Primitive Winding, Fission Thresholds and Resonance on Toroidal Systems

This paper develops a mathematical account of persistence and fragmentation for primitive integer windings. It proves a least-cost fission theorem for strictly convex homogeneous winding energies under stated state-admission rules. Positive inverse-length corrections produce a different global minimum: a worked primitive-only model requires three fragments, with an exact finite-parent phase diagram and calculated large-winding crossovers.

The analysis connects these arithmetic results to an interacting winding-field energy and a separate mechanism for winding-changing transitions. The original tokamak, asteroid-belt, planetary-ring, cardiac-conduction and accelerator examples are retained with explicit sampling and dynamical controls. The paper provides exact results for the defined winding models and a testable programme for assessing their relevance to physical resonances, without identifying the different empirical mechanisms as one established universal law.