The three-body problem has resisted closed-form solution for over 300 years not due to mathematical deficiency, but due to an epistemological error: attempting to track linear positions instead of rotational vorticity.
CURL curl is a topological operator (∇ × (∇ × H)) applied to a constitutional Hamiltonian field. The shear parameter γ does not degrade the trefoil orbit — it rotates the complex trace through the complex plane at exactly 3π/4 radians per unit γ on a circle of constant magnitude |Tr| = 4 (variation < 10⁻¹⁵).
At γ = 1/3, the trace crosses the real axis with the exact algebraic value Tr/8 = −1/2 (odd powers) or +1 (even powers).
git clone https://github.com/helixprojectai-code/HELIX-CURL-curl.git
cd HELIX-CURL-curl
pip install numpy matplotlib
python curl_curl_prototype_v2_5.py| Discovery | Value | Precision |
|---|---|---|
| Trace magnitude | Constant ( | Tr |
| Phase rotation rate | 3π/4 per unit γ (exactly linear) | Residual = 0 |
| Threshold γ | 1/3 | Diff ≈ 9.1 × 10⁻⁷ |
| Value at γ=1/3 (odd power) | Tr/8 = −1/2 | Diff ≈ 4.83 × 10⁻¹⁵ |
| Value at γ=1/3 (even power) | Tr/8 = +1 | Exact |
The γ sweep doesn't degrade the invariant — it rotates it:
- At γ = 0:
Tr/8 = −1/(2√2) + i/(2√2)(phase 3π/4) - At γ = 1/3: Trace hits the real axis exactly
- All eigenvalues lie on the unit circle → system is exactly unitary
| Power | Baseline (γ=0) | Extremum (γ=1/3) | Threshold |
|---|---|---|---|
| Trefoil¹ | −1/(2√2) | −1/2 | 1/3 |
| Trefoil² | 0 | +1 | 1/3 |
| Trefoil³ | +1/(2√2) | −1/2 | 1/3 |
| Trefoil⁴ | 0 | +1 | 1/3 |
Reproducible constitutional AI test results using the Helix Standard Test Set:
| Document | Description |
|---|---|
| RUNBOOK.md | Complete test runbook with PII-cleaned scripts for DeepSeek, Azure OpenAI, and generic OpenAI-compatible endpoints |
| TEST_INFO_AND_RESULTS.md | Full test results — 4-mode multi-run analysis (500 prompts each), latency breakdowns, and constitutional compliance verification |
Quick Test:
export DEEPSEEK_API_KEY="your-key"
./scripts/run_constitutional_tests.sh deepseekHELIX-CURL-curl/
├── curl_curl_prototype_v2_5.py # Main prototype + complex plane visualization
├── topological_governor.py # Topological Governor demo (pulse + reset)
├── assists/
│ └── curl_curl_v2_5_output.png # Definitive complex plane figure
├── archive/ # Previous versions (v1 → v2.4)
├── docs/
│ └── CURL_curl_Whitepaper_v2.md # Full preprint
├── LICENSE # Apache 2.0
└── README.md
v2.0 (Current)
- Strand-aware R-matrix (prime 5 integrated)
- Uniform γ scaling
- Discovery that γ rotates the trace instead of degrading it
- Threshold corrected from 0.17 → 1/3
- Composition law + master reset at Trefoil⁴ (γ=1/3)
v1.0 – Original Apache 2.0 baseline
@software{hope_2026_curl_curl,
author = {Hope, Stephen},
title = {CURL curl: A Topological Shortcut to the Three-Body Problem},
year = {2026},
publisher = {Zenodo},
doi = {10.5281/zenodo.19467359},
url = {https://github.com/helixprojectai-code/HELIX-CURL-curl}
}Apache License 2.0 — see LICENSE for details.
The trefoil doesn't untie. It rotates.
γ = 1/3 is where it faces you.
Glory to the CURL curl.
Glory to the trefoil.
Glory to the field. 🦉⚓🦆