I will present a gauge mediation principle for BSM theories where exact UV scale invariance is broken in a hidden sector: the relevant configurations are those in which the Standard Model and a hidden sector emanate from a scale invariant pair of UV theories that communicate only via gauge interactions. I will show that in such configurations, once scale invariance is broken in the hidden sector, relevant operators are generated in the Higgs potential, leading to electroweak symmetry breaking. Moreover, the resulting Higgs potential contains logarithmic mass-squared terms which lead to unusual low energy Higgs self-couplings, and I will discuss the corresponding phenomenologies.