The Three-Body Problem: Interpretations Under Different Theoretical Frameworks

Author: Zhang Suhang (Luoyang, Henan)

I. Complete Comparison of Core Solutions Across Three Frameworks

1. MOC Framework → Solution: Curvature Conservation / Curvature Equilibrium

Core Logic (Geometric Ontology)

When a three-body or multi-body system reaches a steady state, the global scalar curvature R satisfies conservation constraints. Gravitational coupling between celestial bodies is equivalent to mutual balancing of spatial curvatures. The overall configuration stops spontaneous deformation, and orbits tend to be stable.

Fundamental Equation: Unified Curvature Equation (UCE), which describes the equilibrium state of geometric fields.

2. DOG Framework → Solution: Frequency Coupling / Frequency Resonance

Core Logic (Discrete Order Ontology)

The eigenfrequencies of all motion modes within the system lock together to form coherent coupling. Quasi-periodic motion and nested orbits are governed collaboratively by discrete frequency spectra. The steady state manifests as symbiotic resonant frequencies.

Fundamental Equation: Frequency Coupling Equation (FCE), a discrete matrix equation describing coupled states of discrete dynamics.

3. UPGS-UPFE Framework → Solution: Probability Equilibrium (Probabilistic Steady State)

1. Definition

Constrained by the Unified Probability Field Equation (UPFE), the system eventually converges to a state with balanced probability distribution, namely probability equilibrium — a unique steady-state solution of the UPGS framework.

2. Mathematical and Physical Implications

- Axiomatic basis: Three fundamental rules — probability-geometry isomorphism p=e^{-h}, frequency transition probability, and extremal action — converge jointly. No net transport exists in the probability flow, expressed as:

\partial_t p=0,\quad \nabla\cdot(p\boldsymbol{v})=0

- Field equation perspective: For the equation \square \Phi = \big(\alpha (\Delta\nu)^2 + \beta n + \gamma R\big)\Phi, its steady-state solution implies a dynamic balance among local frequency difference \Delta\nu, topological winding number n and scalar curvature R. The probability of each possible state no longer evolves over time.

- Physical interpretation for the three-body system:

Instead of focusing merely on individual orbits, we consider all possible motion patterns of the system — stable revolution, perturbation, chaotic transition, escape and others. The proportional probability of each pattern remains constant, and the overall statistical distribution stays unchanged, which defines probability equilibrium.

3. Distinction and Complementarity with the Other Two Frameworks

- Curvature equilibrium: Steady state of deterministic geometric structures.

- Frequency coupling: Steady state of discrete dynamic motion modes.

- Probability equilibrium: Global steady state of probabilistic fields.

II. Supplementary Elaboration (Adapted for Academic Papers)

1. Formal Nomenclature for Steady-State Solutions

Standard name: Probability Equilibrium State (alternatively: Probabilistic Steady State, Probability Distribution Equilibrium Solution)

2. Characteristic Equation Properties

At the steady state of UPFE, all time derivative terms vanish (\partial_t \Phi=0), and the equation is simplified to:

\nabla^2 \Phi = \big(\alpha (\Delta\nu)^2 + \beta n + \gamma R\big)\Phi

Under this condition:

1. Local frequency difference \Delta\nu, topological winding number n and scalar curvature R are mutually locked;

2. Probability amplitude, probability density and geometric potential remain time-independent;

3. Strict conservation of probability flow is realized across the entire domain, leading to universal probability equilibrium.

3. Linkage of Three Layers of Steady States for the Three-Body Problem

1. Geometric layer: Mutual interactions of the three bodies achieve curvature equilibrium (MOC).

2. Motion mode layer: Orbital oscillations achieve frequency coupling (DOG).

3. Global statistical layer: All motion states of the system achieve probability equilibrium (UPGS-UPFE).

The three frameworks characterize the ultimate steady state of the three-body system from geometric, dynamic and probabilistic statistical perspectives respectively, verifying and complementing one another.

III. Concise Summary (Directly Applicable to Paper Conclusions)

The MOC framework derives solutions for curvature equilibrium, the DOG framework focuses on frequency coupling, and the steady-state solution of the UPFE corresponds to probability equilibrium. Representing steady states in geometry, discrete dynamics and probabilistic statistics respectively, the three frameworks jointly form a complete system for describing the steady state of the three-body problem.