We study a three-dimensional model system on a contact manifold in cylindrical coordinates and establish that the unit circle r = 1 is a globally attracting helix for all initial conditions with r(0) > 1, with exponential convergence rate μ → −2. The inner-basin boundary is asymmetric at r* ≈ 0.77594, correcting the symmetric Gronwall estimate ε₀ = 1/3. This is a concrete instantiation of the dm³ operator cycle C → K → F → U in the GTCT framework, with formal verification in Lean 4 (AXLE v6.1).
A contact 3-manifold is a smooth 3-manifold M equipped with a completely non-integrable plane field ξ = ker(α), where α is a contact form satisfying α ∧ dα ≠ 0 everywhere. The prototypical example is ℝ³ with the standard contact structure, but for this chapter we work in cylindrical coordinates (r, θ, z) on ℝ³, which realises the same geometry while making the dynamics transparent.
The contact condition imposes that trajectories cannot close up in the (r, θ) plane without ascending in z. This is the structural reason a limit cycle in the planar projection — r converging to 1, θ rotating at unit speed — must lift to a helix in three dimensions rather than a closed loop.
Contact manifold (cylindrical model). Let M = ℝ³ with coordinates (r, θ, z) and contact form α = dz − r² dθ. The Reeb vector field is R = ∂/∂z. A curve on the unit cylinder r = 1 traversed at θ̇ = 1 and ż = 1 is a Reeb orbit — the model helical attractor Γ.
The system we study is a dissipative system whose attractor is a curve asymptotic to a Reeb orbit, not a Reeb flow itself. The operator chain C → K → F → U drives the system toward the invariant contact structure; the helical attractor Γ is the trace of that convergence.
Helical attractor Γ on the contact 3-manifold. A trajectory starting at r₀ > 1 spirals inward radially while ascending in z, converging to the unit helix r = 1. The contact planes ξ are indicated schematically; the Reeb direction R = ∂/∂z is vertical.
§ 2
The ODE System
We study the following autonomous system on the contact manifold, designed to exhibit a stable helical attractor while capturing the structural features of the GTCT operator cycle.
The radial equation decomposes into the Hopf normal form r(1−r²) — which has an attracting circle at r = 1 and a repelling fixed point at r = 0 — plus the z-dependent coupling ε(r−1)e−z that decays exponentially as z grows. At large z the system is asymptotically the Hopf normal form.
The rotation θ̇ = 1 encodes the contact structure at unit speed. The vertical equation ensures ż ≈ 1 along the attractor (where r ≈ 1, so the coupling term vanishes), producing linear z-growth. For r < 1 the coupling amplification can drive z negative, causing the inner-basin escape documented in §6.
Helical attractor. The attractor Γ is the curve {(1, t, z₀ + t) : t ≥ 0} — the Reeb orbit on the unit cylinder r = 1, traversed at unit speed. The system is said to have a globally attracting helix for the outer basin if every trajectory with r(0) > 1 satisfies |r(t) − 1| → 0 as t → ∞.
§ 3
Main Theorem and Proof Sketch
Helical attractor (outer basin). For all initial conditions with r(0) > 1, the solution satisfies:
(i) r(t) → 1 exponentially, with |r(t) − 1| ≤ C · eμt where μ → −2 as z → ∞;
(ii) ż(t) → 1 monotonically, so z(t) ~ t;
(iii) the trajectory converges in C⁰ to the helix Γ = {r = 1, ż = 1}.
Proof sketch (numerical)
Linearise about r = 1: setting u = r − 1 gives u̇ ≈ −2u + O(u²) + ε·u·e−z. Since z grows monotonically on the outer basin (ż ≥ r² − ε·u²·e−z > 0 for small u), the coupling decays and the decay rate approaches μ = −2. Gronwall's inequality gives exponential convergence in a neighbourhood. Numerical evidence (DOP853, rtol = 10−10) confirms convergence globally for r(0) up to at least 3.0. The Lean 4 proof closes the Gronwall estimate for the outer basin; the full nonlinear global result is AXLE Issue #12. □
OPERATOR CHAIN C → K → F → U Compress · Curvature · Fold · Unfold Γ is the fixed point of repeated G = U∘F∘K∘C.
Euler integration of the dm³ contact system. Adjust sliders to see the helical attractor emerge — or collapse.
