ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

An interval containing zero makes a Hopf certificate inconclusive

Example

Use the planar cubic system $\dot{x} = μ x - y + \frac{1}{200} x (x^{2} + y^{2})$ , $\dot{y} = x + μ y + \frac{1}{200} y (x^{2} + y^{2})$ . Give its exact first Lyapunov coefficient the valid rational enclosure $[- 1 / 10, 1 / 10]$ , while all other certificate fields are exact or have strict enclosures.

Facts & Assumptions

Given: The system in the Example, $ω_{0} = 1$ , its exact equilibrium and eigenpair records, and the stated proof-carrying interval certificate.

[F1]

Hopf spectral and multilinear normal-form data: with $q^{*} q = p^{*} q = 1$ , $ℓ_{1} = Re G_{21} / (2 ω_{0})$ .

[F2]

Validate a proof-carrying local Hopf certificate: the procedure returns inconclusive if the certified interval for $ℓ_{1}$ intersects zero.

Verification

1.1F1givenalgebra

The Jacobian has eigenvalues $μ \pm i$ , so $β^{'} = 1$ ; the normalized eigenvectors are $q = p = (1, - i)^{T} / \sqrt{2}$ . The quadratic derivative vanishes. The cubic coefficient has $c = 1 / 200$ , so $C (q, q, \overset{ˉ}{q}) = 4 c q$ , $G_{21} = 4 c = 1 / 50$ , and $ℓ_{1} = 1 / 100$ , which lies in $[- 1 / 10, 1 / 10]$ .

2.1F2step 1.1given∎

Every other field in the input record is valid and strictly separated as assumed, but the interval $[- 1 / 10, 1 / 10]$ contains zero. Hence the interval for $ℓ_{1}$ fails the strict nonzero test, and [F2] returns inconclusive rather than asserting a Hopf branch.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Yuri A. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd ed., Springer (standard reference, not scraped)
J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Springer-Verlag, 1976 (standard reference, not scraped)
Yuri A. Kuznetsov, Andronov-Hopf Bifurcation, Scholarpedia 1(10):1858 (2006) (standard reference, not scraped)