/**
 * TropicalBezoutConjecture — the string-math (tropical / toric geometry) sub-class
 * of the Conjecture Engine.
 *
 * String theory AS PHYSICS is unfalsifiable and out of scope (W.530); string theory
 * AS MATHEMATICS gives exactly-checkable proof-carrying geometry. This module verifies
 * the cleanest correctly-implementable instance of the tropical↔classical correspondence:
 *
 *   Bernstein–Kushnirenko / BKK ("tropical Bézout"): the number of solutions of a
 *   generic system of two plane polynomials with Newton polygons P, Q equals the
 *   normalized mixed volume MV(P, Q). For the degree-d Newton triangle T_d, this
 *   recovers classical Bézout: MV(T_{d1}, T_{d2}) = d1·d2.
 *
 * The mixed volume is computed for real — a genuine Minkowski-sum + convex-hull +
 * shoelace pipeline over lattice polygons, in exact integer arithmetic — NOT a formula
 * assertion. The conjecture survives iff the computed MV equals the classical Bézout
 * number across the swept degrees; a deliberately-false companion claim is the
 * discovering falsifier.
 *
 * Receipt-carrying evidence, NOT a Lean proof (W.511). This verifies finite instances
 * of the correspondence; PROVING mirror symmetry / the Gross–Siebert program is
 * Lean-tier (Paper 22). Full Mikhalkin curve-counting (N_d with multiplicities) is the
 * harder next rung; BKK/Bézout is the correct, tractable first slice.
 *
 * Built on the same tropical/toric backbone as the rest of the stack (Paper 29).
 */
export declare const TROPICAL_BEZOUT_V1: "conjecture.stringmath.tropical-bezout.v1";
export type TropicalBezoutSolverType = typeof TROPICAL_BEZOUT_V1;
export type TropicalConjectureStatus = 'survived' | 'falsified' | 'undecided';
export type TropicalProbeStatus = 'pass' | 'fail';
export type LatticePoint = readonly [number, number];
export interface MixedVolumeEvaluation {
    d1: number;
    d2: number;
    mixedVolume: number;
    classicalBezout: number;
    status: TropicalProbeStatus;
    message: string;
}
export interface TropicalCounterexample {
    d1: number;
    d2: number;
    expected: number;
    actual: number;
    reason: string;
}
export interface TropicalScenario {
    id: string;
    role: 'survivor' | 'falsifier';
    statement: string;
    evaluations: ReadonlyArray<MixedVolumeEvaluation>;
    status: TropicalConjectureStatus;
    counterexamples: ReadonlyArray<TropicalCounterexample>;
}
export interface TropicalBezoutReceipt {
    solverType: TropicalBezoutSolverType;
    specVersion: 1;
    maxDegree: number;
    scenarios: ReadonlyArray<TropicalScenario>;
    gate: {
        survivorScenarioId: string | null;
        falsifierScenarioId: string | null;
        passed: boolean;
    };
    receiptKey: string;
}
export interface TropicalBezoutOptions {
    /** Largest degree swept (inclusive), for both factors. Must be >= 1. Default 6. */
    maxDegree?: number;
}
/** The standard degree-d Newton triangle conv{(0,0),(d,0),(0,d)}. */
export declare function degreeTriangle(d: number): LatticePoint[];
/** Twice the Euclidean area of a polygon via the shoelace formula (exact integer). */
export declare function twiceArea(polygon: ReadonlyArray<LatticePoint>): number;
/** Convex hull (CCW, no collinear interior points) via Andrew's monotone chain. */
export declare function convexHull(points: ReadonlyArray<LatticePoint>): LatticePoint[];
/** Minkowski sum of two lattice polygons: convex hull of all pairwise vertex sums. */
export declare function minkowskiSum(p: ReadonlyArray<LatticePoint>, q: ReadonlyArray<LatticePoint>): LatticePoint[];
/**
 * Normalized 2D mixed volume MV(P, Q) = (Area(P⊕Q) − Area(P) − Area(Q)), which by
 * Bernstein–Kushnirenko is the generic intersection number of two plane curves with
 * Newton polygons P, Q. Computed in exact integer arithmetic via twice-areas.
 */
export declare function mixedVolume2D(p: ReadonlyArray<LatticePoint>, q: ReadonlyArray<LatticePoint>): number;
/**
 * Run the tropical-Bézout (BKK) string-math conjecture sub-class end to end:
 * a survivor (mixed volume == classical Bézout, computed for real) and a discovering
 * falsifier (the false sum-claim). Deterministic receipt key via the shared
 * `buildConjectureStableKey` primitive. The gate passes when the survivor is fully
 * survived AND the falsifier is refuted by at least one discovered counterexample.
 */
export declare function runTropicalBezoutConjecture(options?: TropicalBezoutOptions): TropicalBezoutReceipt;
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