import { type QuadraticIterationOptions, type ResolvedQuadraticIterationOptions } from './complex-quadratic.js';
import { type FieldEvaluation4, type ImplicitField4, type Vec4f64 } from './types.js';
export interface QuaternionJuliaOptions extends QuadraticIterationOptions {
    /** Quaternion parameter in [i,j,k,real] order. */
    parameter: Vec4f64;
}
export interface QuaternionJuliaEvaluation extends FieldEvaluation4 {
    readonly derivativeBound: number;
    readonly continuousIteration: number;
}
/** Rotation fixing the (real,i) plane and turning the (j,k) plane. */
export declare function rotateQuaternionJuliaSymmetry(point: ArrayLike<number>, angle: number): Vec4f64;
/** Quadratic quaternion Julia field q -> q^2 + c. */
export declare class QuaternionJuliaField implements ImplicitField4<QuaternionJuliaEvaluation> {
    readonly id = "quaternion-julia-quadratic";
    readonly parameter: Vec4f64;
    readonly options: ResolvedQuadraticIterationOptions;
    readonly symmetries: readonly [{
        readonly id: "jk-circle";
        readonly description: "For parameters in the (real,i) plane, rotations of the (j,k) plane commute with iteration.";
    }];
    readonly sliceTheorems: readonly [{
        readonly id: "complex-plane";
        readonly description: "Restricting to the (real,i) plane reproduces the complex quadratic Julia iteration.";
    }, {
        readonly id: "symmetry-plane-suspension";
        readonly description: "Every hyperplane containing the (real,i) plane has the same section up to the circle symmetry.";
    }];
    readonly distanceEstimator: {
        readonly certificate: "provenNDA";
        readonly description: "Normed-division-algebra escape estimate using an upper bound on the iterate derivative.";
        readonly recommendedStepSafety: 0.72;
    };
    constructor({ parameter, ...iterationOptions }: QuaternionJuliaOptions);
    evalCPU(point: ArrayLike<number>): QuaternionJuliaEvaluation;
}
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