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The Neil Gehrels Swift observatory observe Gamma-Ray bursts (GRBs) plateausin X-rays. We test the reliability of the closure relations through thefireball model when dealing with the GRB plateau emission. We analyze 455 X-raylightcurves (LCs) collected by \emph{Swift} from 2005 (January) until 2019(August) for which the redshift is both known and unknown using thephenomenological Willingale 2007 model. Using these fits, we analyze theemission mechanisms and astrophysical environments of these GRBs through theclosure relations within the time interval of the plateau emission. Finally, wetest the 3D fundamental plane relation (Dainotti relation) which connects theprompt peak luminosity, the time at the end of the plateau (rest-frame), andthe luminosity at that time, on the GRBs with redshift, concerning groupsdetermined by the closure relations. This allows us to check if the intrinsicscatter \sigma_{int} of any of these groups is reduced compared to previousliterature. The most fulfilled environments for the electron spectraldistribution, p>2, are Wind Slow Cooling (SC) and ISM Slow Cooling for cases inwhich the parameter q, which indicates the flatness of the plateau emission andaccounts for the energy injection, is =0 and =0.5, respectively, both in thecases with known and unknown redshifts.  We also find that for the sGRBs All ISM Environments with $q=0$ have thesmallest \sigma_{int}=0.04 \pm 0.15 in terms of the fundamental plane relationholding a probability of occurring by chance of p=0.005. We have shown that themajority of GRBs presenting the plateau emission fulfil the closure relations,including the energy injection, with a particular preference for the Wind SCenvironment. The subsample of GRBs that fulfil given relations can be used aspossible standard candles and can suggest a way to reduce the intrinsic scatterof these studied relationships.

Closure relations during the plateau emission of Swift GRBs and the fundamental plane