import { type ExactValue } from '../coxeter/exact.js';
import { CellComplex } from '../geometry/cell-complex.js';
import { type DoubledE8Vector, type DoubledIcosian } from './e8.js';
import { ConvexWindow, ModelSet, type ExactHalfspace, type ModelPoint, type WindowBoundaryPolicy, type WindowLocation } from './model-set.js';
/**
 * Integer action of multiplication by phi on coefficients in the fixed
 * E8 root basis used by `createElserSloaneModelSet`.
 */
export declare const elserSloaneInflationMatrix: ReadonlyArray<ReadonlyArray<bigint>>;
/**
 * The canonical 720 vertices of the Elser-Sloane acceptance window.
 *
 * Coordinates are scaled by `6/kappa`, where the common orthogonal-projection
 * normalizer cancels from every membership decision. With doubled icosian
 * input coordinates this means points are compared as `3 x*`.
 */
export declare function elserSloaneWindowVertices(): readonly (readonly ExactValue[])[];
/**
 * Exact H-description of the four-dimensional Elser-Sloane window.
 *
 * The ten seeds are the H4 orbits of the 1,200 supporting hyperplanes of
 * the 720-vertex hull. Every inequality is `normal dot x <= 12/phi` in
 * the same `6/kappa` coordinate convention as `elserSloaneWindowVertices`.
 */
export declare function elserSloaneWindowHalfspaces(): readonly ExactHalfspace[];
export declare function createElserSloaneWindow(): ConvexWindow;
export type ElserSloanePhasonOffset = readonly [
    ExactValue,
    ExactValue,
    ExactValue,
    ExactValue
];
export interface ElserSloaneModelSetOptions {
    boundaryPolicy?: WindowBoundaryPolicy;
    /** Exact internal translation in eleventh-coordinate units. */
    phasonOffsetElevenths?: ElserSloanePhasonOffset;
}
/** Galois conjugation `a+b phi -> (a+b)-b phi`. */
export declare function phiConjugate(value: ExactValue): ExactValue;
/** Fixed unimodular E8 root basis used for bounded coefficient enumeration. */
export declare function elserSloaneLatticeBasis(): readonly (readonly ExactValue[])[];
/** Coordinates of an icosian in the fixed E8 root basis. */
export declare function elserSloaneCoefficients(root: DoubledIcosian): readonly bigint[];
/** The canonical E8 -> R4 + R4 cut-and-project scheme in exact icosian coordinates. */
export declare function createElserSloaneModelSet(options?: WindowBoundaryPolicy | ElserSloaneModelSetOptions): ModelSet;
/** Exact E8 lattice automorphism: phi in physical space, 1-phi internally. */
export declare function elserSloaneInflate(coefficients: readonly bigint[]): readonly bigint[];
/** Inverse inflation; the identity S^-1 = S-I follows from S^2 = S+I. */
export declare function elserSloaneDeflate(coefficients: readonly bigint[]): readonly bigint[];
export interface ElserSloanePatchOptions {
    /** Symmetric coefficient box `[-r,r]^8`. Default 1; radius 2 has 390,625 candidates. */
    coefficientRadius?: number;
    /** Optional radial crop in physical R4 after exact window selection. */
    physicalRadius?: number;
    /** Candidate safety cap passed to `ModelSet.sample`. */
    maxCandidates?: number;
    boundaryPolicy?: WindowBoundaryPolicy;
    /** Exact internal translation in eleventh-coordinate units. */
    phasonOffsetElevenths?: ElserSloanePhasonOffset;
}
export interface ElserSloaneNormPatchOptions {
    /** Complete E8 shells through this quadratic norm. Default 6. */
    maxE8Norm?: number;
    /** Optional radial crop in physical R4 after exact window selection. */
    physicalRadius?: number;
    boundaryPolicy?: WindowBoundaryPolicy;
    /** Exact internal translation in eleventh-coordinate units. */
    phasonOffsetElevenths?: ElserSloanePhasonOffset;
}
export interface ElserSloanePatch {
    readonly points: readonly ModelPoint[];
    readonly candidateCount: number;
    readonly acceptedBeforePhysicalCrop: number;
    readonly boundaryCount: number;
}
/**
 * A deterministic finite coefficient-box sample of the infinite model set.
 *
 * @param options - Coefficient bound, crop, safety cap, boundary policy, and phason.
 */
export declare function elserSloanePatch(options?: ElserSloanePatchOptions): ElserSloanePatch;
/**
 * A symmetry-preserving finite sample built from complete ambient E8 shells.
 * This is the preferred bounded source for explanatory renders.
 *
 * @param options - Shell bound, crop, boundary policy, and exact phason.
 */
export declare function elserSloaneNormPatch(options?: ElserSloaneNormPatchOptions): ElserSloanePatch;
/** Exact section by the fourth physical coordinate; zero is the icosahedral section. */
export declare function elserSloaneSection(points: readonly ModelPoint[], coordinate?: ExactValue): readonly ModelPoint[];
/** Exact metric-nearest graph within one fixed-coordinate 3D section. */
export declare function elserSloaneSectionEdges(points: readonly ModelPoint[]): Uint32Array;
/**
 * Integer Galois product for a point difference in the unnormalized icosian
 * coordinates: |delta_parallel|^2 |delta_perp|^2. The orthonormal E8
 * projection convention multiplies this integer by 1/20.
 */
export declare function elserSloaneGaloisProduct(left: ModelPoint, right: ModelPoint): bigint;
/** Internal coordinate `6 x*` for a doubled icosian representing `2x`. */
export declare function elserSloaneInternalCoordinate(root: DoubledIcosian): readonly ExactValue[];
export declare function classifyElserSloaneIcosian(root: DoubledIcosian): WindowLocation;
export type ElserSloaneSourceShell = 'root' | 'second-shell';
export interface ElserSloaneGermPoint {
    readonly sourceShell: ElserSloaneSourceShell;
    readonly standard: DoubledE8Vector;
    readonly icosian: DoubledIcosian;
    readonly windowLocation: 'inside' | 'boundary';
    readonly parallel: Float64Array;
    readonly perpendicular: Float64Array;
}
export interface ElserSloaneGerm {
    readonly points: readonly ElserSloaneGermPoint[];
    /** Exact metric skeletons of the accepted root and norm-4 600-cells. */
    readonly edges: Uint32Array;
    readonly rootCount: number;
    readonly secondShellCount: number;
    readonly boundaryCount: number;
}
/** The exactly accepted norm-2 and norm-4 germ of the canonical model set. */
export declare function elserSloaneGerm(): ElserSloaneGerm;
export interface ElserSloaneGermComplexOptions {
    /** Which Galois embedding supplies the R4 positions. Default `parallel`. */
    embedding?: 'parallel' | 'perpendicular';
    /** Uniform scale after embedding. Default 1. */
    scale?: number;
    /** Accepted source shell whose exact metric edges are included. Default `both`. */
    sourceShell?: ElserSloaneSourceShell | 'both';
}
/**
 * A renderable 4D view of the two accepted 600-cell shells.
 *
 * @param options - Embedding, scale, and source shells to include.
 */
export declare function createElserSloaneGermComplex(options?: ElserSloaneGermComplexOptions): CellComplex;
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