1. Electromagnetism Refresher

Why a simple toroidal solenoid fails catastrophically via $\nabla B$ drift, and why field lines must twist ($\iota \neq 0$).

2. 3D Geometry, Helices & Knots

Why a stellarator is a closed toroidal helix (Călugăreanu writhe), when field lines form $(m,n)$-torus knots, and what parameters shape it.

3. The Four Great Perils

Magnetic islands (KAM breakdown), Neoclassical $1/\nu$ banana losses, Microturbulence (ITG/TEM), and Coil Manufacturing Tolerances.

Lorentz Force & Guiding-Center Drift

A particle of mass $m$, charge $q$ in a magnetic field $\mathbf{B}$ obeys $m\dot{\mathbf{v}} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})$. To lowest order, it gyrates around a field line at cyclotron frequency $\Omega = qB/m$ with Larmor radius $\rho = v_\perp / \Omega$, conserving magnetic moment $\mu = \frac{m v_\perp^2}{2B}$.

By the Hairy Ball / Poincaré–Hopf theorem, a nowhere-vanishing tangent field requires a torus $T^2$. But bending a solenoid into a torus of major radius $R$ forces $B \propto 1/R$ (by Ampère's Law $\oint \mathbf{B}\cdot d\mathbf{l} = \mu_0 I_{\text{tf}}$). This gradient & curvature drive a cross-field drift:

• The Fatal Trap (Untwisted): Ions drift UP, electrons drift DOWN $\implies$ vertical electric field $\mathbf{E}_{\text{vert}}$ $\implies$ outward $\mathbf{E} \times \mathbf{B}$ expulsion into the wall in microseconds!
• The Cure (Helical Twist $\iota$): Twisting field lines from top to bottom short-circuits the charge separation!

Macroscopic Force Balance as Differential Forms

Viewing the plasma macroscopically as a conducting fluid with pressure $p(x)$, current density $\mathbf{J}(x)$, and magnetic field $\mathbf{B}(x)$, static equilibrium requires:

Taking the inner product with $\mathbf{B}$ and $\mathbf{J}$ immediately yields two geometric orthogonality laws:

• Arnold–Poincaré–Hopf Theorem: At every regular value where $\nabla p \neq 0$, the compact level surface $\Sigma_p = p^{-1}(c)$ supports two linearly independent tangent vector fields $(\mathbf{B}, \mathbf{J})$.
• Therefore, the tangent bundle $T\Sigma_p$ is trivial ($\chi(\Sigma_p) = 0$), forcing every regular flux surface $\Sigma_\psi$ to be diffeomorphic to a 2-torus $T^2$!

Rotational Transform $\iota(\psi)$ as a Rotation Number

Label the nested flux surfaces by toroidal magnetic flux $\psi$. On each torus $\Sigma_\psi \cong S^1_\theta \times S^1_\phi$, the solenoidal field $\mathbf{B}$ defines a volume-preserving flow.

How do we generate $\iota(\psi) \neq 0$?

• Tokamak (2D Axisymmetric): Drives megamperes of internal plasma current $\mathbf{J}_\phi$ ($\implies$ transformer pulses & violent current-driven disruptions).
• Stellarator (3D Non-Axisymmetric): Generates $\iota(\psi)$ purely from external 3D coils and 3D surface geometry with zero net plasma current ($I_p = 0$).

1. Why It Is NOT a Regular Circular Helix

• An infinite straight helix has constant curvature $\kappa_0$ and constant torsion $\tau_0$, enjoying exact continuous screw symmetry in $\mathbb{R}^3$.
• Closing the Helix into a Torus: Bending the helix into a closed loop $\gamma_0 : S^1 \to \mathbb{R}^3$ breaks Euclidean symmetry! Curvature $\kappa(s)$ and torsion $\tau(s)$ become periodic functions of arc length $s$.
• By the Garren–Boozer Theorem, exact quasisymmetry cannot hold to all orders in distance $r = \sqrt{\psi}$ from a closed curve (it becomes overdetermined at $O(r^3)$), turning stellarator design into a deep geometric optimization problem.

