Chern number minus one, giving total one. Thus $N=1$, and the grading in question is modulo two.

Define the merged homology in each parity by

$${\overline{H}}_{\bar{i}}(W;\mathbb{Z})=\underset{k\equiv i(\mathrm{mod}2)}{⨁}{H}_k(W;\mathbb{Z}),$$

and the cyclic integral quantity by

$$\begin{array}{c}{\beta }_{\mathbb{Z}}^{\mathrm{cyc}}(W)=b(W)+2({\bar{t}}_0+{\bar{t}}_1),\\ {\bar{t}}_i=d(\mathrm{Tor}{\overline{H}}_{\bar{i}}(W;\mathbb{Z})).\end{array}$$

Merging degrees can combine torsion of different primes into fewer generators. For instance, a factor of order two and a factor of order three in distinct degrees of the same parity require one generator in each integer degree, but one generator after those degrees are merged. Accordingly ${\beta }_{\mathbb{Z}}^{\mathrm{cyc}}\le \lambda$, possibly strictly. This is the numerical distinction behind compatibility with the cyclic integral bound attributed to Bai–Xu in the papers. (OpenAI, 2026d)

For the twenty-two-dimensional examples,

$$\begin{array}{rl}{\beta }_{\mathbb{Z}}^{\mathrm{cyc}}(M_m)& =80+1920m,\\ \#\mathrm{Fix}({\varphi }_m)& =80+1952m,\\ \mathrm{SM}(M_m)& =80+1968m.\end{array}$$

The fixed-point count is therefore above the cyclic bound, by $32m$, while below the stable Morse number. (OpenAI, 2026c)

The sharpness companion changes the profiles so that the first inequality becomes equality. Put $L=1661$. In each parity list, the difference $\delta =a-q$ equals $-1$ throughout indices $1,\dots ,L$, except that it is $+1$ at $2$ and $L-1$; outside the interval it is zero. Then

$$(e_C\ast \delta {)}_j={\delta }_j+{\delta }_{j-1}\le 0.$$

Thus the $3$-primary profile dominates in every degree of $C$. But convolution with $e_M$ has positive values precisely at $j=3,L$, each equal to four, because

$${\delta }_3+6{\delta }_2+{\delta }_1=-1+6-1=4.$$

The extra degreewise generator cost is eight in each parity. Two parities and the factor two in $\lambda$ turn it into a stable-Morse excess of $32$. The construction realizes profiles with

$$\begin{array}{c}b(X)=174281,\\ \sum _j{q}_j=14937.\end{array}$$

Therefore

$$\begin{array}{rl}4\lambda (C)& =8b(X)+32\sum _j{q}_j\\ & =1872232\\ & ={\beta }_{\mathbb{Z}}^{\mathrm{cyc}}(M),\\ \mathrm{SM}(M)& =1872232+32.\end{array}$$

Finite-order transfer realizes the first number exactly. The manifold has complex dimension $1666$, hence real dimension $3332$. This companion optimizes equality with the cyclic lower bound, rather than minimizing dimension. (OpenAI, 2026d)

🏷️ The ordinary Morse-number companion and compact insertion

The twelve-dimensional companion uses a different obstruction. Record its excluded group-theoretic input as follows: there are closed symplectic four-manifolds $X$ with arbitrarily large minimum generator number $d$ for ${\pi }_1(X)$, together with a constructed Morse function on $X\times {\mathbb{R}}^8$ having exactly $\chi (X)+8$ critical points and equal outside a compact set to

$$\begin{array}{c}Q(u,v)=\frac{|v{|}^2-|u{|}^2}{2},\\ u,v\in {\mathbb{R}}^4.\end{array}$$

The manifold and function are supplied by the manuscript’s group-theoretic and relative-handle arguments. The latter retain precisely two handles in each of indices five and seven, preserve quadratic collars, and use exactly eight added variables. We do not replace this geometric assertion by the existence of an abstract chain complex. (OpenAI, 2026a)

The dynamical task is to insert sixteen copies of the resulting small-time return map into

$$M=X\times (S^2{)}^4.$$

There is one product cap for each north/south choice on the four spheres. Identify the quadratic variables with four symplectic planes, with two negative and two positive signs. The small quadratic flow is rotation in each plane. A global product rotation can be chosen so that its angle is strictly between zero and $2\pi$ away from the poles and its endpoint agrees with the quadratic return map in every cap.

