import { BivectorN } from '../math/bivector.js';
import { Rotor4 } from '../math/rotor4.js';
export type BaseMap<Base> = (state: Base) => Base;
export type RotorCocycle<Base> = (state: Base) => BivectorN;
export interface SkewProductFlowOptions<Base> {
    readonly id?: string;
    readonly baseMap: BaseMap<Base>;
    readonly cocycle: RotorCocycle<Base>;
    /** Scalar epsilon in `exp(epsilon B(x))`. Default 1. */
    readonly coupling?: number;
}
export interface SkewProductState<Base> {
    readonly base: Base;
    readonly fiber: Rotor4;
    readonly step: number;
}
/**
 * A discrete compact-group extension `(x,R) -> (f(x), exp(epsilon B(x)) R)`.
 *
 * The base map owns its state representation. The fiber is always a `Rotor4`,
 * so each increment is an SO(4) isometry and pair-quaternion normalization is
 * the complete drift repair after composition.
 */
export declare class SkewProductFlow<Base> {
    readonly id: string;
    readonly baseMap: BaseMap<Base>;
    readonly cocycle: RotorCocycle<Base>;
    readonly coupling: number;
    constructor({ id, baseMap, cocycle, coupling }: SkewProductFlowOptions<Base>);
    initial(base: Base, fiber?: Rotor4): SkewProductState<Base>;
    /** The state-dependent fiber increment `exp(epsilon B(x))`. */
    increment(base: Base): Rotor4;
    step(state: SkewProductState<Base>): SkewProductState<Base>;
    iterate(state: SkewProductState<Base>, count: number): SkewProductState<Base>;
}
/** Cover-independent Frobenius distance from a rotor's SO(4) matrix to identity. */
export declare function rotorIdentityResidual(rotor: Rotor4): number;
export interface PeriodicOrbitHolonomyOptions<Base> {
    readonly period: number;
    readonly baseDistance: (left: Base, right: Base) => number;
    /** Maximum closure error accepted for the supplied periodic orbit. Default 1e-12. */
    readonly closureTolerance?: number;
    /** SO(4) identity residual below which holonomy counts as identity. Default 1e-12. */
    readonly identityTolerance?: number;
}
export interface PeriodicOrbitHolonomy<Base> {
    readonly flow: string;
    readonly period: number;
    readonly orbit: readonly Base[];
    readonly returnedBase: Base;
    readonly closureError: number;
    readonly closureTolerance: number;
    readonly closed: boolean;
    readonly holonomy: Rotor4;
    readonly identityResidual: number;
    readonly identityTolerance: number;
    readonly nontrivial: boolean;
    /** A finite obstruction to the cocycle being a coboundary on this base system. */
    readonly essentialWitness: boolean;
}
/** Ordered fiber product around a claimed periodic base orbit. */
export declare function periodicOrbitHolonomy<Base>(flow: SkewProductFlow<Base>, initialBase: Base, { period, baseDistance, closureTolerance, identityTolerance }: PeriodicOrbitHolonomyOptions<Base>): PeriodicOrbitHolonomy<Base>;
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