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Math-2227 · mathematics · Dynamical systems

Arnold and Arnold–Givental conjectures

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
For a Hamiltonian diffeomorphism of a closed symplectic manifold, prove the Arnold lower bound on its number of fixed points in terms of Morse-theoretic data. In the Lagrangian Arnold–Givental form, prove the corresponding lower bound for intersections of a Lagrangian submanifold with its Hamiltonian image under the standard hypotheses.

Formulation references: [S1], [S2].

Short English statement

For a Hamiltonian diffeomorphism of a closed symplectic manifold, prove the Arnold lower bound on its number of fixed points in terms of Morse-theoretic data. In the Lagrangian Arnold–Givental form, prove the corresponding lower bound for intersections of a Lagrangian submanifold with its Hamiltonian image under the standard hypotheses.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2227-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2227

Degenerate Arnold fixed-point obstruction: The paper refutes the critical-number Hamiltonian branch, but the bundled record also asks the separate Lagrangian Arnold-Givental intersection assertion.

Ordinary Morse-number obstruction: The bundle record includes the Morse-theoretic Arnold branch and a separate Lagrangian Arnold-Givental branch. The paper disproves a Morse or stable Morse fixed-point lower bound only; the other branch is not addressed.

Stable Morse obstruction from torsion: The bundle record includes the Morse-theoretic Arnold branch and a separate Lagrangian Arnold-Givental branch. The paper disproves a Morse or stable Morse fixed-point lower bound only; the other branch is not addressed.

Stable Morse deficits in dimension twenty-two: The bundle record includes the Morse-theoretic Arnold branch and a separate Lagrangian Arnold-Givental branch. The paper disproves a Morse or stable Morse fixed-point lower bound only; the other branch is not addressed.

Sharp cyclic Floer count below stable Morse: The bundle record includes the Morse-theoretic Arnold branch and a separate Lagrangian Arnold-Givental branch. The paper disproves a Morse or stable Morse fixed-point lower bound only; the other branch is not addressed.

This page records partial support. Complete resolution of the catalogue question is not established by this record.

Arnold and Arnold–Givental conjectures

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; additionally, Ordinary Morse-number obstruction: The paper claims nondegenerate Hamiltonian maps on closed twelve-dimensional symplectic manifolds with fewer fixed points than their ordinary Morse numbers; the shortfall can be made arbitrarily large and all associated orbit loops are contractible.

Stable Morse obstruction from torsion: The first stable-Morse companion claims a simply connected closed Kähler example with sixteen fewer nondegenerate Hamiltonian fixed points than its stable Morse number; every associated orbit is contractible.

Stable Morse deficits in dimension twenty-two: In fixed real dimension 22, the paper claims simply connected Kähler examples with stable Morse number $80+1968m$ and exactly $80+1952m$ nondegenerate Hamiltonian fixed points for each $m\ge1$; hence the deficit is unbounded.

Sharp cyclic Floer count below stable Morse: The manuscript claims a simply connected Kähler manifold of dimension 3332 with 1,872,232 nondegenerate Hamiltonian fixed points and stable Morse number 1,872,264; its minimal Chern number is one, and the cyclic integral Floer lower estimate is exactly attained by that count.

