The Higgs quartic coupling λ has now been indirectly measured at the electroweak scale. Assuming no new low-scale physics, its running is known and, together with gauge and Yukawa couplings, it is a crucial new piece of information constraining UV completions of the Standard Model. In particular, supersymmetry broken at an intermediate or high energy scale with (i.e. λ=0) is consistent with present data and has an independent theoretical appeal. We analyze the possible string-theoretic motivations for (including both the shift-symmetry and the more economical variant of a Z2 symmetry) in a Higgs sector realized on either 6- or 7-branes. We identify specific geometries where λ≃0 may arise naturally and specify the geometrical problems which need to be solved to determine its precise value in the generic case. We then analyze the radiative corrections to λ. Finally we show that, in contrast to naive expectations, λ<0 at the SUSY breaking scale is also possible. Specifically, string theory may produce an MSSM plus chiral singlet at a very high scale, which immediately breaks to a non-SUSY Standard Model with λ<0. This classically unstable theory then becomes metastable through running towards the IR.