Arnold Conjecture on Hamiltonian Fixed Points

Arnold Conjecture on Hamiltonian Fixed Points

OpenAI's claim

Degenerate Arnold fixed-point obstruction: A Hamiltonian map of the complex quadric threefold is claimed to have exactly three fixed points although its critical number and rational cup-length lower bound are four. At least one fixed point is degenerate.

Mathematical statement of the claimed result

Mathematical claim 1.

For the complex quadric $Q^3=\{[z]\in\mathbb{CP}^4:\sum_{i=0}^4z_i^2=0\}$ with restricted Fubini--Study symplectic form, the claim constructs a smooth Hamiltonian diffeomorphism $\phi$ with \[\#\operatorname{Fix}(\phi)=3<4=\operatorname{Crit}(Q^3)=\operatorname{cuplength}(Q^3;\mathbb Q).\] At least one fixed point is degenerate. Here $\operatorname{Crit}$ minimizes over all smooth functions, including degenerate ones; the cup-length convention is the longest nonvanishing product of positive-degree classes, with one added to that count.

Outline of the argument

Degenerate Arnold fixed-point obstruction: A finite-order Hamiltonian map is perturbed by the flow of an invariant extension of a function on its fixed locus. A power identity excludes new fixed points elsewhere, and symmetric criticality identifies the remaining fixed points with critical points on that locus. A suitably chosen function on the quadric surface then supplies the three-point count.

Formal verification

Family 347 describes Lean coverage for the degenerate quadric critical-number obstruction; the nondegenerate ordinary and stable-Morse constructions are separate.

Degenerate Arnold fixed-point obstruction: The manuscript links ArnoldCounterexample.lean (declaration OAI.ArnoldCounterexample.main).

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

• OpenAI, Three fixed points on the symplectic quadric threefold, 2026-09-23; citation key OAI:A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:counterexample.
• Family 347: Lean scope.
• Comparator ArnoldCounterexample.json.
• Comparator ArnoldCounterexample.lean.
• OpenAI catalogue entry.

Arnold Conjecture on Hamiltonian Fixed Points

Definitions and mathematical statement

A symplectic manifold $(M,\omega)$ carries a closed nondegenerate two-form. A Hamiltonian diffeomorphism is the time-one map of a time-dependent vector field $X_{H_t}$ satisfying $\iota_{X_{H_t}}\omega=-\,\mathrm d H_t$. For a closed connected $M$, put \[ \operatorname{Crit}(M)=\min_{f\in C^\infty(M,{\mathbb{R}})}\#\{x:\,\mathrm d f(x)=0\}. \] The selected assertion is $\#\operatorname{Fix}(\phi)\ge\operatorname{Crit}(M)$ for every Hamiltonian diffeomorphism $\phi$. This minimum permits degenerate critical points; it is not the minimum restricted to Morse functions. Likewise the fixed points need not be nondegenerate. An infinite fixed-point set satisfies the cardinal lower bound automatically.

Formulation and concept references: [S1].

Short English statement

Does every Hamiltonian motion on a closed symplectic manifold leave at least as many points fixed as any smooth real function on that manifold must have critical points?

Research attempt

Current frontier.

Editorial scope note. The formulation source is entitled A $C^0$ counterexample to the Arnold conjecture. Its counterexamples are Hamiltonian homeomorphisms, not the smooth Hamiltonian diffeomorphisms of the present target [S1], Theorem 1. They do not disprove this entry. The same source distinguishes the degenerate $\operatorname{Crit}(M)$ statement from nondegenerate Morse or Floer bounds and recalls established special cases. A lower bound by a cup-length or by a Betti-number sum is not automatically the requested $\operatorname{Crit}(M)$ bound.

Attack route.

A Floer-theoretic route would need a lower bound strong enough for the minimum over all smooth functions, including degenerate ones. A generating- function route can achieve that exact comparison if the fixed points become the critical points of one function on $M$. We pursue the latter route near the diagonal and then test a possible global extension by eliminating auxiliary variables. The missing input is a global reduction without branching or caustics, not merely the existence of finite-dimensional approximations to a Hamiltonian action functional.

Attempt: a graph lemma with the exact critical-point count.

In $(M\times M,-\omega\oplus\omega)$, the diagonal $\Delta$ is Lagrangian, and so is the graph of a symplectomorphism. The Lagrangian neighborhood theorem identifies a neighborhood of $\Delta$ with a neighborhood of the zero section in $T^*M$ [E1]. Fix such an identification. Assume, additionally, that a Hamiltonian isotopy $\phi_t$ from the identity has all its graphs inside this neighborhood and that projection of each transformed graph to $M$ is a diffeomorphism. Then the time-one graph is the graph of an exact one-form $dF$, and \begin{equation}\label{wsref-0-65-71-31-37-38-2d-6c-6f-63-61-6c} \#\operatorname{Fix}(\phi_1) =\#\operatorname{Crit}(F) \ge\operatorname{Crit}(M). \end{equation} No nondegeneracy hypothesis is needed.

