DOI: 10.5281/zenodo.17635887
title: Three-Body Trust Catalog v1.0: Hash-Verified High-Precision Periodic Orbits

description: **Three-Body Trust Catalog v1.0: Hash-Verified High-Precision Periodic Orbits**

This record provides a high-precision, hash-anchored catalog of selected planar Newtonian three-body periodic orbits with equal masses and gravitational constant \( G = 1 \). The aim is trust rather than novelty: a reproducible, falsifiable reference implementation for several historically important periodic solutions, suitable as a benchmark for numerical integrators, chaos studies, and future methodological work.

Two Jupyter notebooks are provided:

1. **`three_body_trust_catalog_v1_full.ipynb`**     This is the canonical high-precision implementation. It uses `mpmath` with 80-digit working precision and a very strict ODE tolerance to integrate each orbit over one nominal period. For each trajectory it computes:   - the nominal period \( T \);     - the total energy drift (max–min);     - the closure error \( \lVert y(T) - y(0) \rVert \);     - a SHA-256 hash of the full trajectory, obtained from a JSON serialization of all sampled states.  

   These hashes are collected in `three_body_trust_catalog_v1_summary.json`, forming the **canonical trust anchor**: anyone can rerun this notebook and verify bitwise agreement with the published trajectories, given the same software environment.

2. **`three_body_trust_catalog_v1_fast.ipynb`**     This is a lighter, faster notebook built from the same equations of motion and initial conditions, but with modestly relaxed precision and ODE tolerances. It reproduces the same set of orbits and diagnostics (period, energy drift, closure error) with substantially reduced runtime, making it convenient for quick tests or exploratory work. When the canonical `three_body_trust_catalog_v1_summary.json` is present, the fast notebook can optionally load it and print side-by-side comparisons of its diagnostics against the canonical values. The fast notebook also computes its own SHA-256 hashes as **diagnostic hashes only**; these are not intended to replace the canonical hashes produced by the full notebook.

The catalog currently includes:- The equal-mass, zero–angular-momentum **figure-8 orbit** originally proven by Chenciner & Montgomery (Annals of Mathematics, 2000), with initial conditions based on the high-accuracy values of Carles Simó.  - Three collisionless, zero–angular-momentum orbits from Suvakov & Dmitrašinović, *“Three classes of Newtonian three-body planar periodic orbits”* (Phys. Rev. Lett. 110, 114301, 2013, arXiv:1303.0181):    - Butterfly I (class I.A.1)    - Moth I (class I.B.1)    - Yin–Yang I, branch 2a (class II.C.2a)  

Following Table I of their paper, we set  \( x_1(0) = -1 \), \( x_2(0) = +1 \), \( x_3(0) = 0 \), all \( y_i(0) = 0 \),  with velocities determined from a single pair \((\dot x_1(0), \dot y_1(0))\) via  \( \dot x_2(0) = \dot x_1(0) \), \( \dot x_3(0) = -2 \dot x_1(0) \),  \( \dot y_2(0) = \dot y_1(0) \), \( \dot y_3(0) = -2 \dot y_1(0) \),  and the period \( T \) taken directly from their Table I.

Together, the full and fast notebooks define a small but carefully curated **trust framework** for the equal-mass three-body problem: the full notebook fixes a canonical, hash-verifiable reference dataset, while the fast notebook provides a practical tool for day-to-day validation and experimentation.

### Included files (flat layout)

This Zenodo record includes the following files, all in a single (flat) directory:

- `three_body_trust_catalog_v1_full.ipynb`    Canonical high-precision notebook that generates the trusted reference trajectories, diagnostics, and SHA-256 hashes.

- `three_body_trust_catalog_v1_fast.ipynb`    Optimized fast notebook that reproduces the same orbits and diagnostics with reduced runtime, and optionally compares against the canonical summary.

- `three_body_trust_catalog_v1_summary.json`    Canonical trust summary produced by the full notebook. Contains, for each orbit:  - nominal period \( T \)    - energy drift    - closure error    - SHA-256 hash of the full trajectory

- `figure8_trajectory.json`  - `butterfly_I_trajectory.json`  - `moth_I_trajectory.json`  - `yinyang_I_branch2a_trajectory.json`    High-precision state samples for each orbit over one nominal period, serialized as JSON (optional but included here for convenience).

- `README.md`    Human-readable overview of the project, usage instructions, and references.

- `methods.md`    Short methods document describing the numerical setup, precision choices, and diagnostic definitions.

- `verification.py`    Standalone verification script that can re-integrate the orbits and check numerical diagnostics (and, if desired, recompute SHA-256 hashes for independent verification).

- `zenodo.json`    Metadata file mirroring the record’s title, authorship, description, keywords, and related identifiers.

- `CITATION.cff`    Citation metadata in Citation File Format, providing a standard way to cite this dataset.

- `LICENSE`    License file (MIT), specifying reuse and attribution terms.

- `checksums.txt`    SHA-256 checksums for all files in this record, enabling integrity checks after download.

**Related record:**  High-precision replication of the Meijaard et al. (2007) benchmark self-stable bicycle model — Zenodo DOI 10.5281/zenodo.17634914.