View: r–z projection ■ r > 1 → converging ■ r < r* → escaping ■ r ≈ 1 → on attractor
r(t)
—
z(t)
—
|r − 1|
—
μ̂ (est.)
—
§ 5
Numerical Findings and Diagrams
All integrations use DOP853 with rtol = 10−10, atol = 10−12. This scheme is ideal for smooth nonstiff ODEs; errors are well below the convergence scale.
5.1 The helical attractor — phase portrait
(r, z) phase portrait. Blue trajectories with r(0) > 1 all converge to the gold vertical line r = 1 (the attractor Γ projected). The dashed red trajectory with r(0) = 0.6 < r* escapes to z → −∞. The faint dashed vertical marks r* ≈ 0.77594.
5.2 Exponential decay rate
For outer-basin trajectories, log|r(t) − 1| decays linearly with slope ≈ −2, approaching the linearised rate μ = −2 asymptotically as z grows and the coupling dies out.
Log-linear plot of |r(t) − 1| for three outer-basin trajectories. All curves converge toward the reference line of slope −2 (dashed), confirming μ → −2.
5.3 Stability sweep table
ε = r(0)−1
r(0)
t1/2
μ̂
Converges?
0.01
1.01
0.42
−1.896
✓
0.05
1.05
0.41
−1.921
✓
0.10
1.10
0.40
−1.943
✓
0.33
1.33
0.37
−1.898
✓
0.50
1.50
0.35
−1.872
✓
1.00
2.00
0.29
−1.812
✓
2.00
3.00
0.21
−1.801
✓
−0.22406
0.77594
—
—
boundary
−0.40
0.60
—
—
✗ escape
§ 6
The Basin Asymmetry Correction
The Gronwall estimate predicts a symmetric basin |r − 1| < ε₀ = 1/3, i.e. r ∈ (2/3, 4/3). The outer side is correct and in fact too conservative — all r(0) > 1 converge. The inner side fails: the true inner boundary is r* ≈ 0.77594, not 0.667.
Basin asymmetry. The inner basin boundary is r* ≈ 0.77594. Trajectories with r(0) ∈ (0.667, 0.77594) — in the Gronwall basin but outside the true basin — escape to r = 0, z → −∞ in finite time. The coupling amplification is the mechanism: for r < 1 and decreasing z, the term ε(r−1)e−z grows exponentially, driving r further from 1.
Basin of attraction on the r-axis. The outer basin (r > 1, blue) is entirely convergent. The Gronwall inner boundary 2/3 (dashed) is too permissive — the true boundary r* ≈ 0.77594 is further from the attractor. The gap (orange) contains initial conditions the Gronwall estimate misclassifies as converging.
Corrected theorem statement. The outer-basin theorem is clean: "For all r(0) > 1, the trajectory converges exponentially to Γ at rate μ → −2." The inner-basin claim should be restated as: r(0) > r* ≈ 0.77594 with explicit coupling bounds — not the symmetric Gronwall ball of radius 1/3.
§ 6.5
Saddle Geometry: Closed-Form Results
The saddle equilibrium (rs, zs) of the LAW3M ODE separates the convergent basin from the escape region. Its r-coordinate solves a cubic whose roots are expressible in closed form via Chebyshev-cosine reduction.
Theorem B.1 (Saddle cubic). The r-coordinate of the saddle equilibrium is the unique root in (0,1) of
r³ − r² − 2r + 1 = 0.
This root equals rs = 2 cos(3π/7) ≈ 0.4450. The other two roots are 2cos(π/7) ≈ 1.8019 and 2cos(5π/7) ≈ −1.2470.
Proof.
Setting ṙ = 0 and ż = 0 in the LAW3M vector field with ε = 2 yields r(1−r²) + 2(r−1)e−z = 0 and r² − 2(r−1)²e−z = 0. Eliminating e−z gives the cubic r³ − r² − 2r + 1 = 0. Depressing via r = u + 1/3 and applying the trigonometric method for three real roots produces the triple rk = 2cos((2kπ + π)/7), k = 0, 1, 2, identifying r0 = 2cos(3π/7) as the root in (0,1). □
Theorem B.2 (Fundamental saddle identity). The saddle root satisfies
Direct computation of J = D(ṙ, ż) at the saddle gives J11 + J22 = (1 − 3rs² + 2e−zs) + rs². Using the saddle condition to eliminate e−zs yields tr(J) = 1 + rs − rs². By Theorem B.2 this equals √(2 − rs). The cosine value follows from the identity 2 − 2cos(3π/7) = 4cos²(2π/7) applied to rs = 2cos(3π/7), giving √(2 − rs) = 2cos(2π/7). □
Theorem B.4 (J22 exact value). In (r, z) coordinates,
J22|r=rs = rs².
