Modules
The library root imports every module below.
| Module | Contents |
|---|---|
ScalarPitchfork.Cubic |
The exact cubic $\alpha\tau A-\beta A^3$: nonzero zeros $\pm\sqrt{\alpha\tau/\beta}$ on the side $\tau\beta>0$, derivative $-2\alpha\tau$ there |
ScalarPitchfork.Perturbed |
At a given nonzero zero of a perturbed cubic: side, derivative sign, square-root amplitude error |
ScalarPitchfork.Existence |
The field pitchforkField; both signed branches exist under the remainder bounds (intermediate value theorem), side and stability of every small zero |
ScalarPitchfork.Branch |
Uniqueness in a fixed window, classification of small zeros, continuity of the branch functions; packaged in pitchfork_branch_curves |
ScalarPitchfork.StrongRemainder |
The stronger remainder bound implies the weak one; the branch theorems transfer |
ScalarPitchfork.Jet |
PitchforkJet: $C^4$, the axis condition and five vanishing jets give the remainder bounds; the stronger bound holds iff $\partial_zR(\tau,0)=0$; branch theorems from jets |
ScalarPitchfork.JetCounterexample |
$R=\tau^2z$: the jets do not give the stronger bound; the axis condition from an exact linearization |
ScalarPitchfork.Global.Alternatives |
The three global alternatives as hypotheses: elimination, closedness, unbounded parameter, connectedness propagation, exclusion for strictly monotone profiles |
ScalarPitchfork.Global.ComponentClosure |
Parent component, one-sided remainder and selected component of a solution set; closedness |