Extended Electron Structures and the Mechanical Origin of Quantum Length Scales

This preprint series explores the mechanical basis of quantum localization and intrinsic length scales in electrons. Using a variational continuum framework, the electron is modeled as an extended, gradient-stiff entity coupled to its environment, revealing how the Compton wavelength and Bohr radius emerge naturally from internal mechanical consistency. The work also reinterprets the Heisenberg uncertainty principle as a structural consequence of localization costs. This series establishes a baseline for extended electron structures and identifies where additional physical structure is necessary to reconcile stability with motion.