  We report spin-polarized tunneling density of states measurements of the
proximity modulated superconductor-insulator transition in ultra thin Be-Al
bilayers. The bilayer samples consisted of a Be film of varying thickness,
$d_\mathrm{Be}=\,$0.8-4.5 nm, on which a 1 nm thick capping layer of Al was
deposited. Detailed measurements of the Zeeman splitting of the BCS coherence
peaks in samples with sheet resistances $R\sim h/4e^2$ revealed a super-linear
Zeeman shift near the critical field. Our data suggests that critically
disordered samples have a broad distribution of gap energies and that only the
higher portion of the distribution survives as the Zeeman critical field is
approached. This produces a counter-intuitive field dependence in which the gap
apparently increases with increasing parallel field.
