THREE-BODY PHASE STRUCTURE AND TRANSITION CLASSIFICATION

A Falsifiable Structural Protocol for Predicting Finite-Horizon Orbital Outcomes

Kevin L. Brown
Independent Researcher | Creation Unified | October 2026

The Three-Body Phase Structure and Transition Classification paper presents a revised, falsifiable computational methodology for improving finite-horizon orbital prediction in gravitational three-body systems. Its central hypothesis is that physically interpretable structural representations can improve predictive accuracy, computational efficiency, or early transition detection relative to established numerical and statistical methods operating under comparable information and resource constraints.

Newtonian gravity provides the governing equations for three interacting point masses, but their trajectories can exhibit periodic, quasi-periodic, resonant, hierarchical, scattering, collisional, and chaotic behavior. This work does not propose a new gravitational force, claim a universal closed-form solution, or replace direct N-body integration.

Instead, it investigates whether evolving orbital structure can be represented in a compact form that makes future dynamical behavior easier to classify.

Mathematical Foundation

The framework nondimensionalizes the Newtonian equations using a characteristic length (R_0), total mass (M), and characteristic timescale:

[t_0=\sqrt{\frac{R_0^3}{GM}}.]

A time-dependent structural state vector is constructed from normalized diagnostics describing:

• Close-encounter compression and geometric separation.
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• Pairwise orbital-energy changes and Jacobi-coordinate energy structure.
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• Hierarchical organization and orbital stability.
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• Historical finite-time dynamical instability.
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• Recurrence and regularity in reduced phase space.
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• Resonance locking and preregistered phase coherence.

The revised methodology corrects an earlier interpretation of a weighted "structural pressure" variable. The structural representation is now treated strictly as a predictive model input, not a newly discovered physical pressure, conserved quantity, or independent dynamical law.

Total energy and angular momentum conservation remain numerical quality controls.

Information and Predictability

A central mathematical refinement distinguishes information content from computational accessibility.

For deterministic Newtonian evolution with a completely known initial state (X_0), a trajectory-derived representation (Z) cannot create additional information about an outcome (Y):

[I(Y;Z\mid X_0)=0.]

The proposed contribution therefore concerns improved recovery of predictive information under finite observational, computational, and model-capacity constraints, rather than information absent from the exact initial state.

This distinction provides a mathematically consistent foundation for evaluating structural-temporal prediction.

Orbital Outcome Classification

The revised protocol separates physical events from dynamical regimes.

Terminal-event classification distinguishes collisions, escapes, and systems without a terminal event during the observation horizon, with explicit treatment of censoring.

Dynamical-regime classification separately evaluates hierarchical organization, regular or periodic motion, quasi-periodicity, resonance, chaos, mixed behavior, and unresolved states.

This separation prevents overlapping physical descriptions from being incorrectly treated as mutually exclusive outcomes.

Primary Predictive Hypothesis

The principal hypothesis is prospective: a frozen structural-temporal model may achieve measurable improvements in held-out classification, transition timing, or computational efficiency over conventional approaches.

The experimental design compares six model classes, M0–M5, spanning initial-state baselines, conventional orbital diagnostics, structural-temporal classifiers, flexibility-matched controls, observation-matched temporal models, and computation-budget comparisons.

Generalization is evaluated across six trajectory families:

1. Equal-mass nonhierarchical systems.
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3. Hierarchical triples.
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5. Restricted three-body configurations.
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7. Near-resonant systems.
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9. Known periodic-orbit families.
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11. Random scattering configurations.

Group-aware and leave-one-family-out validation are intended to distinguish transferable predictive structure from family-specific memorization.

Secondary THD and Phase-Locking Hypotheses

A separate hypothesis examines whether preregistered resonance relations and changes in circular phase coherence provide useful early indicators of resonance capture, resonance escape, or dynamical transition.

The Triune Harmonic Dynamics (THD) interpretation additionally tests whether the proposed sequence

Emergence → Contrast → Integration

provides predictive value beyond conventional change-point detection and alternative temporal orderings.

Neither resonance coherence nor THD ordering is assumed to be universally present. Both remain independently falsifiable hypotheses requiring controlled statistical evaluation.

Falsification and Independent Replication

The protocol specifies causal observation windows, conservation-quality controls, numerical convergence checks, matched-information comparisons, held-out validation, calibration measures, and safeguards against outcome leakage and retrospective model adjustment.

The accompanying reproducibility package provides computational specifications, executable demonstrations, unit tests, and documentation supporting further implementation and independent review.

Passing software tests establishes implementation consistency within the tested scope; it does not establish the predictive superiority of the proposed framework.

Current Research Status

This publication is a theoretical and computational research protocol, not a completed empirical validation study.

The full six-family benchmark and independent replication remain outstanding. Validation requires frozen experimental settings, execution of the benchmark corpus, held-out predictive scoring, uncertainty and calibration analysis, computational-efficiency comparisons, phase-surrogate testing, and cross-family transfer evaluation.

The proposed contribution is a testable method for determining whether physically interpretable structural representations can make gravitational three-body behavior more predictably and efficiently classified without modifying Newtonian mechanics.

Keywords: Three-body problem; celestial mechanics; Newtonian gravity; orbital stability; chaotic dynamics; finite-horizon prediction; structural classification; resonance; phase coherence; computational efficiency; machine learning; Triune Harmonic Dynamics; falsifiability; reproducibility.