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.

A constant imperfection removes the pitchfork's trivial branch

Example

For a fixed $ε > 0$ , consider the perturbed equilibrium equation $F (x, μ, ε) = x^{3} + μ x + ε = 0.$ This is a symmetry-breaking perturbation of the odd pitchfork equation $x^{3} + μ x = 0$ .

Here $ε^{1 / 3}$ denotes the unique positive real number whose cube is $ε$ .

Facts & Assumptions

Given: $F (x, μ, ε) = x^{3} + μ x + ε$ with fixed $ε > 0$ .

[F1]

Scalar fold, transcritical, and symmetric pitchfork equilibrium branches: a pitchfork at $x = 0$ requires the trivial branch $F (0, μ) = 0$ and odd symmetry in $x$ .

[F2]

The parametrized implicit function theorem with $C^{k}$ regularity: if $F_{x}$ is nonzero at an equilibrium, there is a unique smooth local equilibrium branch through it.

Verification

1.1F1givenalgebra

We have $F (0, μ, ε) = ε \neq 0$ , and $F (- x, μ, ε) + F (x, μ, ε) = 2 ε \neq 0$ . Thus both the trivial-branch and odd-symmetry hypotheses in [F1] fail at the former pitchfork point.

2.1F2givenalgebra∎

At $μ = 0$ , $x_{*} = - ε^{1 / 3}$ is an equilibrium and $F_{x} (x_{*}, 0, ε) = 3 ε^{2 / 3} > 0$ . By [F2] it continues as a unique smooth simple equilibrium branch near $(x_{*}, 0)$ ; the equilibrium at the origin has disappeared.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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)