Given experimental evidence at the LHC for physics beyond the standard model,
how can we determine the nature of the underlying theory? We initiate an
approach to studying the "inverse map" from the space of LHC signatures to the
parameter space of theoretical models within the context of low-energy
supersymmetry, using 1808 LHC observables including essentially all those
suggested in the literature and a 15 dimensional parametrization of the
supersymmetric standard model. We show that the inverse map of a point in
signature space consists of a number of isolated islands in parameter space,
indicating the existence of "degeneracies"--qualitatively different models with
the same LHC signatures. The degeneracies have simple physical
characterizations, largely reflecting discrete ambiguities in electroweak-ino
spectrum, accompanied by small adjustments for the remaining soft parameters.
The number of degeneracies falls in the range 1<d<100, depending on whether or
not sleptons are copiously produced in cascade decays. This number is large
enough to represent a clear challenge but small enough to encourage looking for
new observables that can further break the degeneracies and determine at the
LHC most of the SUSY physics we care about. Degeneracies occur because
signatures are not independent, and our approach allows testing of any new
signature for its independence. Our methods can also be applied to any other
theory of physics beyond the standard model, allowing one to study how model
footprints differ in signature space and to test ways of distinguishing
qualitatively different possibilities for new physics at the LHC.Comment: 55 pages, 30 figure