= Solution
In the magnetic <Aharonov-Bohm effect>, coherent electron paths enclose inaccessible <magnetic flux> even though the field vanishes along the paths. The <magnetic vector potential> contributes relative <phase>
$$
\boxed{\Delta\phi=\frac q\hbar\oint\mathbf A\cdot d\boldsymbol\ell
=\frac{q\Phi_B}{\hbar}\pmod{2\pi}.}
$$
Recombination turns this <phase> into a fringe displacement. For an electron $q=-e$, changing <flux> by $h/e$ gives one full fringe period. Reversing the loop orientation reverses the <phase> sign. A gauge change modifies each path's <phase> convention but leaves the closed-loop <phase> and observed <interference> unchanged. The accessible region surrounds excluded <flux>, so a globally trivial gauge cannot remove that <phase>.
\Image[/past-exam-of-the-optics-and-electrodynamics-course-of-the-university-of-cambridge/2014/paper-1-oe2014-aharonov-bohm.png]
{title=Original two-path Aharonov-Bohm schematic and the half-period fringe shift from half an electron flux period; paths avoid the flux region}
A shielded toroidal magnet provides a concrete test: <electron holography> compared waves passing through its hole and outside it. Superconducting and copper coatings confined the field and excluded electrons from the magnet; the measured relative <phase> tested enclosed <flux> rather than a classical Lorentz deflection along the paths.<Evidence for Aharonov-Bohm effect with magnetic field completely shielded from electron wave by Akira Tonomura and others>[Physical Review Letters 56, 792 (1986), primary abstract: shielded toroid and electron-holographic relative phase]{ref}
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