CURL curl: A Topological Shortcut to the Three-Body Problem and Constitutional AI Phase-Locking

Topological Shortcut to the Three-Body Problem "The three-body problem isn't unsolvable — it's a rotation. γ = 1/3 is where the trefoil faces you. " Abstract 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). Quick Start git clone https: //github. com/helixprojectai-code/HELIX-CURL-curl. git cd HELIX-CURL-curl pip install numpy matplotlib python curlcurlₚrototypeᵥ2₅. py Key Results (Verified to Machine Epsilon) 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 Complex Plane Discovery 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 Composition Law 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 Repository Structure HELIX-CURL-curl/ ├── curlcurlₚrototypeᵥ2₅. py # Main prototype + complex plane visualization ├── topologicalgovernor. py # Topological Governor demo (pulse + reset) ├── assists/ │ └── curlcurlᵥ2₅ₒutput. png # Definitive complex plane figure ├── archive/ # Previous versions (v1 → v2. 4) ├── docs/ │ └── CURLcurlWhitepaperᵥ2. md # Full preprint ├── LICENSE # Apache 2. 0 └── README. md Version History v2. X (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 License 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. 🦉⚓🦆