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A constructive scheme for determining pure states (clusters) at very lowtemperature in the 3-spins glass model on a random lattice is provided, in fullagreement with Parisi's one step replica symmetry breaking (RSB) scheme. Proofis based on the analysis of an exact decimation procedure. When the number c ofcouplings per spin is smaller than some critical value c_d, all spins areeliminated at the end of decimation (RS phase). In the range c_d<c<c_s, areduced Hamiltonian is left; each ground state (GS) of the latter is a "seed"from which a cluster of GS of the original Hamiltonian can be reconstructed.Above c_s, GS are frustrated with an energy per spin larger than -c. The numberof GS in each cluster, the number of clusters, the distances between GS arecalculated and correspond to RSB predictions

Rigorous Decimation-Based Construction of Ground Pure States for Spin-Glass Models on Random Lattices