2. How Torus Knots & Writhe Govern the Twist

By the Călugăreanu–White–Fuller Theorem, the self-linking of a ribbon around the magnetic axis $\gamma_0$ decomposes into geometric Writhe and frame Twist:

• 3D Axis Writhe ($\mathrm{Wr}$): A helical axis winding $(1, N_{\text{fp}})$ around a torus is topologically an unknot, but its 3D writhe generates rotational transform $\iota_0$ without any plasma current!
• Rational Field Lines = $(m,n)$-Torus Knots: Whenever $\iota(\psi) = n/m \in \mathbb{Q}$, every closed magnetic field line on $\Sigma_\psi$ is literally an $(m,n)$-torus knot!

The Poincaré Return Map on a Poloidal Slice

In straight-field-line (Boozer) coordinates $(\psi, \theta, \phi)$, the magnetic field 2-form is $\omega_B = d\psi \wedge (d\theta - \iota(\psi)d\phi)$, so field lines satisfy the linear flow:

• Irrational Diophantine $\iota \notin \mathbb{Q}$: A single field line densely fills the 2-torus $T^2$ (Kronecker foliation), forming a robust KAM flux surface that confines plasma pressure.
• Rational $\iota = n/m \in \mathbb{Q}$: The field line closes on itself after $m$ toroidal and $n$ poloidal turns as an $(m,n)$-torus knot.
• Peril #1 (Resonant Magnetic Islands): At rational $\iota = n/m$, any tiny resonant magnetic perturbation $B_{m,n}$ tears the flux surface into a chain of magnetic islands, short-circuiting radial confinement!

1. Near-Axis Curve Parameters

Build outward from the magnetic axis $\gamma_0(s)$ in powers of $r = \sqrt{\psi}$:

• $N_{\text{fp}} \in \mathbb{N}$: Number of discrete field periods around the torus.
• Axis Harmonics $(R_{0,n}, Z_{0,n})$: Define curvature $\kappa(s)$, torsion $\tau(s)$, and axis writhe $\mathrm{Wr}(\gamma_0)$.
• Elongation $\bar{\eta} > 0$: Controls $|B| = B_0(1 + r\bar{\eta}\cos\vartheta)$ and elliptical eccentricity via a 1D periodic Riccati ODE.

2. Outer Boundary Fourier Modes

Double Fourier series on $T^2$:

• $R_{0,0}, R_{1,0}$: Major & minor radius.
• $R_{0,1}, Z_{0,1}$: 3D helical axis excursion.
• $R_{1,1}, Z_{1,1}$: Rotating ellipse twist.
• $R_{2,1}, Z_{2,1}$: "Bean" / triangularity indentation (creates magnetic well!).

3. 3D Modular Coil Filaments

External Biot–Savart hardware curves $\gamma_k : S^1 \to \mathbb{R}^3$:

• Unlinked poloidal loops with 3D out-of-plane warping reproduce helical fields without linked torus-knot coils.
• Governed by clearance $d_{\min} > 0$ and max curvature $\kappa_{\max}$.

What is Microturbulent Transport?

In Wendelstein 7-X, once 3D optimization suppressed neoclassical $1/\nu$ transport, radial heat loss was dominated by microturbulence—small-scale ($\sim \rho_i$) drift-wave instabilities driven by steep temperature and density gradients ($\nabla T_i, \nabla T_e, \nabla n$):

• Cross-Field $\mathbf{E} \times \mathbf{B}$ Eddies: Microinstabilities create fluctuating potentials $\delta\phi$, driving radial turbulent advection $\delta\mathbf{v}_E = \frac{\mathbf{B} \times \nabla\delta\phi}{B^2}$.
• ITG (Ion Temperature Gradient): Driven when $\eta_i = \frac{L_n}{L_{T_i}} = \frac{d\ln T_i}{d\ln n} > \eta_{\text{crit}}$ in regions of "bad" magnetic curvature ($\kappa_n < 0$) where $\nabla B$ and $\nabla p$ point in the same direction. Caused "ion temperature clamping" ($T_i \le 1.5\text{ keV}$) in early W7-X experiments!
• TEM (Trapped Electron Mode): Driven when bounce-trapped electrons in magnetic wells precess in the same direction as drift waves, resonantly feeding them energy.

1. The Maximum-$J$ Criterion ($\partial_\psi \mathcal{J}_\parallel < 0$)

The longitudinal second adiabatic bounce invariant for particles trapped between mirror points $l_\pm$ is:

Trapped particles precess toroidally at rate $\Delta\alpha = -\frac{1}{q}\frac{\partial \mathcal{J}_\parallel}{\partial \psi}$, where:

• When bean shaping ($R_{2,1} > 0$) and finite $\beta$ create a magnetic well ($\partial_\psi B > 0$), we get $\partial_\psi \mathcal{J}_\parallel < 0$ (Maximum-$J$).
• Result: Trapped electrons precess backwards relative to drift waves, breaking the TEM resonance!

