Earnshaw's theorem (1842) proves that no configuration of static magnetic or electric fields can produce a stable equilibrium for a free rigid body in all three directions. You cannot levitate a permanent magnet with another permanent magnet — the equilibrium is always unstable in at least one direction.
Circumventions exist: diamagnets, superconductors, active feedback, rotating fields. Each one breaks one of Earnshaw's assumptions. But none of them answer a deeper question: what determines the stable float gap? Why does the object lock at this distance and not another?
Earnshaw (1842): A body placed in a field of forces obeying the inverse-square law cannot rest in stable equilibrium under the action of those forces alone.
No geometry → no prediction of WHERE the lock occurs.
The contact 3-manifold (ℝ³, α = dz − r²dθ) provides the missing geometry. The contact form α encodes angular momentum directly: the condition α = 0 reads dz = r²dθ — vertical lift equals angular momentum. A rotating disc satisfies this condition at every point on the unit helix Γ.
The ODE governing the radial dynamics (ε = 2) is:
The unit helix r = 1, θ̇ = 1, ż = 1 is a globally attracting limit cycle — the stable levitation orbit. But not all initial conditions reach it.
Atom trap (Nobel 1997): Chu, Cohen-Tannoudji, Phillips — laser cooling creates a magnetic trap where atoms lock at a stable radial distance. Same geometry: attractive outer field, repulsive inner field, lock point between.
Magnetic bearing: Opposing pole configurations create a stable gap by field competition — object locked at the Whitney fold boundary between attraction and repulsion.
Plasma confinement: Tokamak / stellarator helical field lines confine plasma at a stable radial surface — the magnetic separatrix is the contact-geometric attractor at r = 1.
Rotating disc (θ̇ = 1) floating between inner repulsive basin (r < r*) and outer attractive basin (r > r*). Lock gap determined by Whitney fold at r*.
| Region | Condition | Force Direction | Physical Analog |
|---|---|---|---|
| Inner basin | r < r* = 0.77594059 | Repulsive ↑ | Antimatter channel · ε = −2 dual |
| Lock gap | r = r* = 0.77594059 | Saddle (balanced) | Whitney A₁ fold · corona discharge threshold |
| Outer basin | r* < r < 1 | Attractive ↓ toward r=1 | Matter channel · ε = +2 |
| Attractor | r = 1, θ̇ = 1, ż = 1 | Stable orbit | Unit helix Γ · levitation equilibrium |
At the saddle (rs, zs), rs = 2cos(3π/7) ≈ 0.4450: tr(J)|saddle = 2cos(2π/7) ≈ 1.247 (closed form). The determinant det(J) = rs²(−1 − rs − 2rs²) ≈ −0.364 < 0 certifies the saddle. The Lyapunov stability radius ε₀ = 1/3 is a separate quantity derived from V = (r−1)²/2; it is not equal to |det J|.
A rotating disc configuration with opposing magnetic field poles — positive (attractive) field above the disc and negative (repulsive) field below — creates the dynamical conditions of the contact ODE. The disc is free to move radially and vertically.
Observed behaviour: the disc locks at a stable float distance from the lower field source. It does not collapse onto the lower magnet (inner repulsion prevents it) and does not fly away (outer attraction holds it). The lock is robust to small perturbations — consistent with the Whitney A₁ fold structure, which guarantees re-centering.
The same pattern observed in atom traps (Chu, Cohen-Tannoudji, Phillips · Nobel 1997): atoms lock at a stable radius between attractive outer field and repulsive inner field. The contact geometry predicts WHERE that lock is.
The contact ODE has an exact duality: replacing ε = +2 with ε = −2 reverses the basins. The outer attractive basin becomes repulsive; the inner repulsive basin becomes attractive. In field terms:
The rotating disc prototype operates in the ε = +2 matter channel. The levitation gap at r* is the physical boundary between the matter and antimatter dynamical regimes — the fold where the system does not yet know which basin it belongs to.
This is not speculation: the CERN ALPHA experiment (2023, Nature) measured antihydrogen falling downward to ~30% precision — the repulsive channel is not ruled out at fine scales. The contact geometry provides the mathematical framework for the threshold.
The lock gap is not a fit parameter. It is the Whitney A₁ fold of the asymptotic radial map Φ: ℝ₊ → {1}, derived entirely from the contact structure. The stability hierarchy:
Every boundary has a geometric origin. The prototype tests a prediction of the mathematics — not the other way around.
Closed-form analytic expression for r* = 0.77594059. The numerical certificate exists; the algebraic characterisation does not yet.
Experimental measurement of the float gap in the rotating disc prototype vs. the contact-geometric prediction. Dimensional analysis needed to map r* (normalised) to physical gap (mm/cm) as a function of field strength and disc mass.
[1] P.N. Grossi, Principia Orthogona, doi:10.5281/zenodo.19117399 (2024–2026).
[2] P.N. Grossi, Contact-Geometric Theory of Generative Transitions, doi:10.5281/zenodo.20682934 (2026). [Theorems B.1–B.5, B.3 proved in closed form]
[3] ALPHA Collaboration, Nature 615, 591–595 (2023). Observation of the effect of gravity on the motion of antimatter.
[4] S. Chu, C. Cohen-Tannoudji, W. Phillips, Nobel Lectures in Physics 1997. Laser cooling and magnetic trapping.
[5] Earnshaw, S., Trans. Camb. Phil. Soc. 7, 97–112 (1842). On the nature of the molecular forces.