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Counterexamples

Search for a candidate violation and rigorously re-check a supplied witness. Certification applies to the declared function, property, point, and assumptions.

geometric_function_atlas.counterexamples

Certified re-evaluation of supplied counterexample witnesses.

CounterexampleResult dataclass

Result of checking one claimed violation point.

property: str instance-attribute

coefficients: tuple[float, ...] instance-attribute

point: tuple[float, float] instance-attribute

interval_lower: float instance-attribute

interval_upper: float instance-attribute

threshold: float instance-attribute

certified: bool instance-attribute

__init__(property: str, coefficients: tuple[float, ...], point: tuple[float, float], interval_lower: float, interval_upper: float, threshold: float, certified: bool) -> None

direction() -> str

to_dict() -> dict[str, Any]

WitnessSearchResult dataclass

Outcome of a grid search followed by interval certification.

property: str instance-attribute

coefficients: tuple[float, ...] instance-attribute

certified: bool instance-attribute

point: tuple[float, float] | None instance-attribute

interval_lower: float | None instance-attribute

interval_upper: float | None instance-attribute

threshold: float instance-attribute

margin: float | None instance-attribute

screen_value: float instance-attribute

grid_points: int instance-attribute

__init__(property: str, coefficients: tuple[float, ...], certified: bool, point: tuple[float, float] | None, interval_lower: float | None, interval_upper: float | None, threshold: float, margin: float | None, screen_value: float, grid_points: int) -> None

to_dict() -> dict[str, Any]

verify_counterexample(coefficients: Sequence[float], *, point: tuple[float, float], property: str = 'starlike') -> CounterexampleResult

Rigorously check a claimed pointwise counterexample.

coefficients are [a2, a3, ...] for f(z) = z + a2*z**2 + a3*z**3 + .... Supported violations are:

  • starlike: Re(z*f'(z)/f(z)) <= 0;
  • becker_univalent: (1-|z|^2)|f''(z)/f'(z)| > 1;
  • nehari_univalent: (1-|z|^2)^2|S_f(z)| > 2.

find_counterexample(coefficients: Sequence[float], *, property: str = 'starlike', witness_hint: tuple[float, float] | None = None, grid_r: int = 120, grid_theta: int = 180) -> WitnessSearchResult

Search for a violation point and certify it in interval arithmetic.

coefficients are [a2, a3, ...] for f(z) = z + a2*z**2 + .... A float grid locates the most-violating point; the final verdict is certified only when the interval enclosure at that exact point clears the threshold. witness_hint, when supplied, is tried before the grid candidate.