2. Helander–Mackenbach Available Energy ($\widehat{\mathcal{A}}$)

How much thermal free energy can a 3D magnetic geometry possibly release into turbulence while conserving phase-space volume and adiabatic invariants $(\mu, \mathcal{J}_\parallel)$?

• Here $\hat{\omega}_* = \omega_n + (z - \tfrac{3}{2})\omega_T$ is the diamagnetic gradient drive and $\hat{\omega}_\alpha \propto -\partial_\psi \mathcal{J}_\parallel$ is the bounce precession.
• Why This Is Revolutionary for Formal Verification: $\widehat{\mathcal{A}}$ is a closed-form 2D geometric integral on each flux surface that rigorously upper-bounds turbulent energy without solving a 5D nonlinear PDE!

Three Geometric & Hardware Failure Modes

1. Flux-Coordinate Jacobian Folding ($\det D\Phi \le 0$): If Fourier shaping modes $(R_{m,n}, Z_{m,n})$ are pushed too far by an unverified optimizer, interior flux surfaces self-intersect or fold over (violating complex-analytic univalence on the poloidal disk $\mathbb{D}$).
2. Ideal MHD Ballooning & Interchange ($\delta^2 W < 0$): At high plasma pressure $\beta = 2\mu_0 p / B^2$, pressure gradients in bad-curvature regions overcome magnetic field-line bending energy, triggering macroscopic instability.
3. Sub-Millimeter Coil Manufacturing Sensitivity: Because 3D magnetic fields decay rapidly away from modular coils, a $\pm 1.5\text{ mm}$ assembly warp $\delta\gamma_k$ can destroy quasisymmetry or create resonant islands unless certified via Biot–Savart Fréchet Lipschitz bounds:
   $$\|\mathbf{B}(\gamma+\delta\gamma) - \mathbf{B}(\gamma)\| \le L_{\text{BS}}(d_{\min})\,\|\delta\gamma\|_{C^1}$$

What Each Slider & Input Controls

• Field Periods $N_{\text{fp}}$ & Rotational Transform $\iota_0$: Set the discrete dihedral symmetry and field-line twist. Watch the Field-Line Topology card switch between Irrational KAM Surfaces and Closed $(q,p)$-Torus Knots!
• Helical Axis Excursion $R_{0,1}$: Bends the central magnetic axis $\gamma_0$ into a 3D toroidal helix, increasing Călugăreanu writhe $\mathrm{Wr}(\gamma_0)$.
• Bean / Triangularity $R_{2,1}$ & Magnetic Well $\partial_\psi B$: Indent the cross-section to make $\partial_\psi \mathcal{J}_\parallel < 0$ (Maximum-$J$), stabilizing TEM turbulence.
• Gradients $a/L_{T_i}$ and $a/L_n$: Crank up $a/L_{T_i}$ or flatten $a/L_n$ to exceed $\eta_{\text{crit}}$ and trigger ITG Temperature Clamping (reproducing the W7-X wake-up call!).

Live Custom Function Formulas to Try

In Panel 3 of the explorer, you can type live mathematical expressions in phi, theta, and s:

• Figure-8 / Spatial Writhe Axis:
  0.28 * cos(2 * phi)
• Reversed-Shear Rotational Transform Profile:
  iota - 0.25 * s + 0.35 * s * s
• Quasi-Helically Symmetric $|B|$ Modulation:
  1 - 0.22 * cos(theta - N * phi)
• Unoptimized Toroidal Mirror Ripple (High $1/\nu$ & TEM):
  1 - 0.25 * cos(theta) - 0.20 * cos(N * phi)

1. Rosetta Stone →

16-row dictionary mapping Stellarator Physics $\leftrightarrow$ Pure Geometry $\leftrightarrow$ Lean 4 Kernel Contracts.

2. 10-Min Kernel Audit →

Inspect the small Lean 4 verification kernel and non-vacuity sanity checks.

3. Pure Math Bounties →

6 self-contained Lean 4 challenge files in Topology, Diff. Geometry & Complex Analysis.