A counterexample to the nearby Lagrangian conjecture

Source and verification

This article explains family 340 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.

🏷️ Preliminaries: cotangent geometry and exact Lagrangians

The cotangent bundle $T^{\ast }Q$ consists of pairs $(q,p)$, where $p$ is a linear functional on the tangent space at $q$. Its canonical one-form is

$${\lambda }_{(q,p)}(\xi )=p(d\pi (\xi )),$$

where $\pi :T^{\ast }Q\to Q$ is projection. The manuscript uses the symplectic form $\omega =d\lambda$ and Hamiltonian convention ${\iota }_{X_H}\omega =-dH$.

A submanifold $L\subset T^{\ast }Q$ is Lagrangian if its dimension is $\dim Q$ and $\omega {|}_L=0$. It is exact if $\lambda {|}_L=df$ for a globally defined real function on $L$. The zero section, consisting of all zero covectors, is an exact Lagrangian.

A Hamiltonian isotopy is motion generated by a smooth time-dependent Hamiltonian. For compact Lagrangians its Hamiltonian can be cut off outside a neighborhood of the compact trace of the motion. Thus requiring one fixed compact support does not restrict the usual isotopy question here.

The nearby Lagrangian conjecture asks whether every closed exact embedded Lagrangian is Hamiltonian isotopic to the zero section. Closed means compact without boundary.

Manuscript result

For some sufficiently large even integer $N$, let

$$Q=S^9\times S^{N-1}.$$

There is a closed exact smoothly embedded Lagrangian $L\subset T^{\ast }Q$, diffeomorphic to $Q$, which no compactly supported Hamiltonian isotopy carries to the zero section. (OpenAI, 2026)

Both manifolds are connected and simply connected. The claimed obstruction concerns the embedding in the cotangent bundle, not a difference in their abstract smooth types.

The prerequisites for the explicit construction are differential forms, Morse theory, and transversality. A high-dimensional topological input supplies a family of functions that cannot be simplified coherently. Its stable homotopy calculation is stated as an input, without developing algebraic $K$-theory.

🏷️ Graphs and generating families

For a smooth function $f:Q\to \mathbb{R}$, the graph of $df$ is exact Lagrangian. The path of graphs of $tdf$ is Hamiltonian, so this elementary construction cannot produce the claimed example.

A generating family introduces additional variables:

$$F:Q\times {\mathbb{R}}^r\to \mathbb{R}.$$

Its vertical critical locus and generated map are

$$\begin{array}{c}\mathrm{Crit}(F)=\{(q,w):d_{w}F=0\},\\ {\Gamma }_F(q,w)=(q,d_{q}F).\end{array}$$

Assume that zero is a regular value of the full map $d_{w}F$, allowing both $q$ and $w$ to vary. This makes the critical locus a smooth manifold of dimension $\dim Q$. It does not require every fiber Hessian $F_{ww}$ to be invertible.

On the critical locus,

$${\Gamma }_F^{\ast }\lambda =d(F{|}_{\mathrm{Crit}(F)}).$$

Hence the induced two-form vanishes and exactness is automatic. Regularity also gives immersion: a vector in the kernel of $d{\Gamma }_F$ has $\delta q=0$ and lies in the kernels of both $F_{ww}$ and $F_{qw}$. Symmetry of the Hessian identifies it with a vector in the kernel of the transpose of the surjective full derivative of $d_{w}F$, so it must vanish.

An immersion can have double points. The construction therefore needs a separate global injectivity argument.

🏷️ A family at infinity that retains smooth information

Start with the indefinite quadratic form

$$q_{k,l}(x,y)=-|x{|}^2+|y{|}^2.$$

Its negative domain on the unit sphere is a tube diffeomorphic to $S^{k-1}\times D^l$. More generally, take a smooth angular function $h$ joined to that quadratic restriction by a path of functions with regular zero level, and extend it by

$$g(ru)=r^{2}h(u),r>0.$$

This extension is smooth away from the origin and $C^1$ at the origin. Its gradient is

$$\nabla g(ru)=r(2h(u)u+{\nabla }_{S}h(u)).$$

The radial and tangential terms are orthogonal. Regularity of the zero level makes the gradient nonzero away from the origin. Over a compact family, this gives uniform bounds

$$c|w|\le |\nabla g_b(w)|\le C|w|.$$

Adding fixed positive and negative quadratic variables is called stabilization. It changes the auxiliary dimension without changing the stable geometric information carried by the tubes.

The topological input is a smooth family $g_b$, indexed by $b\in S^9$, whose stable tube class $a$ is nonzero, even though the associated family of sphere homotopy types is stably trivial. In other words, forgetting the smooth tube information loses the obstruction.

The parameter $b$ will become part of the base of one Lagrangian. It does not index a family of entire Lagrangians in a fixed cotangent bundle.