The endpoint qualification matters. North and south symplectic coordinates reverse the longitude angle, so the two local rotation paths can differ by a full turn even though their endpoint multipliers agree. The paper tracks these full turns when proving contractibility.

The local correction path is

$${\delta }_t={\Phi }_F^{\epsilon t}\circ {\Phi }_Q^{-\epsilon t}.$$

It is the identity outside the compact core where $F$ differs from $Q$. Transplant its Hamiltonian into each cap and extend by zero. Composing this correction with the global rotations gives endpoint ${\Phi }_F^{\epsilon }$ inside every cap. Outside the caps, the rotation still moves at least one nonpolar coordinate through an angle strictly between zero and a full turn, so there are no fixed points there. The short-period estimate and the nonsingular Hessians count exactly the critical points of $F$ inside each cap, all nondegenerate.

For a critical point, the actual Hamiltonian loop is a small $F$-flow applied to an auxiliary loop of full rotations in the Euclidean fiber. Contract that fiber loop within the cap and apply the same flow to the contraction. Since the base point is an equilibrium, the contraction remains based there. This proves contractibility even though $X$ need not be simply connected. Thus

$$\begin{array}{rl}\#{\mathrm{Fix}}_0({\varphi }_H^1;H)& =16(\chi (X)+8)\\ & =\chi (M)+128.\end{array}$$

On the other hand, any Morse function on an even-dimensional closed manifold needs at least $d$ index-one handles and at least $d$ handles of the next-to-top index: apply the generator bound to the function and its negative. If $m_i$ counts index-$i$ critical points, Euler’s identity gives

$$\begin{array}{rl}\sum _i{m}_i& =\chi (M)+2\sum _{i\text{ odd}}{m}_i\\ & \ge \chi (M)+4d.\end{array}$$

Choosing $d>32$ makes the ordinary Morse bound larger than the constructed fixed-point count. Letting $d$ grow gives arbitrary deficits in the same real dimension twelve. The homology in these examples is torsion-free and concentrated in even degrees, so its total rank is $\chi (M)$, below the fixed-point count. This companion isolates an ordinary Morse obstruction that the simpler homological count does not detect; it does not itself prove a stable-Morse counterexample. (OpenAI, 2026a)

🏷️ Key steps and proof outlines

A uniform short-period bound prevents extra fixed points

Let $V$ be a smooth vector field on a compact manifold. Choose finitely many nested coordinate balls so that trajectories of sufficiently short duration remain in one larger chart, where $V$ has a common Lipschitz constant $L$. If a trajectory $y$ closes after time $T$, put $m={\max }_t|V(y(t))|$ and choose a maximizing time $t_0$. Since ${\int }_0^{T}V(y(t))dt=0$ and $|y(t)-y(t_0)|\le Tm$,

$$m\le \frac{L}{T}{\int }_0^T|y(t)-y(t_0)|dt\le LTm.$$

For $T<1/L$, this forces $m=0$. Thus sufficiently short periodic trajectories are equilibria, uniformly over the manifold. This elementary lemma is what makes the fixed-point counts exact rather than perturbative lower bounds. (OpenAI, 2026c)

Squaring the perturbed involution identifies its entire fixed set

Extend the chosen Morse functions from the fixed components of an involution $g$ and average the extension with its pullback by $g$. The resulting Hamiltonian $f$ is invariant, so its flow ${\psi }^t$ commutes with $g$. Consequently $({\psi }^{\epsilon }g{)}^2={\psi }^{2\epsilon }$.

A fixed point of the perturbed map lies on a period-$2\epsilon$ orbit of $X_f$, hence is an equilibrium by the short-period lemma. It must then be fixed by $g$ itself. On each fixed component, invariance makes the normal derivative of $f$ vanish, so the equilibrium condition is exactly criticality of the chosen component Morse function.