Here are the exactness details. A graph over $M$ in $T^*M$ is Lagrangian exactly when its defining one-form $\beta_t$ is closed, by pulling back the canonical symplectic form. Its cohomology class remains zero along the Hamiltonian isotopy. To see this without invoking a flux slogan, write $\lambda$ for the canonical one-form in the neighborhood and $j_t$ for a parametrization of the moving Lagrangian. Cartan's formula gives \[ \frac{d}{dt}j_t^*\lambda =j_t^*(\iota_{V_t}d\lambda)+d\left(j_t^*(\iota_{V_t}\lambda)\right). \] The first term is exact because the motion is Hamiltonian; the second is explicitly exact. Reparametrizing the graph by its base projection does not change vanishing of its cohomology class. At $t=0$ the one-form is zero, so $\beta_1=dF$. Intersections with the zero section are precisely zeros of $dF$, while the original graph intersects $\Delta$ precisely at fixed points. The neighborhood identification is bijective, proving (eq178-local) with cardinalities, not multiplicities.

This explains why a sufficiently small Hamiltonian isotopy for which the graph condition holds admits the exact bound. The additional isotopy and projection assumptions have not been inferred just from an arbitrary Hamiltonian time-one map. If there are infinitely many fixed points, (eq178-local)'s lower bound is automatic, as in the original statement.

Attempt: an auxiliary-variable elimination criterion.

Suppose a more global construction produced a smooth generating function $G:M\times{\mathbb{R}}^k\to{\mathbb{R}}$ whose critical points are in bijection with the fixed points under study. Assume, for the moment, that for every $x\in M$ the fiber function $\eta\mapsto G(x,\eta)$ has exactly one critical point $\eta(x)$ and has invertible Hessian in $\eta$ there. The implicit function theorem gives smooth local branches; uniqueness makes them agree on overlaps, so $\eta(x)$ is globally smooth. Define \[ F(x)=G(x,\eta(x)). \] The chain rule and $\partial_\eta G(x,\eta(x))=0$ imply $dF=\partial_xG(x,\eta(x))$. Hence \begin{equation}\label{wsref-0-65-71-31-37-38-2d-65-6c-69-6d-69-6e-61-74-65} x\in\operatorname{Crit}(F) \quad\Longleftrightarrow\quad (x,\eta(x))\in\operatorname{Crit}(G). \end{equation} This proves a bijection and therefore the required count, even when the remaining critical points of $F$ are degenerate. Fiberwise proper strict convexity with positive definite fiber Hessian is one sufficient way to obtain the stated uniqueness and existence; those are extra hypotheses, not properties asserted for Hamiltonian generating functions in general.

This suggests an ambitious global route: subdivide a Hamiltonian isotopy into small pieces, build a finite-dimensional generating family, and eliminate all the intermediate variables. The last operation is where the proof stops. Composition generally leads to indefinite generating families; there is no reason for each fiber to have a unique critical point or an invertible fiber Hessian everywhere. Also, a global generating function on the trivial bundle $M\times{\mathbb{R}}^k$ with a fixed-point bijection has not been supplied for arbitrary $M$ by the local graph calculation.

Attempt: an explicit caustic test.

The elementary family \[ G(x,\eta)=\eta^3/3-x\eta \] has fiber critical equation $\eta^2=x$. It has no real fiber critical point for $x<0$, one degenerate point for $x=0$, and two points for $x>0$. Thus even a smooth finite-dimensional generating expression need not admit a single smooth elimination branch. Choosing one branch where possible would discard critical points and need not define a function on all of the base. The example is a test of the proposed elimination mechanism, not a Hamiltonian counterexample on a closed manifold.

A different possible shortcut is to prove a lower bound after generic perturbation and pass to the limit. Critical or fixed points may coalesce; compactness only supplies limit points, not the same number of distinct ones. In a coordinate model the functions $x^3/3-tx$ have two critical points for $t>0$ and one at $t=0$. Thus the distinction between nondegenerate and degenerate bounds in [S1] cannot be bypassed by an unqualified limiting argument. Local Floer homology or a stronger critical-point invariant might retain information through coalescence, but that additional theory has not been converted here into the exact $\operatorname{Crit}(M)$ comparison.

Stress test.

The Hamiltonian condition was used for exactness, not merely for closedness of $\beta_t$. A closed nonexact one-form could have a different zero count, so replacing Hamiltonian by symplectic isotopy would invalidate the proof. Likewise, a graph in the product is not necessarily a graph of a one-form under the fixed cotangent-neighborhood projection.

The source's $C^0$ example demonstrates that an unrestricted extension to Hamiltonian homeomorphisms would be false [S1]. None of our lemmas extends the smooth generating-function or implicit-function arguments to that category. Finally, the minimum in $\operatorname{Crit}(M)$ is not replaced by the Morse number: the functions obtained in (eq178-local) and (eq178-eliminate) may have degenerate critical points and still give exactly the comparison needed.

Outcome.

Outcome: Exploratory approach with unresolved gap.

The attempt proves the exact bound under a local Hamiltonian graph hypothesis and gives a precise sufficient criterion for eliminating auxiliary variables globally. It then isolates failures of the naive composition and perturbation arguments. The missing step is a global critical-point comparison that survives caustics and degeneracy on every closed symplectic manifold. These are standard local mechanisms developed as a research test, not a new proof of the full Arnold statement or a counterexample to it.

Sources

• S1 Formal statement, abstract. https://arxiv.org/abs/1609.09192. Catalogue formulation source.
• E1 Alan Weinstein, Symplectic manifolds and their Lagrangian submanifolds, Advances in Mathematics 6 $1971$, 329--346, Lagrangian neighborhood theorem. https://doi.org/10.1016/0001-8708(71)90020-X. Consulted bibliographic record 27 September 2026. The neighborhood theorem is the standard geometric input; exactness and the critical-point comparison used here are spelled out in the attempt.