🏷️ Coupling to an additional sphere

Choose a height function $\theta$ on $S^9$. The tube family can be arranged to equal one fixed nondegenerate quadratic form near the two critical points of $\theta$. Outside those neighborhoods,

$$|d\theta |\ge {\delta }_0>0.$$

Set

$$R(b)=\Lambda e^{K\theta (b)}.$$

After smoothing $g_b$ near the origin to obtain $G_b$, define

$$\begin{array}{rl}& F(b,v,w)\\ & =G_b(w)-R(b)\beta (|w|/T)⟨v,w⟩,\end{array}$$

where $v\in S^{N-1}$ and $\beta$ is one on $[0,1]$ and zero on $[2,\infty )$.

The parameters are chosen in order: first $K$ for momentum separation, then $\Lambda$ to exclude critical points near the smoothed origin, then $T$ large enough to exclude them in the cutoff region. Every vertical critical point consequently lies where both modifications are inactive and satisfies exactly

$$\nabla g_b(w)=R(b)v.$$

Homogeneity parametrizes the entire critical locus by $(b,u)\in S^9\times S^{N-1}$:

$$\begin{array}{c}v=\frac{\nabla g_b(u)}{|\nabla g_b(u)|},\\ w=\frac{R(b)u}{|\nabla g_b(u)|}.\end{array}$$

The inverse recovers $u=w/|w|$. Thus the critical locus is compact and diffeomorphic to $Q$.

For regularity, variation of $v$ spans $v^{\perp }$ in the critical equation, while radial variation $\delta w=w$ contributes $\nabla g_b(w)=Rv$. Together they span all vertical target directions. This proves full regularity even when an individual fiber Hessian is singular.

🏷️ The two momentum tests eliminate collisions

Suppose two critical points have the same image under ${\Gamma }_F$. They have the same base $(b,v)$. Their momenta in the sphere direction satisfy

$$d_{v}F(\dot{v})=-R(b)⟨w,\dot{v}⟩,\dot{v}\perp v.$$

Equality therefore forces

$$w-w^{\prime }=tv$$

for some real $t$. Sphere momenta eliminate every possible difference except one radial direction.

The base momenta eliminate that last direction. Smooth angular dependence gives

$$\Vert d_w{d}_b{g}_b(w)\Vert \le C_1|w|.$$

Both critical points, and the straight segment between them, have norm at most $R(b)/c$. Hence

$$\begin{array}{rl}& \Vert d_b{g}_b(w)-d_b{g}_b(w^{\prime })\Vert \\ & \le (C_1/c)R(b)|t|.\end{array}$$

The difference of the complete base momenta is

$$d_b{g}_b(w)-d_b{g}_b(w^{\prime })-KR(b)td{\theta }_b.$$

Away from the critical neighborhoods of $\theta$, its norm is at least

$$R(b)|t|(K{\delta }_0-C_1/c).$$

Choose $K{\delta }_0>C_1/c$. Equality of momenta then forces $t=0$. Near the two exceptional neighborhoods, $g_b$ is quadratic, so the critical equation has a unique solution for each $v$ directly.

The generated map is therefore injective everywhere. A compact injective immersion is an embedding. Euler’s homogeneous identity additionally gives the global primitive

$$F{|}_{\mathrm{Crit}(F)}=-g_b(w).$$

This collision argument is the step that distinguishes an embedded cotangent example from a construction whose projection has double points.

🏷️ The one-critical-point obstruction

The topological part proves the following interface. Consider a smooth family of functions on ${\mathbb{R}}^{2p}$ over $S^9$, with $p\ge 20$, agreeing outside one fixed ball with regular degree-two homogeneous data. Suppose each fiber has exactly one nondegenerate critical point, varying smoothly, and the stable spherical data at infinity are trivial. Then its stable tube class must be trivial.

The proof first moves the unique critical point to the origin and makes its local quadratic model constant after suitable stabilization. Cutoffs are chosen so their derivative errors are dominated by the existing gradient, preventing new critical points.

Between a small quadratic sphere and a large homogeneous sphere, the zero level is regular. It gives a family of cobordisms: manifolds interpolating between the two boundary levels. Morse theory identifies both boundary inclusions as homotopy equivalences.

Knowing that each individual cobordism is a product does not produce a product that varies over $S^9$. The source thickens the entire family by an interval and places it inside a fixed cylinder with a product complement. A stable injectivity result then gives a coherent product for the original family.

This allows the outer tubes to retract to neighborhoods of their cores. In sufficiently high codimension, a parameter-dependent general-position argument deforms the core embeddings to one fixed core. The whole tube family becomes stably trivial, contradicting $a\ne 0$ when the interface is applied to the selected infinity data.

The family-level product statement and the stable tube computation are substantive topological inputs. Ordinary Morse theory for one function would not establish this obstruction.

🏷️ Hamiltonian transport closes the contradiction

Assume a Hamiltonian isotopy carries the constructed $L$ back to the zero section. The source transports its generating family through finitely many small Hamiltonian steps. Each step adds a fixed split quadratic form and preserves the stable homogeneous tube data at infinity.

