The TOTEM experiment at the LHC has performed the first measurement at √s = 13 TeV of the ρ parameter, obtaining the following results: ρ =0.09±0.01 and ρ =0.10±0.01, depending on different physics assumptions and mathematical modelling. The unprecedented precision of the ρ measurement, combined with the TOTEM total cross-section measurements in an energy range larger than 10 TeV (from 2.76 to 13 TeV), has implied the exclusion of all the models classified and published by COMPETE. The ρ results obtained by TOTEM are compatible with the theoretical model predictions. The very low-|t| reach allowed also to determine the absolute normalisation using the Coulomb amplitude for the first time at the LHC and obtain a new total proton-proton cross-section measurement σtot = (110.3±3.5) mb, completely independent from the previous TOTEM determination. Combining the two TOTEM results yields σtot = (110.5±2.4) mb.\n