STEPANOV GEOMETRY FOR MAGNETIC CONFINEMENT FUSION: ANALYTIC FLUX SURFACES, FINITE-β EQUILIBRIUM, AND TURBULENT TRANSPORT COEFFICIENTS

We present an analytic geometric framework for toroidal magnetic confinement fusion based on the Stepanov substitution method. The framework expresses the coupled problems of magnetohydrodynamic equilibrium, bootstrap current, and turbulent transport in a unified geometric formalism, with all necessary theorems provedwithin the paper.The central object is the Stepanov Jacobian J on toroidal flux surfaces. We prove the following results:(1) Existence of analytic flux surfaces (Theorem 2.2): For any irrational rotational transform and smooth toroidal boundary, nested flux surfaces with straight field lines exist and are constructed via the Stepanov transformation.(2) Optimal coil geometry (Theorem 3.2): The external coil configuration minimising magnetic energy satisfies a boundary flatness condition, yielding an explicit analytic coil shape.(3) Regularisation of rational surfaces (Theorem 6.3): The divergence of the Stepanov transformation at rational rotational transform values is regularised bythe finite width of magnetic islands. A regularised transformation with a complex rotational transform ιε = ι + iε is smooth everywhere, with the imaginary partcontrolled by the island width. This provides an analytic formula for the resonant harmonic amplitudes bmn in terms of the coil geometry, enabling the systematic suppression of magnetic islands through optimal coil design.(4) Finite-β equilibrium (Theorem 5.1): The 3D Grad–Shafranov equation in Stepanov coordinates is a nonlinear elliptic equation for the Jacobian. Convergence of the Stepanov iteration is proved via the Nash–Moser implicit function theorem (The-orem 5.3).(5) Bootstrap self-consistency (Theorem 7.3): The coupled system of equilibrium, heat transport, and bootstrap current converges to a unique self-organised state.(6) Geometric transport coefficients (Theorem 8.1): All geometric quantities entering the gyrokinetic equation—magnetic drift, bounce time, precession frequency—are expressed analytically in terms of J.The method provides a rigorous geometric foundation for stellarator optimisation, reducing the computational complexity of the equilibrium problem from iterative numerical solution to an analytic construction, with convergence guarantees established by the Nash–Moser theorem.