This requires an explicit transport construction for varying homogeneous infinity data. A generating-family theorem stated only for one fixed quadratic form at infinity would not automatically apply.

The small-step construction keeps the critical-locus parametrization a diffeomorphism onto the moving Lagrangian. It also restores exact homogeneity outside a larger uniform ball without creating critical points. The subdivision is chosen from the Hamiltonian and its support; subsequent changes of family estimates alter cutoff radii rather than requiring an endless refinement of the time subdivision.

At the zero section, projection of the critical locus to the base is a diffeomorphism. There is therefore exactly one critical point in each fiber. Its vertical Hessian is nonsingular: a Hessian-kernel vector would be tangent to the critical locus and vertical for this diffeomorphic projection.

Restricting to one slice $S^9\times \{v_0\}$ now gives exactly the one-critical-point family ruled out above. Transport preserves its nontrivial stable tube class and its trivial spherical class. The obstruction forces the same tube class to be trivial, a contradiction.

The result is claimed only for some sufficiently large even $N$; this argument does not provide a low-dimensional counterexample or an explicit smallest dimension.

🏷️ Key steps and proof outlines

The extra sphere variables make the critical locus regular

Let $g_b$ be the homogeneous degree-two family carrying the prescribed infinity data, with $b\in S^9$. Choose a height function $\theta$ and set $R(b)=\Lambda e^{K\theta (b)}$. The generating family has the form

$$F(b,v,w)=G_b(w)-R(b)\beta (|w|/T)⟨v,w⟩,v\in S^{N-1},$$

where $G_b$ smoothly fills the homogeneous function near zero and $\beta$ is one on the inner region and zero outside a larger ball. Parameters are chosen in the order $K$, then $\Lambda$, then $T$.

The lower gradient bound $|\nabla g_b(w)|\ge c|w|$ and a sufficiently large $\Lambda$ exclude critical points near zero. A sufficiently large $T$ excludes them in the cutoff annulus and outside it. Every critical point therefore satisfies exactly $\nabla g_b(w)=R(b)v$ in the region where the formula is homogeneous.

Varying $v$ spans the $v^{\perp }$ directions in this equation. Radial variation of $w$ contributes $\nabla g_b(w)=Rv$, supplying the remaining direction. The full critical equation is thus regular even if an individual fiber Hessian is degenerate. Homogeneity also gives the explicit parametrization

$$(b,u)⟼\left(b,\frac{\nabla g_b(u)}{|\nabla g_b(u)|},\frac{R(b)u}{|\nabla g_b(u)|}\right).$$

Its inverse recovers $u=w/|w|$, proving that the critical locus is diffeomorphic to the desired product. (OpenAI, 2026)

Two momentum comparisons rule out double points

Suppose two critical points have the same image in $T^{\ast }Q$. They have the same $(b,v)$. Equality of sphere momenta, $d_{v}F=-R⟨w,\cdot ⟩$ on $v^{\perp }$, implies $w-w^{\prime }=tv$.

The mixed derivative bound $\Vert d_w{d}_b{g}_b(w)\Vert \le C_1|w|$ and the critical-point radius bound yield

$$\Vert d_b{g}_b(w)-d_b{g}_b(w^{\prime })\Vert \le (C_1/c)R(b)|t|.$$

But the difference of base momenta also contains $-KR(b)td{\theta }_b$. Away from neighborhoods of the two critical points of $\theta$, one has $|d\theta |\ge {\delta }_0>0$. Choosing $K{\delta }_0>C_1/c$ makes this term dominate, forcing $t=0$. Near those critical points the family was chosen to be one fixed nondegenerate quadratic form, so the critical equation already has a unique solution. This proves global injectivity; compactness upgrades the immersion to an embedding.

Hamiltonian transport preserves the obstruction at infinity

The compact exact embedding is only half the argument. The infinity family has a stable tube class chosen to be nontrivial while its spherical obstruction vanishes. Small Hamiltonian steps transport generating families after fixed quadratic stabilizations and restore homogeneous behavior outside a uniform larger ball. Thus the stable class survives the entire proposed isotopy.

At the zero section, the transported critical locus projects diffeomorphically to the base. Each fiber then has exactly one critical point with nonsingular vertical Hessian. Restriction to a fixed sphere slice gives a family of one-critical-point functions with the original stable infinity data. The family-level cobordism theorem would trivialize that data, contradicting its choice. The source’s stable topological computation and coherent product theorem are indispensable inputs; the explicit momentum calculation proves embeddedness, not the nontriviality of that stable class.

See Also

Taming and compatibility in four dimensions studies another distinction between symplectic existence and stronger geometric structure. There the issue is closedness of a compatible form; here it is the ambient Hamiltonian isotopy class of an exact embedded Lagrangian.

🏷️ Manuscripts and verification scope

• A counterexample to the nearby Lagrangian conjecture — pinned manuscript and companion source.

No family scope page was present at lean/docs/340.md in the pinned snapshot. This does not establish that all related formalization code is absent. 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 2026. A counterexample to the nearby Lagrangian conjecture.