Result 143, Dynamical systems and ergodic theory

Hilbert's sixteenth problem: uniform bounds for limit cycles

Resolves the uniform boundedness assertion in Hilbert's sixteenth problem: the number of isolated periodic orbits of a real planar polynomial vector field is bounded by a finite constant depending only on its degree. For classical quintic Liénard systems, the exact maximum is two limit cycles.

Lean formalization Proof

The bigger picture

Why it matters

Polynomial equations can describe motion that repeats indefinitely. These manuscripts claim a ceiling on the number of isolated repeating motions in a plane based only on polynomial degree, limiting how complicated such dynamics can become.

What changes?

The unreviewed manuscripts report a finite bound, depending only on degree, for isolated periodic orbits of every real planar polynomial vector field: two polynomial rules governing motion in a plane. These orbits, called limit cycles, are closed trajectories isolated from other closed trajectories. For classical Liénard systems, where x changes at rate y minus F(x) and y at rate minus x, the claimed exact maximum is two when F is any real polynomial of degree at most five.

What does that help mathematicians do?

The uniform claim is stronger than saying each individual system has finitely many limit cycles. It would prevent researchers from producing arbitrarily many cycles by varying coefficients while keeping the degree fixed. The quintic result separately rules out three coexisting limit cycles in the specified Liénard family and says two can occur. It does not give an exact maximum for all degree-five planar systems.

Are there practical applications?

The immediate value is foundational: these claims would constrain the possible arrangements of periodic motion in polynomial dynamical systems and give a sharp counting target for quintic Liénard examples. The abstracts provide neither numerical bounds for general degrees nor a procedure for computing them, so they do not establish a practical cycle-counting tool.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Uniform bounds for planar polynomial limit cycles

September 24, 2026 160 pages

For every degree, we prove that the number of isolated periodic orbits of a real planar polynomial vector field is bounded by a finite constant depending only on that degree. This establishes the uniform boundedness assertion in the second part of Hilbert's sixteenth problem. The proof uses separation of asymptotic expansions on nested complex domains and a finite-dimensional counting argument.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026,
  author = {{OpenAI}},
  title = {{Uniform bounds for planar polynomial limit cycles}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026/uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026.pdf}{OAI:uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026}},
  year = {2026}
}

Two limit cycles for quintic Liénard systems

September 24, 2026 40 pages Main result formalized in Lean

Every classical Liénard system x˙=y−F(x)\dot x=y-F(x), y˙=−x\dot y=-x, with F an arbitrary real polynomial of degree at most five, has at most two geometrically distinct isolated periodic orbits, and the bound is attained. This proves the degree-five case of the Lins Neto–de Melo–Pugh conjecture.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:two-limit-cycles-for-quintic-lienard-systems-September-24-2026,
  author = {{OpenAI}},
  title = {{Two limit cycles for quintic Li{\'e}nard systems}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/two-limit-cycles-for-quintic-lienard-systems-September-24-2026/two-limit-cycles-for-quintic-lienard-systems-September-24-2026.pdf}{OAI:two-limit-cycles-for-quintic-lienard-systems-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/143.md.

Hilbert's sixteenth problem: uniform bounds for limit cycles

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

For the quintic Liénard system x′=y−F(x)x'=y-F(x), y′=−xy'=-x, the formalized result proves that every real polynomial FF of degree at most five yields at most two limit cycles, and that some such FF yields exactly two. Limit cycles are isolated images of nonconstant periodic solutions. No sign, parity, hyperbolicity, or amplitude restriction is imposed.

Comparator links

Result Comparator statement
Exact two-cycle bound for quintic Liénard systems QuinticLienard.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.