/**
 * Exact arithmetic for Coxeter group enumeration.
 *
 * Group elements and vertex orbits are identified by exact
 * mirror-distance tuples — never by Float64 comparison — so enumeration
 * cannot invent or merge chambers. The unit-normal mirror basis keeps
 * the required constants tiny: every doubled-Gram entry is 2, 0, −1,
 * −√2, or −φ, so three quadratic rings cover all built-in groups:
 *
 *   ℤ        (A, D families — simply laced)
 *   ℤ[√2]    (B, F families — the 4-marked link)
 *   ℤ[φ]     (H families — the 5-marked link, φ² = φ + 1)
 *
 * Elements are pairs (a, b) of bigints meaning a + b·ρ for the ring's
 * radical ρ; the integer ring simply keeps b = 0.
 */
/** An element a + b·ρ of one of the supported quadratic rings. */
export interface ExactValue {
    readonly a: bigint;
    readonly b: bigint;
}
export type ExactRingKind = 'integer' | 'sqrt2' | 'phi';
export interface ExactRing {
    readonly kind: ExactRingKind;
    readonly zero: ExactValue;
    readonly one: ExactValue;
    fromInt(n: number): ExactValue;
    add(x: ExactValue, y: ExactValue): ExactValue;
    sub(x: ExactValue, y: ExactValue): ExactValue;
    neg(x: ExactValue): ExactValue;
    mul(x: ExactValue, y: ExactValue): ExactValue;
    /** The radical ρ itself (√2 or φ); throws for the integer ring. */
    radical(): ExactValue;
    /** Exact hash key; equal keys ⇔ equal ring elements. */
    key(x: ExactValue): string;
    keyTuple(xs: readonly ExactValue[]): string;
    /** Exact order in the selected real embedding: −1, 0, or +1. */
    sign(x: ExactValue): -1 | 0 | 1;
    compare(x: ExactValue, y: ExactValue): -1 | 0 | 1;
    /** One-time conversion to Float64 (a + b·ρ numerically). */
    toNumber(x: ExactValue): number;
}
/** ℤ: ρ unused (b stays 0). */
export declare const integerRing: ExactRing;
/** ℤ[√2]: ρ² = 2. */
export declare const sqrt2Ring: ExactRing;
/** ℤ[φ]: ρ² = ρ + 1 (the golden ratio). */
export declare const phiRing: ExactRing;
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