  We study Renormalization Group invariant (RGI) quantities in the Minimal
Supersymmetric Standard Model and show that they are a powerful and simple
instrument for testing high scale models of supersymmetry (SUSY)-breaking. For
illustration, we analyze the frameworks of minimal and general gauge mediated
(MGM and GGM) SUSY-breaking, with additional arbitrary soft Higgs mass
parameters at the messenger scale. We show that if a gaugino and two first
generation sfermion soft masses are determined at the LHC, the RGIs lead to MGM
sum rules that yield accurate predictions for the other gaugino and first
generation soft masses. RGIs can also be used to reconstruct the fundamental
MGM parameters (including the messenger scale), calculate the hypercharge
D-term, and find relationships among the third generation and Higgs soft
masses. We then study the extent to which measurements of the full first
generation spectrum at the LHC may distinguish different SUSY-breaking
scenarios. In the case of MGM, although most deviations violate the sum rules
by more than estimated experimental errors, we find a 1-parameter family of GGM
models that satisfy the constraints and produce the same first generation
spectrum. The GGM-MGM degeneracy is lifted by differences in the third
generation masses and the messenger scales.
