Helical Attractors on Contact 3-Manifolds
and the Inner Basin Boundary

Helical Attractors on Contact 3-Manifolds: Basin Geometry, Whitney Singularities, and the Law of Geometric Space Dynamics (LAW3M)

We study a three-dimensional ODE on the contact 3-manifold (ℝ³, α = dz − r²dθ) in cylindrical coordinates, where α is the standard contact form and r²dθ encodes angular momentum as a one-form. The system — which we call the Law of Geometric Space Dynamics (LAW3M) — takes the form

with coupling constant ε = 2. The contact condition α = 0 reads dz = r²dθ, identifying vertical lift directly with angular momentum — a geometric identity rather than a force law. We prove that the unit helix Γ = {r = 1, θ̇ = 1, ż = 1} is a globally attracting limit cycle for all initial conditions with r(0) > r* and z(0) ≥ log 2, with exponential convergence rate μ = −2 (Lyapunov exponent).

The principal result is the numerical certification of the inner basin boundary r* via adaptive bisection with the DOP853 integrator (rtol = 10⁻¹², atol = 10⁻¹⁴). We find

This value lies strictly above the Lyapunov bound 2/3 ≈ 0.667, establishing a basin asymmetry driven by the coupling term ε(r−1)²e⁻ᶻ. The stability constant hierarchy is confirmed: ε₀ = 1/3 < 2/3 < r* = 0.77594 < κ* = √(7/9) ≈ 0.882 < 1.

We identify r* as a Whitney A₁ singularity (standard fold) of the asymptotic radial map Φ: ℝ₊ → {1}. The fold is the structural origin of the basin asymmetry: two qualitatively different convergence paths — from above (outer basin, r > 1) and from below (inner basin, r* < r < 1) — fold onto the single attractor value r = 1. At r* the coupling term reaches the ionisation threshold; in a rotating electromagnetic realisation, this corresponds to plasma corona discharge.

The system connects to the dm³ / Principia Orthogona operator chain G = U∘F∘K∘C, where the fold F is the Whitney A₁ singularity at r*, and the embodiment threshold τ = 2 equals the coupling constant ε. LAW3M sits at the Δ (tetranacci) rung of the n-bonacci recurrence ladder, where the dominant root ≈ 1.927 first approaches τ = 2 with order-4 memory. The geometry selects a circular cross-section, constant angular velocity, and vertical lift proportional to r² as inevitable consequences of the contact form — not engineering choices.

Saddle geometry. A second equilibrium — the saddle (r_s, z_s) separating escape from convergence — is now analytically resolved. Its r-coordinate satisfies the cubic r³ − r² − 2r + 1 = 0 with closed-form solution r_s = 2cos(3π/7) ≈ 0.4450 (Theorem B.1). The Jacobian trace at the saddle equals tr(J) = 2cos(2π/7) (Theorem B.3) and the eigenvalues are λ_± = cos(2π/7) ± ½√(32r_s²+15r_s−10), numerically ≈ 1.1097 and −0.2443 (Theorem B.5). These results are proved in closed form. The basin boundary r* ≈ 0.775940575502295 is a distinct object (the Whitney A₁ fold threshold) and its closed-form expression remains an open analytic problem.

The certification script (Python 3, SciPy) is publicly available and runs in under two minutes on standard hardware. All results are archived at doi:10.5281/zenodo.19117399.