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Math-2654 · mathematics · Geometry

Stable Morse lower bound for Hamiltonian fixed points

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every closed connected symplectic manifold M and every Hamiltonian diffeomorphism $\varphi $ with all fixed points nondegenerate, must $\#Fix(\varphi )\geq SM(M)$? Here SM(M) is the minimum number of critical points of Morse functions on $M×R^k$ that agree outside a compact set with a nondegenerate quadratic form in the $R^k$ variables, minimized over $k\geq 0$.

Short English statement

For every closed connected symplectic manifold M and every Hamiltonian diffeomorphism $\varphi $ with all fixed points nondegenerate, must $\#Fix(\varphi )\geq SM(M)$? Here SM(M) is the minimum number of critical points of Morse functions on $M×R^k$ that agree outside a compact set with a nondegenerate quadratic form in the $R^k$ variables, minimized over $k\geq 0$.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2654-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2654

Stable Morse obstruction from torsion: The universal stable-Morse lower bound is contradicted by the announced nondegenerate examples with contractible orbit loops. The different constructions retain their stated dimensions and topology; no claim about the symplectically aspherical special case follows from them.

Stable Morse deficits in dimension twenty-two: The universal stable-Morse lower bound is contradicted by the announced nondegenerate examples with contractible orbit loops. The different constructions retain their stated dimensions and topology; no claim about the symplectically aspherical special case follows from them.

Sharp cyclic Floer count below stable Morse: The universal stable-Morse lower bound is contradicted by the announced nondegenerate examples with contractible orbit loops. The different constructions retain their stated dimensions and topology; no claim about the symplectically aspherical special case follows from them.

The claimed counterexample lies within the question’s hypotheses and contradicts its universal conclusion.

Stable Morse lower bound for Hamiltonian fixed points

OpenAI's claim

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.
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 2.
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 3.
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

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.

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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