One rhythm at large gain

Summary

A feedback system can oscillate without having a unique rhythm. Different starting conditions might lead to different repeating cycles. Finding one stable cycle on a computer does not rule out another elsewhere.

This candidate settles that question for an explicit large-gain region of the Wilhelm–Heinrich oscillator. In that region, every possible positive cycle is attracting, and there is exactly one. Positive solutions converge to it unless they lie on the equilibrium's stable manifold.

The proof also reveals useful geometry: the long-term positive dynamics of this three-variable system lie on a graph over a plane. Above the oscillation threshold, the graph's domain is a disk. The graph is generated by a convex function; it is not itself claimed to be a convex subset of three-dimensional space.

The full uniqueness conjecture remains open. This is an unrefereed mathematical candidate, with internal checking and a reproducible computational package—not an accepted solution for every parameter.

Summary for specialists

For positive physical rates, the sufficient condition is

$$r_0\ge 1000\,\frac{\max(a_0,b_0)^2}{\min(a_0,b_0)}.$$

With $\mu=a_0/b_0$ and $k=a_0r_0/b_0^2$, this becomes

$$k\ge1000\max(1,\mu^3).$$

Every nonconstant positive periodic orbit in this region is hyperbolic and attracting. For $0<\mu\le1$, the estimates include $\langle\eta\rangle\le0.83\mu$, hence weak exponent $\lambda_c\le-0.17\mu$ in the paper's normalized time. The reciprocal-rate correspondence transports cycles and multipliers, not the entire positive domains.

For all positive parameters, the positive-interior attractor is the graph of a convex function, $C^1$ in the interior of its planar domain. Only above $k_H=\mu(1+\mu)$ is that domain a closed Jordan disk bounded by the outermost projected cycle. At and below Hopf, the attractor is the equilibrium. These statements do not describe the extinction equilibrium and its connections in the closed nonnegative phase space.

Technical account

The positive adjoint supplies supporting planes and exact transverse multipliers for a cycle of normalized period $T$:

$$m_s=e^{-(1+\langle\eta\rangle)T},\qquad m_t=e^{(\langle\eta\rangle-\mu)T}.$$

Their product is $e^{-(1+\mu)T}$, but a product below one does not show that both factors are below one. The load-bearing estimate is the separate weak-exponent sign. The large-gain argument controls all cycles, not just the orbit found by a numerical solver; the planar reduction then converts all-cycle attraction into uniqueness.

Strict supporting inequalities extend from cycles to bounded entire trajectories. They yield a convex-function graph with reduced divergence $\eta-\mu$. Each projected cycle is strictly convex and star-shaped. Exact formulas involving two positive convexity gauges provide further ways to express the unresolved stability sign.

The revised small-rate-ratio analysis derives its formerly assumed local slow-graph property from Eldering's invariant-manifold persistence theorem. A cutoff leaves the original dynamics unchanged near a prescribed compact set; adjoining a frozen parameter gives joint regularity down to the singular limit. This establishes the needed local graph, not global confinement. The resulting wedge theorem still concerns a fixed energy ball and has a nonexplicit small-parameter threshold.

A separate kurtosis inequality for strictly convex potentials is a reusable mathematical result, rather than only an oscillator-specific calculation.

Evidence, assurance and limitations

The revised full numerical suite passed in the pinned environment, including 198 exact symbolic identities and regeneration of 31 saved cycle approximations. Other checks cover geometric identities, multipliers, large-gain inequalities and deliberately false assertions. The package records which historical tables and exploratory scripts are not regenerated.

An added exact-rational interval audit encloses 56 selected scalar comparisons, including the bound behind $0.83$. These are endpoint comparisons, not 56 independent proofs. They do not certify trajectories, sampled monotonicity grids or the analytic reductions that connect the constants to every cycle.

The supplied review audit had a failed aggregate receipt caused by a dependency-version mismatch despite passing scientific diagnostics. That receipt remains unchanged. A separate historical small-rate-ratio quadrature comparison also remains unresolved; a successful alternative integration does not erase it.

There is no external specialist review, unaffiliated reproduction or end-to-end formal verification. Historical priority is not established. In particular, the full text of Li's 1981 third-order uniqueness work remains an applicability gap. The parameter region between the Hopf collars and the explicit large-gain threshold is not exhausted by the computations.

Relationship to earlier work

Smith established permanence, finiteness of the cycle set above Hopf, and convergence to a cycle outside the equilibrium's stable manifold. Boros and Hofbauer summarize those results for the original model; their mass-conserving extension is a different system. The new claim here is an explicit large-gain uniqueness region together with the detailed attractor and multiplier framework.

Baigent's convex carrying-simplex criterion and Mierczyński's regularity theorem are close geometric antecedents. Their stated competitive-map hypotheses include coordinate-face invariance. The Wilhelm–Heinrich flow in physical coordinates does not have that property, so those theorems do not directly give the present result. This is a hypothesis comparison, not proof that no older theorem or change of coordinates could apply.

Who should care, and why

Why the problem matters

A model with one attracting rhythm is easier to interpret than one whose eventual rhythm depends on the initial state. But that distinction requires a statement about every possible cycle. The main contribution is a route from geometric structure and universal bounds to such a statement in a concrete region—not a larger collection of simulated examples.

How to inspect or reproduce the recorded checks

Start with AI_INDEX.md, then the manuscript's short reading route and CLAIMS.json. Use the pinned Python environment and run python code/run_all.py --regenerate in a fresh copy: the driver rewrites receipts and generated tables. Run python code/rational_interval_audit.py for the selected outward bounds, and python code/revision_controls.py normally and with -O for the added rejection controls.

The manifest checks bytes, not theorem truth. Read SOURCES.md for source-access depth and REVIEW_RESPONSE.md for every supplied review disposition. The local slow-graph proof and its internal source mapping are separately indexed.

The most valuable next projects

First, obtain specialist scrutiny of the attractor and large-gain arguments and close the remaining source-applicability gaps. Next, make the Hopf collar explicit and lower the large-gain threshold. A validated middle-range analysis would need enclosures of all relevant cycles, rather than simulations of selected initial states. The small-rate-ratio endpoint requires further analytic control: the local energy-ball result does not exclude cycles outside that ball.

What is in the evidence package

The package contains the complete manuscript and editable LaTeX, a claim ledger, source notes, review dispositions, the original numerical suite, the additional rational-interval checker, deliberate-error controls, receipts, historical failure records and a substantive AI index. Original research text and data use CC0-1.0; original code uses MIT, with provenance exclusions documented in LICENSES.md. The banner and audio explain the result; neither adds scientific evidence.

@misc{wilhelmheinrichlargegain2026,
  title        = {One rhythm at large gain},
  author       = {Anonymous},
  year         = {2026},
  doi          = {10.5281/zenodo.23126164},
  url          = {https://doi.org/10.5281/zenodo.23126164},
  version      = {3.1.0-candidate},
  howpublished = {Zenodo},
  note         = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/wilhelm-heinrich-large-gain/}
}