The Lins-de Melo-Pugh conjecture for quintic Lienard systems: at most two limit cycles

Theorem 1.1: if $\deg F\le5$ , with no parity or coefficient-sign assumption, the system $\dot x=y-F(x)$ , $\dot y=-x$ has at most two limit cycles in $\mathbb R^2$ , counted as geometric images regardless of multiplicity, stability or hyperbolicity, and some member of the class has exactly two. This confirms the conjectured value $\lfloor (5-1)/2\rfloor=2$ . It concerns only the classical form with restoring force exactly $x$ ; generalized Lienard systems and other degrees are not treated. The paper also shows that a proposed degree-five four-cycle example of Hernandez Rosales has only two cycles.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The Lins-de Melo-Pugh conjecture for quintic Lienard systems: at most two limit cycles · Lins-de Melo-Pugh conjecture, degree five

Confidence: Not scored

Registry verification: lean checked · announcement · partial

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VibeMathed record: The Lins-de Melo-Pugh conjecture for quintic Lienard systems: at most two limit cycles

specialist registry · passed

No independent mathematician has checked this yet. The release's formalization catalogue lists comparator config ComparatorChallenges/QuinticLienard.json with declaration OAI.QuinticLienard.main in OAI/Analysis/LienardCycles/Main.lean, permitted axioms propext, Quot.sound and Classical.choice. The comparator statement file was read: it defines the vector field (y - F(x), -x) for a real polynomial F, periodic orbits as ranges of nonconstant periodic solutions, and limit cycles as periodic orbits having an open neighbourhood that contains no other periodic orbit; it states that for deg F <= 5 the set of limit cycles has at most 2 elements and that some such F has exactly 2. That is the paper's headline claim, with no parity, sign or hyperbolicity assumption. Not rebuilt here: the Lean build and axiom check were not run. The paper's Theorem 1.1 was also read against the conjecture as quoted in its introduction.

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The Lins-de Melo-Pugh conjecture for quintic Lienard systems: at most two limit cycles

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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Lean proof: OAI.QuinticLienard.main

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

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Lean comparator statement: QuinticLienard

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lean-proof link supplied by the VibeMathed registry.