# Planar oscillator cycle-count assurance

Family 143: Hilbert's sixteenth problem: uniform bounds for limit cycles. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Control-model researchers can compare a quintic Liénard system with the exact at-most-two isolated-cycle statement.

## Applicability and commercial boundary

The general polynomial degree bound has no effective formula in the introduction. The selected formal scope covers only the quintic Liénard specialization, not general Hilbert-sixteenth boundedness.

## Initial business decision

Research or conditional engineering only until an effective implementation and a recurring buyer problem are identified. No commercial demand or profitability is established. Where a direct product bridge is weak, the legitimate use is a research reference or evidence adapter, rather than a new standalone company.

## Next verification action

Extract the specific system and independently validate any computed cycle rather than treating a simulation as exhaustive.

## Evidence scope

The catalog statement was individually reviewed. Main-paper abstracts and available selected formal scopes were additionally inspected; explicit effectiveness passages were inspected for 076, 143 and 178. No independent proof verification was run. Pinned source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

[Uniform bounds for planar polynomial limit cycles](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026/uniform-bounds-for-planar-polynomial-limit-cycles-September-24-2026.pdf).