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.
Mathematical claim 2.
A claimed example uses a connected compact symplectic manifold $(M,\omega)$ without boundary, of real dimension twelve. A smooth Hamiltonian $H$ of time period one has only nondegenerate time-one fixed points, and \[\#\operatorname{Fix}_0(\phi_H^1;H)=\#\operatorname{Fix}(\phi_H^1)=\chi(M)+128<\operatorname{Morse}(M).\] The subscript $0$ counts contractible Hamiltonian orbits; $\operatorname{Morse}(M)$ is the minimum critical-point number among ordinary Morse functions. Nondegenerate means the derivative has no eigenvalue $1$.
Mathematical claim 3.
The claimed $1412$-dimensional example uses a Kähler manifold $M$ which is closed and simply connected. A smooth Hamiltonian of period one has nondegenerate fixed points, all represented by contractible periodic trajectories, with \[\operatorname{SM}(M)=318968,\qquad\#\operatorname{Fix}(\phi_H^1)=318952=\operatorname{SM}(M)-16.\] The stable Morse number $\operatorname{SM}$ minimizes critical points of a Morse function on $M\times\mathbb R^k$, for some $k\ge0$, agreeing with a nondegenerate auxiliary quadratic form outside a compact set.
Mathematical claim 4.
For each integer $m\ge1$, the real dimension is twenty-two. The claimed manifold $M_m$ is closed, simply connected and Kähler, and a smooth period-one Hamiltonian has only nondegenerate fixed points with contractible trajectories. Their numbers satisfy \[\operatorname{SM}(M_m)=80+1968m,\qquad\#\operatorname{Fix}(\phi_{H_m}^1)=80+1952m.\] Thus arbitrarily large stable Morse numbers occur with the fixed-point count at most $(1-1/128)\operatorname{SM}(M_m)$. The stable Morse convention permits stabilization by Euclidean variables with quadratic behavior outside a compact set.
Mathematical claim 5.
In complex dimension $1666$, the claimed $M$ is closed, simply connected and Kähler, with $c_1(TM)(\pi_2(M))=\mathbb Z$. A smooth Hamiltonian of time period one has only nondegenerate fixed points and contractible associated orbits, and the counts are \[\#\operatorname{Fix}(\phi_H^1)=\beta_{\mathbb Z}^{\mathrm{cyc}}(M)=1872232,\qquad\operatorname{SM}(M)=1872264.\] Here $\beta_{\mathbb Z}^{\mathrm{cyc}}=\sum_i\operatorname{rank}H_i(M;\mathbb Z)+2\sum_{\bar i\in\mathbb Z/2}d(\operatorname{Tor}\bigoplus_{k\equiv i\pmod2}H_k(M;\mathbb Z))$, where $d$ is the minimum number of generators. The parity-grouped integral homology count is attained while the geometric stable Morse number exceeds it by $32$.

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. Ordinary Morse-number obstruction: Finite groups with increasing group-generator number but bounded augmentation-ideal generation create a gap between ordinary and stabilized Morse counts. A relative twelve-dimensional handle construction realizes a small critical-point model while preserving quadratic collars. Local Hamiltonian realization on a closed product transfers that reduced count to nondegenerate fixed points.

Stable Morse obstruction from torsion: Integral homology at two distinct primes gives a stable-Morse lower bound sensitive to the degree-wise torsion generator counts. A Hamiltonian involution on a blown-up Kähler surface has four fixed components. An invariant small perturbation realizes Morse functions on those components, and the resulting product count lies below the stable-Morse benchmark. Stable Morse deficits in dimension twenty-two: Torsion of different prime orders is distributed across degrees so that taking products changes the maximum generator multiplicity. Blowups and product constructions realize the needed homology while keeping the dimension fixed. An involution with controlled fixed components and an invariant Morse perturbation give the explicit stable-Morse and Hamiltonian counts.

Sharp cyclic Floer count below stable Morse: The construction distinguishes degree-wise integral torsion counts from their cyclic reduction modulo twice the Chern number. Complex blowups and a Hamiltonian finite-order action realize the desired topology and fixed components. A controlled invariant perturbation computes the nondegenerate count exactly, showing that equality in the cyclic bound can coexist with a stable-Morse deficit.

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.

Ordinary Morse-number obstruction: No manuscript-specific Comparator is listed in the pinned inventory.

The documented selection does not cover the entire manuscript assertion relevant here.

Stable Morse obstruction from torsion: No manuscript-specific Comparator is listed in the pinned inventory.

The documented selection does not cover the entire manuscript assertion relevant here.

Stable Morse deficits in dimension twenty-two: No manuscript-specific Comparator is listed in the pinned inventory.

The documented selection does not cover the entire manuscript assertion relevant here.

Sharp cyclic Floer count below stable Morse: No manuscript-specific Comparator is listed in the pinned inventory.

The documented selection does not cover the entire manuscript assertion relevant here.

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

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