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The electron is re-examined as a single electromagnetic wave that closes upon itself with a half-twist, forming a Möbius ribbon whose circumference equals one reduced Compton wavelength. This topological closure traps the invariant rest-energy and forces the field to return to its initial phase only after a 4π rotation, reproducing spin one-half without invoking abstract spinors. Adding an integer number N of radial nodes generates a standing-wave ladder where frequency and radius scale as f_N = f₀/(N +1) and r_N = (N +1)r_inner. The ladder matches the Bohr--Rydberg spectrum when n=N+1 and links the fine-structure constant to the Compton--Bohr radius ratio α = r_inner/a₀. We give a concrete sketch for the Schwinger anomaly from core--envelope interference and outline how point-like scattering can coexist with a finite geometric core. The geometry yields two near-term tests: an enhanced h/2e harmonic in mesoscopic Aharonov--Bohm rings (with the usual h/e present but relatively suppressed in a specific regime), and a forward-Compton form-factor minimum at a momentum transfer set by Qr_inner≈ x₁ with x₁ the first Bessel zero (x₁ ≈ 2.4048), i.e. Q ≃ 1.2 MeV/c to 1.6 MeV/c depending on the aperture kernel (ring-like J₀ vs spherical sinc). We also streamline the orbital taxonomy (S,P,D,...) as emergent angular families from azimuthal node counts; the music-note analogy is offered only as a light heuristic.

Competing interests

The authors declare that they have no competing interests.

Use of Artificial Intelligence (AI)

The authors declare that they did not use generative AI to come up with new ideas for their review.