# 8. Topology and Discreteness When source-free flow closes on itself, closure imposes integer winding. Integer winding yields discrete mode families. Discreteness is geometric before it is spectral. A closed surface does not automatically support smooth continuous circulation. On a sphere, a continuous nowhere-vanishing tangential flow is impossible: any attempt to comb it smoothly leaves at least one zero or defect, by the hairy-ball theorem. A torus avoids that obstruction. It is the simplest closed surface on which continuous circulation can close on itself without enforced stagnation points. On a closed loop of such a surface, a continuous pattern must match itself after one circuit. That requires $$ n \lambda = L, $$ where $L$ is the circuit length and $n$ is an integer. On a torus there are two independent non-contractible cycles, so a closed flow is labeled by an integer pair $(m,n)$. Those winding classes do more than label allowed modes. They classify the circulation itself. The same closed winding that forces discrete matching also carries angular momentum about the center of the bounded mode. If $\mathbf{X}$ is that center, write the relative position within the mode as $$ \boldsymbol{\rho}=\mathbf{x}-\mathbf{X}. $$ Then the angular momentum of the closed circulation is $$ \mathbf{L}=\int \boldsymbol{\rho}\times\frac{\mathbf{S}}{c^2}\,dV. $$ Spin is this angular momentum of the self-closing toroidal circulation. Charge, introduced in chapter 10, is a different global aspect of the same mode: the signed through-hole flux across the torus aperture. The torus therefore supports more than one discrete global aspect of one circulation, because it has more than one non-contractible cycle. The sign of spin is set by handedness: the orientation of the circulation relative to the direction of advance. Different winding classes $(m,n)$, together with that handedness, define different discrete circulation classes. The allowed wavelengths and frequencies are therefore discrete. Different integers label different global modes. The key point comes before any specific spectrum: once source-free transport closes on itself, continuity and single-valuedness force integer classes of solutions. Specific spectral laws require additional geometry and come later. The present step is narrower and stronger: quantization begins as closure.