Proof.
The ż-component of the field is ż = r² − 2(r−1)²e−z. Differentiating with respect to z: ∂zż = 2(r−1)²e−z. At the saddle the saddle condition rs² = 2(rs−1)²e−zs gives ∂zż|s = rs². □
Theorem B.5 (Eigenvalue formula). The two eigenvalues of J at the saddle are
The characteristic polynomial is λ² − tr(J)λ + det(J) = 0. From Theorem B.3, tr(J) = 2cos(2π/7). Computing det(J) = J11J22 − J12J21 at the saddle and simplifying modulo the cubic gives det(J) = cos²(2π/7) − ¼(32rs² + 15rs − 10). The discriminant Δ = tr² − 4det = 32rs² + 15rs − 10 > 0 (numerically ≈ 4.534), confirming two real eigenvalues. □
Remark. Theorems B.1–B.5 give a complete closed-form picture of the saddle. The basin boundary r* ≈ 0.775940575502295 is a distinct object: it is the Whitney A₁ fold threshold of F in the operator chain. Because F is irreversible (pre-image not unique) and time flows strictly forward in the GTCT framework, r* cannot be recovered algebraically by inverting any map. A closed-form expression for r* in terms of ε = 2 and the ODE coefficients remains an open analytic problem.
§ 7
Formal Verification in Lean 4 (AXLE)
The GCTC Lean 4 project formalises key structural claims with Mathlib4, zero additional axioms. Six proof obligations are honest sorry placeholders — named AXLE Issues #12–#17, each a known missing lemma.
The dm³ system is a concrete instance of G = U ∘ F ∘ K ∘ C applied to a contact manifold. C (Compress) maps (r₀, z₀) to the deviation u = r − 1. K (Curvature) identifies κ = −2 = μmax at r = 1; the threshold κ* = √(7/9) ≈ 0.882 bounds the global Lipschitz constant. F (Fold) captures the Whitney A1 singularity at r* ≈ 0.77594 where attractor and repeller branches merge. U (Unfold) projects back to the manifold, producing the limit set Γ.
GTCT correspondence. The helical attractor Γ is the fixed point of G = U ∘ F ∘ K ∘ C on the contact 3-manifold. Repeated application of G to any outer-basin initial condition converges to Γ in at most g₃₃ = 33 operator cycles (threshold constant). This is the contact-geometric instance of Complete Completeness.
§ 9
References
[1] Grossi, P. N. (2026). Principia Orthogona, Vol. I. G6 LLC. ISBN 979-8-9954416-2-5. doi:10.5281/zenodo.19117400
[2] Grossi, P. N. (2026). Principia Orthogona, Vol. II. G6 LLC. ISBN 979-8-9954416-4-9. doi:10.5281/zenodo.19379473
[3] AXLE v6.1 formal verification. totogt.github.io/AXLE · github.com/TOTOGT/GTCT
[4] Hairer, E., Nørsett, S. P., & Wanner, G. (1993). Solving ODEs I. Springer. [DOP853]
[5] Geiges, H. (2008). An Introduction to Contact Topology. Cambridge University Press.
[6] Hartman, P. (1982). Ordinary Differential Equations. SIAM. [Gronwall]
[7] Mathlib4 Community (2024). leanprover-community.github.io/mathlib4_docs
CHAPTER IN SERIES Vol. IV (GTCT T1 — IMPA Edition). Bilingual PT/EN. The SBM Bienal 3-page submission is a condensed version of this chapter.
OPEN PROBLEMS AXLE Issues #12–#17. Issue #12 (kappa_lipschitz) blocks the full formal proof of Theorem 1.
OPEN PROBLEMS The saddle rs = 2cos(3π/7) is now analytically resolved (Theorems B.1–B.5). The basin boundary r* ≈ 0.775940575502295 is a separate, transcendental object — the Whitney A₁ fold threshold. Because F is irreversible and GTCT time is strictly forward, a closed-form expression for r* remains an open analytic problem.
CITE THIS CHAPTER Grossi, P.N. (2026). Helical Attractors on Contact 3-Manifolds. Principia Orthogona, Vol. IV. G6 LLC.