  Supersymmetric extensions of the Standard Model with small R-parity and
lepton number violating couplings are naturally consistent with primordial
nucleosynthesis, thermal leptogenesis and gravitino dark matter. We consider
supergravity models with universal boundary conditions at the grand unification
scale, and scalar tau-lepton or bino-like neutralino as next-to-lightest
superparticle (NLSP). Recent Fermi-LAT data on the isotropic diffuse gamma-ray
flux yield a lower bound on the gravitino lifetime. Comparing two-body
gravitino and neutralino decays we find a lower bound on a neutralino NLSP
decay length, $c \tau_{\chi^0_1} \gsim 30 cm$. Together with gravitino and
neutralino masses one obtains a microscopic determination of the Planck mass.
For a stau-NLSP there exists no model-independent lower bound on the decay
length. Here the strongest bound comes from the requirement that the
cosmological baryon asymmetry is not washed out, which yields $c
\tau_{\tilde\tau_1} \gsim 4 mm$. However, without fine-tuning of parameters,
one finds much larger decay lengths. For typical masses, $m_{3/2} \sim 100 GeV$
and $m_{NLSP} \sim 150 GeV$, the discovery of a photon line with an intensity
close to the Fermi-LAT limit would imply a decay length $c\tau_{NLSP}$ of
several hundred meters, which can be measured at the LHC.
