The contact form α = dz − r²dθ defines a plane field on the 3-dimensional dm³ manifold that is maximally non-integrable: no surface exists everywhere tangent to the planes defined by ker(α). This is what prevents the helical flow from collapsing into a planar, non-rotating state.
The Reeb vector field R = ∂z, defined by ι_R α = 1 and ι_R dα = 0, generates the persistent helical motion. Every orbit of R is a Legendrian curve — the flow follows the contact geometry inevitably, without external forcing.
The transverse Lyapunov exponent μ_max = −2 quantifies how strongly perturbations away from the limit cycle Γ are damped. Combined with the fold geometry at r*, it fixes the sigmoid Hill coefficient n ≈ 3.64 — steep enough to act as a flood gate, shallow enough to avoid cavitation.
The basin hierarchy ε₀=1/3 < r*≈0.776 < κ*≈0.882 < 1 demarcates three nested zones: the inner laminar core, the transition annulus, and the outer turbulent boundary. All three are determined by the contact geometry alone.
| Contact form | α = dz − r² dθ |
| Non-integrability | α ∧ dα ≠ 0 everywhere |
| Reeb field | R = ∂z generates persistent rotation |
| Lyapunov exponent | μ_max = −2 |
| Hill coefficient | n ≈ 3.64 |
| Basin hierarchy | ε₀=1/3 < r*≈0.776 < κ*≈0.882 < 1 |