import { type Complex2, type ComplexQuadraticEvaluation, type QuadraticIterationOptions, type ResolvedQuadraticIterationOptions } from './complex-quadratic.js';
import { type FieldEvaluation4, type ImplicitField4, type Vec4f64 } from './types.js';
export interface BicomplexJuliaOptions extends QuadraticIterationOptions {
    /** Parameter in [i1,i2,i1*i2,real] order. */
    parameter: Vec4f64;
}
export interface BicomplexIdempotentPair {
    readonly first: Complex2;
    readonly second: Complex2;
}
export interface BicomplexJuliaEvaluation extends FieldEvaluation4 {
    readonly factors: readonly [ComplexQuadraticEvaluation, ComplexQuadraticEvaluation];
}
/** Exact linear change from [i1,i2,i1*i2,real] to the two complex factors. */
export declare function bicomplexToIdempotent(point: ArrayLike<number>): BicomplexIdempotentPair;
/** Inverse idempotent transform in [i1,i2,i1*i2,real] order. */
export declare function idempotentToBicomplex(first: Complex2, second: Complex2): Vec4f64;
/** Quadratic bicomplex Julia field, evaluated as two independent complex factors. */
export declare class BicomplexJuliaField implements ImplicitField4<BicomplexJuliaEvaluation> {
    readonly id = "bicomplex-julia-quadratic";
    readonly parameter: Vec4f64;
    readonly factorParameters: BicomplexIdempotentPair;
    readonly options: ResolvedQuadraticIterationOptions;
    readonly symmetries: readonly [{
        readonly id: "idempotent-product";
        readonly description: "The bicomplex quadratic map is the direct product of two complex quadratic maps.";
    }];
    readonly sliceTheorems: readonly [{
        readonly id: "factor-diagonalization";
        readonly description: "In idempotent coordinates every slice is an affine slice of a product of complex Julia sets.";
    }];
    readonly distanceEstimator: {
        readonly certificate: "provenProduct";
        readonly description: "Per-factor complex estimates combined in the orthogonal product metric.";
        readonly recommendedStepSafety: 0.2;
    };
    constructor({ parameter, ...iterationOptions }: BicomplexJuliaOptions);
    evalCPU(point: ArrayLike<number>): BicomplexJuliaEvaluation;
}
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