Can a simple feedback system settle into more than one repeating rhythm?

The Wilhelm–Heinrich oscillator is a small mathematical model connected to chemical reactions and delayed population growth. Above a critical setting, it oscillates. Previous research established that only finitely many repeating cycles can occur, but did not settle whether there is always exactly one.

This candidate proves a substantial part of that claim. When the feedback gain is sufficiently large, every possible cycle is attracting, and there is exactly one. The threshold is explicit, so the result identifies a parameter region rather than merely saying that large enough values should work.

The proof uses geometry. Although the system has three state variables, its long-term positive dynamics lie on the graph of a convex function. Above the oscillation threshold, that graph sits over a disk. The authors derive exact formulas describing whether disturbances away from a cycle grow or shrink. Those formulas, together with bounds that apply to every cycle, establish uniqueness in the large-gain region.

The distinction between every cycle and one simulated cycle matters. A computer can find an attracting rhythm without ruling out another one elsewhere. Here, the written argument supplies that missing universal step within its stated region.

The revised package includes exact algebraic checks, rigorous interval bounds for selected scalar constants, and numerical experiments. It also repairs a previously assumed local invariant-manifold step. That repair strengthens a separate small-parameter result, but does not make it a global theorem.

The intermediate parameter region remains open. The work is an unrefereed candidate, not a formally verified proof or an independently accepted solution of the full problem.

This is Evidence Press, One rhythm at large gain, dated the third of October twenty twenty-six. The paper and evidence package are linked on the release page. This synthetic AI voice explains the research; it is not additional scientific evidence.