Nondegeneracy is checked in two blocks. On the tangent block, $[e^{\epsilon A}-I]/\epsilon \to A$, where the Morse Hessian makes the Hamiltonian linearization $A$ invertible. On the normal block, the involution is $-I$, so the fixed-point linearization tends to $-2I$. Both remain invertible for one sufficiently small $\epsilon$ at all finitely many critical points. Smooth time cutoffs concatenate the Hamiltonian path to $g$ with the short flow and give a smooth periodic Hamiltonian.

The numerical deficit is separate from its geometric realization

The stable-Morse companions apply that transfer to four fixed components. Their integer-graded lower bound counts free homology ranks plus twice the minimum torsion generator counts in each degree. Product factors convolve the prime multiplicity profiles before those generator counts are taken. Since combining coprime cyclic factors takes a maximum of multiplicities, convolution and degreewise maximum need not commute. The displayed profiles exploit exactly this failure; the free ranks cancel in the deficit.

The sharp cyclic companion merges integer degrees of the same parity before counting generators. This can combine torsion contributions that integer grading keeps apart, explaining how the constructed fixed-point count can meet the cyclic integral bound while falling below the stable Morse number. The geometric handle-attainment and manifold-realization theorems remain necessary inputs; arithmetic profiles alone do not construct the examples. (OpenAI, 2026d)

The twelve-dimensional ordinary-Morse companion uses compact insertion instead. Its correction is identity outside the prescribed quadratic core, while the background sphere rotations have no fixed points away from the selected caps. Full turns in the actual paths are tracked to contract the resulting loops. The degenerate three-point companion retains the finite-order transfer idea but uses a smooth function with fewer critical points than a Morse function can have; the nondegeneracy block argument therefore does not apply to all its points. These distinctions keep the different Arnold-type bounds and the different examples separate.

🏷️ Dependencies and further reading

The finite-order transfer argument is an elementary dynamical mechanism once its symplectic fixed locus is supplied. The stable-Morse results additionally require geometric Morse realization and the asserted manifolds with the prescribed integral homology. The twelve-dimensional result requires relative handle cancellation with a fixed quadratic exterior and a verified compact insertion. The degreewise calculations explain the numerical gaps but do not independently establish those geometric inputs.

• Symplectic ball packings develops another use of Hamiltonian rearrangement, where matching local constructions and controlling the region outside them are essential.
• Exact Lagrangian embeddings gives related symplectic preliminaries and emphasizes the hypotheses under which exactness imposes geometric restrictions.
• The collection guide places this family among the differential geometry articles and records the intended graduate prerequisites.

🏷️ Manuscripts and verification scope

• Hamiltonian Fixed Points Below the Stable Morse Number in Dimension Twenty-Two — pinned manuscript and companion source.
• Sharpness of the Cyclic Integral Floer Bound Below the Stable Morse Number — pinned manuscript and companion source.
• Hamiltonian Fixed Points Below the Stable Morse Number — pinned manuscript and companion source.
• A nondegenerate counterexample to the Morse-number Arnold bound — pinned manuscript and companion source.
• Three fixed points on the symplectic quadric threefold — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. Despite its broad heading, the scope page selects only Three fixed points on the symplectic quadric threefold: three Hamiltonian fixed points, a critical-number lower bound of four, and at least one degenerate fixed point. It explicitly excludes the separate nondegenerate Morse-number construction and does not document coverage of the three stable-Morse manuscripts. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026a. A nondegenerate counterexample to the Morse-number Arnold bound.

🐻  OpenAI 2026b. Hamiltonian Fixed Points Below the Stable Morse Number.

🐻  OpenAI 2026c. Hamiltonian Fixed Points Below the Stable Morse Number in Dimension Twenty-Two.

🐻  OpenAI 2026d. Sharpness of the Cyclic Integral Floer Bound Below the Stable Morse Number.

🐻  OpenAI 2026e. Three fixed points on the symplectic quadric threefold.