import { type ExactValue } from '../coxeter/exact.js';
import { type DoubledIcosian } from '../lattice/e8.js';
import type { ModelPoint } from '../lattice/model-set.js';
import type { Vec4f64 } from '../field/types.js';
import type { Decoration, DecorationEquivarianceGenerator } from './decoration.js';
export type ExactVector4 = readonly [ExactValue, ExactValue, ExactValue, ExactValue];
/** An exact four-dimensional parameter and its single Float64 rendering. */
export interface ExactParameter4 {
    readonly exact: ExactVector4;
    readonly denominator: bigint;
    readonly value: Vec4f64;
}
/** Canonical model-set decoration: retain the coordinate discarded by physical rendering. */
export declare const elserSloanePerpendicularParameterDecoration: Decoration<ModelPoint, ExactParameter4>;
/** Exact key for a parameter, including its coordinate scale. */
export declare function exactParameter4Key(parameter: ExactParameter4): string;
export declare function exactParameter4Equals(left: ExactParameter4, right: ExactParameter4): boolean;
/** Exact key for a doubled icosian source point. */
export declare function doubledIcosianKey(point: DoubledIcosian): string;
/** Reflection in a unit H4 root, with both point and root stored in doubled coordinates. */
export declare function reflectDoubledIcosian(point: DoubledIcosian, root: DoubledIcosian): DoubledIcosian;
/** Apply Galois conjugation coordinate-wise. */
export declare function galoisTwistIcosian(point: DoubledIcosian): DoubledIcosian;
/** The perpendicular-space parameter of an exact icosian source `x`: `c=x*`. */
export declare const elserSloaneGermParameterDecoration: Decoration<DoubledIcosian, ExactParameter4>;
/** Four deterministic exact simple roots with Coxeter links H4 = 5-3-3. */
export declare function elserSloaneH4ReflectionGenerators(): readonly DoubledIcosian[];
/** H4 on physical provenance paired with its Galois-conjugate internal action. */
export declare function elserSloaneDecorationGenerators(): readonly DecorationEquivarianceGenerator<DoubledIcosian, ExactParameter4>[];
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