# Discrete Electric Charge as a Topological Invariant of Source-Free Maxwell Fields ### Abstract (<=120 words) State that a pair of integer windings $Q=(n_{1},n_{2})\in\pi_{1}(T^{2})\cong\mathbb Z^{2}$ reproduces the observed discreteness of electron charge without sources: each winding contributes a fixed flux quantum; stable field lines force integer multiples. Quantised Coulomb energy emerges from mode geometry, not from Gauss-law sources. ### Skeleton 1. **Motivation** – experimental charge quantisation; limitations of source-based explanations. 2. **Geometry of a toroidal vacuum mode** – radii $(R,r)$, harmonic forms, definition of $Q$. 3. **Topological theorem** – smooth Maxwell evolution preserves $Q$; sketch homotopy proof. 4. **Flux quantisation** – each unit of $Q$ carries flux $\Phi_{0}$; derive from Stokes + single-valued vector potential. 5. **Mapping to the electron** – match $\Phi_{0}$ with $\alpha$ (fine-structure) to fix $R/r$. 6. **Observable predictions** – (i) half-integer anomalies forbidden; (ii) allowed annihilation channels require $\Delta Q=0$; list spectroscopic tests. 7. **Discussion** – relation to Dirac monopole quantisation and $\theta$-vacua; why no free parameters remain. ## Notes * Keep “light-based metric” as a practical ruler only; emphasise metric-free statements of observables. * Cite prior “energy–energy attraction” derivation and clearly state when that argument is reused. * Append detailed proofs or numerical code; main text stays lean, \~6–8 pages each. These outlines isolate the two messages you want: **(I) charge quantisation from topology, (II) inverse-square attraction from energy**, making the conceptual difference unmistakable. ## chatgpt convo https://chatgpt.com/c/6892797a-7af0-8330-aeb1-2f1757fa704b