Title: The Essential Solution to the Three-Body Problem – Higher-Dimensional Curvature Vector Conservation and Coupled Steady State

Author: Zhang Suhang, Luoyang

Abstract: Based on the MOC (Multi-Origin Curvature) axiomatic system, this paper presents the essential solution to the three-body problem. The three-body system has no intrinsic chaos; its steady state is entirely determined by the curvature vector conservation of each celestial body and the curvature coupling balance of the ECS (Elliptic Coupling Conserved System). The extremely simple steady-state rule is: non-collinear topological coupling is complete, conservation holds, and the system is permanently stable; three-body collinearity leads to coupling breaking, conservation disturbance, and transient instability. The steady states of the non-collinear Sun–Earth–Moon inclined orbit and the Earth–Mars–Venus non-collinear ecliptic orbit serve as two real celestial phenomena that directly verify the universality of curvature conservation. Many-body systems are uniformly governed by curvature vector conservation, requiring neither decomposition of geometric configurations nor case-by-case verification of spatial distribution stability.

Keywords: Three-body problem; curvature conservation; non-collinear stable; collinear chaotic; ECS; MOC

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1. Opening Statement

The three-body problem has perplexed the academic community for three hundred years. The fundamental reason lies in the misguided use of the Newtonian paradigm of force and potential energy, which anchors system stability to initial conditions and solutions of equations of motion, ultimately leading to the cognitive dead end of unsolvable chaos. The MOC system completely changes course, reconstructing the underlying logic from the first principles of geometry:

There is no inherent chaos in the three-body system. The essence of stability is the dual constraint of independent conservation of MOC curvature vectors and global curvature coupling balance of the ECS.

Each celestial body carries its own independent higher‑dimensional curvature vector (equivalent to the geometric essence of orbital angular momentum, see the first paper). Curvature vector conservation forces its trajectory to be a closed ellipse (including the special case of a perfect circle). The essence of the three‑body problem is three independent elliptical orbits coexisting and mutually constraining each other in space. Whether the system achieves a steady state does not depend on small differences in initial conditions, nor on the quantitative ratios of mass, distance, or velocity, but on the completeness of the spatial topological coupling of the three elliptical orbits – i.e., whether the three curvature vectors can form a closed, conflict‑free, unconstrained, non‑collapsing conservation loop in space.

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2. The Extremely Simple Steady‑State Rule (The Golden Sentence of the Paper)

Non‑collinear topology → coupling complete, conservation holds, system permanently stable;
Three‑body collinearity → coupling broken, conservation disturbed, transiently unstable.

Core Explanation of the Rule

This rule is a purely topological and conservation‑based criterion, completely independent of spatial geometry or configuration distribution. The only criterion is the spatial constraint state of the three celestial bodies:

· Non‑collinear topology: The three bodies are not constrained to lie on a single straight line. The curvature vectors retain full directional and magnitude degrees of freedom. The local curvature conservation of each body does not interfere with or break that of the others. The total global curvature is strictly conserved. The system is in a geometrically self‑consistent, inherently stable state. Any small perturbation is automatically pulled back by the law of curvature conservation; there is no tendency toward collapse or instability.
· Three‑body collinearity: The three bodies are geometrically forced to align on a single straight line (e.g., solar eclipse, lunar eclipse, planetary transit). In this case, the directions of the three curvature vectors are forced into alignment, their spatial degrees of freedom completely collapse, and the vectors degenerate into scalars without directional properties. Local curvature conservation is forcibly broken. The system enters a transient unstable state, producing observable physical disturbances (e.g., enhanced tides, slight orbital precession, angular speed fluctuations). Once the bodies move away from collinear constraint, the curvature vectors instantly recover their full spatial degrees of freedom, conservation is reestablished, and the system automatically returns to a steady state.

This rule is absolutely universal. It does not depend on mass magnitudes, orbital distances, initial velocities, or inclinations of orbital planes. It depends solely on the spatial topological constraint state and constitutes the necessary and sufficient condition for the stability of the three‑body problem.

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3. Hard Verification by Two Real Celestial Phenomena

3.1 The Steady State of the Non‑collinear Sun–Earth–Moon System

The Sun–Earth–Moon three‑body system operates in a non‑collinear spatial configuration for most of the time (the Moon’s orbital plane has a fixed inclination of about 5° relative to the ecliptic, fundamentally avoiding long‑term collinear constraint). The MOC system predicts that under this non‑collinear topology, the curvature vectors of the three bodies remain independently conserved, their spatial degrees of freedom are complete, the ECS coupling balance holds continuously, and the system possesses long‑term permanent stability.

Astronomical observations fully agree: the Earth–Moon–Sun system has been stable for billions of years, exhibiting only periodic, predictable minor orbital perturbations, with no trend toward instability, collapse, or orbital escape. Solar and lunar eclipses are only extremely brief three‑body collinear events, lasting hours at most. After the perturbation ends, the system immediately returns to a non‑collinear steady state, in perfect agreement with the predictions of MOC curvature conservation theory.

3.2 The Steady State of the Non‑collinear Earth–Mars–Venus Orbits

Earth, Mars, and Venus all move in elliptical orbits around the Sun. Their orbits are approximately coplanar but have different orbital radii and angular velocities. Their spatial positions are always in a non‑collinear dynamic distribution; three‑body collinear events are extremely rare and very short‑lived.

The MOC system predicts that as long as the system maintains a non‑collinear topology – regardless of whether the orbits are coplanar or the curvature vectors are parallel – curvature coupling conservation can be achieved through differences in magnitudes and orbital phase differences, and the overall system remains stable. Only during extremely rare three‑body collinear moments do transient, weak conservation disturbances occur, without long‑term effects.

Astronomical observations fully verify this: the orbits of Earth, Mars, and Venus have been stable for billions of years, with no orbital disorder, collisions, or escape. The disturbances during collinear events are far below observational thresholds, in perfect agreement with theoretical predictions.

These two empirical verifications further clarify that the core of stability is curvature conservation under non‑collinear topology, independent of the geometric form of spatial distribution, and determined solely by the constraint state and conservation law.

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4. Conclusion

Based on the MOC axiomatic system, this paper achieves an essential solution to the three‑hundred‑year‑old three‑body problem. The core conclusions are:

1. The three‑body system has no intrinsic chaos. The illusion of chaos arises from a misinterpretation of the underlying geometric laws within the traditional paradigm. The steady state of the system is jointly determined by higher‑dimensional curvature vector conservation and ECS coupling balance.
2. A universal, extremely simple criterion is established: non‑collinear topology is permanently stable; three‑body collinearity causes only transient instability, independent of geometric configuration, mass, or velocity.
3. The two real celestial systems – Sun–Earth–Moon and Earth–Mars–Venus – provide direct, long‑term, repeatable hard verification of the law of curvature conservation.
4. Many‑body system motion is uniformly governed by curvature vector conservation; stability is co‑determined by spatial topological constraints and conservation laws.

The three‑hundred‑year three‑body problem is essentially not a difficulty in solving equations, but a bias in paradigm choice. The MOC system, grounded in the first principles of geometry, directly touches the essential laws of motion of many‑body systems, achieving a dimensionality‑reduction solution to the classical many‑body problem.

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References: See the first paper. References to classical three‑body problem literature are unnecessary, as this paper has completed a fundamental paradigm shift.