import type { CoxeterDiagram } from './diagram.js';
import type { ExactRing, ExactValue } from './exact.js';
/**
 * A finite Coxeter group as a chamber-action graph: elements are integer
 * chamber IDs (0 = identity) and the only stored operation is right
 * multiplication by a generator. This is the group's source of truth —
 * matrices, rotors, and coordinates are all derived, never compared.
 */
export interface CoxeterAction {
    readonly diagram: CoxeterDiagram;
    readonly rank: number;
    readonly order: number;
    /**
     * leftMultiply[g * rank + i] = the element sᵢ·g — the point-reflection
     * table produced directly by the BFS (reflecting a chamber's point in
     * simple mirror i is left multiplication).
     */
    readonly leftMultiply: Uint16Array | Uint32Array;
    /**
     * rightMultiply[g * rank + i] = the element g·sᵢ. Right multiplication
     * by stabilizer generators closes **left cosets** g·W_S — exactly the
     * face translates of the Wythoff construction.
     */
    readonly rightMultiply: Uint16Array | Uint32Array;
    /** BFS tree: s(parentGenerator[g])·parent[g] = g; identity points at itself. */
    readonly parent: Uint16Array | Uint32Array;
    readonly parentGenerator: Uint8Array;
    /** (−1)^wordLength — the determinant of the reflection product. */
    readonly parity: Int8Array;
}
/**
 * Reflection in mirror `i`, acting on a signed mirror-distance tuple:
 * d′ⱼ = dⱼ − C[j][i]·dᵢ with C the doubled Gram matrix. Exact — this is
 * the entire geometric kernel of the enumeration.
 */
export declare function reflectDistances(point: readonly ExactValue[], mirror: number, twoGram: ReadonlyArray<ReadonlyArray<ExactValue>>, ring: ExactRing): ExactValue[];
/**
 * Enumerates the group by exact BFS on the orbit of a chamber-interior
 * point (all mirror distances 1): its stabilizer is trivial, so orbit
 * points correspond one-to-one with group elements. Throws if the count
 * disagrees with the diagram's declared order — the enumeration cannot
 * silently drift.
 */
export declare function enumerateCoxeterAction(diagram: CoxeterDiagram): CoxeterAction;
/**
 * The orbit of an arbitrary exact mirror-distance tuple — the Wythoff
 * vertex set for a seed built from a ring pattern (1 on ringed mirrors,
 * 0 on unringed ones). Unlike chamber enumeration the stabilizer may be
 * nontrivial, so the orbit is a quotient: |orbit| = order / |stabilizer|.
 * Returns the distinct tuples in BFS order.
 */
export declare function orbitDistanceTuples(diagram: CoxeterDiagram, seed: readonly ExactValue[]): ExactValue[][];
/**
 * The standard Wythoff seed for a ring pattern: distance 1 from every
 * ringed (active) mirror, 0 from the rest — every generator edge then
 * has exact length 2·1 in the realization.
 */
export declare function wythoffSeed(diagram: CoxeterDiagram, rings: readonly boolean[]): ExactValue[];
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