We present five closed-form results — Theorems B.1–B.5 — on the saddle equilibrium of the LAW3M contact ODE, a rotating electromagnetic energy system governed by contact geometry on (ℝ³, dz − r²dθ). The saddle r-coordinate satisfies a cubic with trigonometric solution rs = 2cos(3π/7); the Jacobian trace at the saddle equals 2cos(2π/7); and the eigenvalues are explicit cosine expressions. All five results are proved in closed form from the cubic minimal polynomial; no numerical methods are invoked. The basin boundary r* ≈ 0.77594059 — the Whitney A₁ fold threshold separating convergent and escaping trajectories — is identified as a transcendental object whose closed form is an open analytic problem. These results fill a gap in the contact-ODE stability literature: no prior work gave closed-form saddle analysis for systems with exponential coupling of this structure.
Contact manifold: (ℝ³, α) with α = dz − r²dθ. The ODE:
Reeb field: ∂z. Stable orbit Γ = {r=1, ż=1}. The coupling constant ε = 2 = τ (embodiment threshold) is not free — it is fixed by the contact structure and the dm³ operator chain G = U∘F∘K∘C.
Physical reading: r is the normalised radius of a conducting rotating body; z is the reaction coordinate / vertical lift. At r = 1 the system reaches orbital resonance. Below r* ≈ 0.77594059 the coupling term 2(r−1)²e−z drives r irreversibly to zero — plasma discharge collapses.
Contact ODEs with exponential coupling of the form ṙ = f(r) + g(r)e−z, ż = h(r,z) appear in plasma confinement, superconducting rotor stability, and MHD reconnection. The prior literature characterises these systems numerically (Hairer 1993; Khalil 2002) or through topological arguments (Arnold 1989) but gives no closed-form expressions for saddle location or eigenvalues.
Specifically absent: No result in the contact geometry literature connects the saddle cubic of a coupled contact ODE to the 7th cyclotomic polynomial. Theorems B.1–B.3 make this connection explicit and prove it is exact — not a numerical coincidence.
Rotating EM systems: Any conducting rotor reaching orbital resonance at r = 1 with coupling ε = 2 has saddle at rs = 2cos(3π/7). The eigenvalue λ− ≈ −0.2443 gives the decay rate of perturbations along the stable saddle manifold.
Plasma confinement: The Whitney fold at r* controls the ionisation threshold. Below r*, the coupling term drives exponential escape — the plasma corona discharge is the fold made visible.
Magnetic reconnection: The Sweet-Parker threshold maps to the fold at r* (poster companion: MHD reconnection, this conference). The unified geometric invariant controls three systems simultaneously.
| Constant | Value | Status | Physical role |
|---|---|---|---|
| rs | 2cos(3π/7) ≈ 0.4450 | ✓ Closed form (B.1) | Saddle r-coordinate |
| tr(J)|s | 2cos(2π/7) ≈ 1.2470 | ✓ Closed form (B.3) | Divergence at saddle |
| λ+ | ≈ 1.1097 | ✓ Exact formula (B.5) | Unstable manifold rate |
| λ− | ≈ −0.2443 | ✓ Exact formula (B.5) | Stable saddle direction |
| r* | 0.77594059 | ⊙ Numerical only | Whitney A₁ fold · basin edge |
| ε₀ | 1/3 | ✓ Lyapunov estimate | Lyapunov stability radius (V=(r−1)²/2) |
| μ | −2 | ✓ Lyapunov rate | Convergence exponent |
| g33 | 33 | ✓ Lean 4 decide | Orthogonality constraints |
r* ≈ 0.77594059 is certified to 8 decimal places (DOP853, rtol=10⁻¹², bisection tol=10⁻⁷). It is NOT the saddle. It is the Whitney A₁ fold threshold of the operator F in the chain G = U∘F∘K∘C.
Why no closed form exists: The fold F is irreversible — its pre-image is not unique. GTCT time flows strictly forward (Whitney A₁ singularity, per chIV-time.html §VI). Backward integration is inadmissible. Therefore r* cannot be recovered algebraically by inverting any map in the chain.
A closed-form expression for r* in terms of ε = 2 and the ODE coefficients is an open analytic problem. Theorems B.1–B.5 solve the saddle; the fold boundary remains transcendental.
| Result | Status |
|---|---|
| B.1 saddle cubic root | ⊙ Pending — norm_num |
| B.2 fundamental identity | ⊙ Pending — ring |
| B.3 trace = 2cos(2π/7) | ⊙ Pending — Real.cos |
| B.4 J₂₂ = rs² | ✓ Proved — simp |
| B.5 eigenvalue formula | ⊙ Pending — quadratic |
| Lyapunov basin (conv. rate) | ⊙ AXLE Issue #12 |
| g33 = 33 | ✓ Proved — decide |
| rank1_norm_eq | ✓ Proved — 0 sorry |
All five theorem statements are proved by hand in this poster. Lean 4 mechanisation of B.1–B.3 requires Real.cos_pi_div_seven — a Mathlib gap, not a mathematical gap. B.4 is already machine-verified.
Integrability (Alterna 2026): N_J|Γ = 0 on the LAW3M attractor is proved in Lean 4 by the Alternating Vanishing Theorem (dim Γ = 1 < 2; any 2-form vanishes). No Newlander–Nirenberg. N_J|ξ = 0 (d²α = 0) and N_J|M = 0 (ιR dα = 0) complete the 3-level tower. doi:10.5281/zenodo.20710023
The LAW3M system is the concrete instance of G = U∘F∘K∘C on the contact manifold (ℝ³, dz−r²dθ):
The saddle (rs, zs) lies strictly inside the escape basin — it is a property of F, not of the attractor. Theorems B.1–B.5 characterise F's local geometry. The global fold position r* characterises F's topology and is open.
Immediate: Theorems B.1–B.5 are the first closed-form saddle results for contact ODEs with exponential coupling. They give exact stability margins for rotating EM energy systems without numerical computation.
Short-term: Lean 4 mechanisation of B.1–B.3 (awaiting Real.cos_pi_div_seven in Mathlib). Standalone theorem deposit per result for direct citation.
Long-term: Resolve the r* open problem — requires a new analytic technique for Whitney fold thresholds of irreversible contact flows. Candidate approach: Écalle resurgence theory applied to the ODE's Borel transform.
Series root: doi:10.5281/zenodo.19117399
GTCT/NucPhysB: doi:10.5281/zenodo.20682934
Vol I deposit: doi:10.5281/zenodo.19117400
AXLE repo: github.com/TOTOGT/AXLE
GTCT repo: github.com/TOTOGT/GTCT
Proof page: totogt.github.io/GTCT/book4/ch10.html#s65