import { VecN } from '@holotope/core';
export interface AnalyzeLinearSimplexMeasureNOptions {
    /** Non-degenerate k-simplex rest state, with `1 <= k <= ambientDimension`. */
    readonly restPositions: readonly VecN[];
    /** Particle positions at normalized trajectory time zero. */
    readonly startPositions: readonly VecN[];
    /** Particle positions at normalized trajectory time one. */
    readonly endPositions: readonly VecN[];
    /** Required intrinsic current/rest k-measure ratio. */
    readonly minimumMeasureRatio: number;
    /** Smallest unresolved normalized-time interval. Default 2^-30. */
    readonly timeTolerance?: number;
    /** Hard de Casteljau subdivision depth. Default 48. */
    readonly maximumDepth?: number;
    /** Relative Bernstein-coefficient tolerance. Default 256 * Number.EPSILON. */
    readonly relativeCoefficientTolerance?: number;
}
/**
 * Evidence shared by every outcome of one linear simplex-measure analysis.
 *
 * The squared measure ratio along a linear trajectory is a polynomial in
 * normalized time, so the question "does this simplex stay above its minimum
 * measure" is a polynomial threshold question. These fields record the
 * polynomial that was analysed and the tolerances it was analysed under,
 * which is what makes any outcome reproducible from the record alone.
 */
export interface LinearSimplexMeasureAnalysisBaseN {
    /** Euclidean dimension the simplex vertices live in. */
    readonly ambientDimension: number;
    /** Intrinsic simplex dimension `k`, counting one less than its vertices. */
    readonly simplexDimension: number;
    /** Degree of the squared-ratio polynomial in normalized time. */
    readonly degree: number;
    /** Required intrinsic current/rest k-measure ratio, as authored. */
    readonly minimumMeasureRatio: number;
    /** Its square, which is the threshold actually compared against. */
    readonly minimumSquaredMeasureRatio: number;
    /** Squared measure ratio at normalized time zero. */
    readonly startSquaredMeasureRatio: number;
    /** Squared measure ratio at normalized time one. */
    readonly endSquaredMeasureRatio: number;
    /** Coefficients of `squaredMeasureRatio(t) - minimumMeasureRatio^2`. */
    readonly monomialCoefficients: Float64Array;
    /** The same squared-ratio threshold polynomial in Bernstein form on [0,1]. */
    readonly bernsteinCoefficients: Float64Array;
    /** Smallest normalized-time interval the subdivision will not split. */
    readonly timeTolerance: number;
    /** Hard de Casteljau subdivision depth, bounding the search regardless. */
    readonly maximumDepth: number;
    /** Bernstein-coefficient tolerance relative to the coefficient scale. */
    readonly relativeCoefficientTolerance: number;
    /** Absolute Bernstein-coefficient tolerance derived from that scale. */
    readonly absoluteCoefficientTolerance: number;
}
export interface LinearSimplexMeasureSafeN extends LinearSimplexMeasureAnalysisBaseN {
    readonly status: 'safe';
    /** Conservative lower bound on the squared-ratio margin. */
    readonly minimumMarginLowerBound: number;
}
export interface LinearSimplexMeasureInitialViolationN extends LinearSimplexMeasureAnalysisBaseN {
    readonly status: 'initial-violation';
    /** Initial squared-ratio margin. */
    readonly initialMargin: number;
}
export interface LinearSimplexMeasurePossibleViolationN extends LinearSimplexMeasureAnalysisBaseN {
    readonly status: 'possible-violation';
    /** Earliest conservative normalized-time enclosure that could meet the threshold. */
    readonly timeBracket: readonly [number, number];
    readonly candidateTime: number;
    readonly marginAtBracketStart: number;
    readonly marginAtBracketEnd: number;
    readonly bernsteinBounds: readonly [number, number];
    readonly reason: 'negative-enclosure' | 'resolution-limit';
}
export type LinearSimplexMeasureAnalysisN = LinearSimplexMeasureSafeN | LinearSimplexMeasureInitialViolationN | LinearSimplexMeasurePossibleViolationN;
/**
 * Conservatively checks intrinsic k-measure along a linear simplex trajectory.
 *
 * For edge matrix `E(t) = A + tB`, the normalized Gram determinant
 * `det(E(t)^T E(t)) / det(Erest^T Erest)` is the squared current/rest
 * k-measure ratio. Its degree is at most 2k. The Float64 reference constructs
 * the three Gram coefficient matrices, expands their determinant by column
 * multilinearity, and classifies the threshold polynomial with Bernstein
 * convex-hull bounds and de Casteljau subdivision.
 *
 * Coefficient construction costs O(3^k k^3 + N k^2). It is an auditable
 * small-k golden path, not a high-dimensional performance backend.
 */
export declare function analyzeLinearSimplexMeasureN(options: AnalyzeLinearSimplexMeasureNOptions): LinearSimplexMeasureAnalysisN;
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