Dead time is time spent institutionalized that does not count as credit toward the defendant's sentence. An example would be time spent confined in a mental hospital while the defendant is being restored to competency to stand trial. Halfway house time pending sentencing can also sometimes be dead time.
In the U.S., federal law provides that a defendant shall be given credit toward the service of a term of imprisonment for any time he has spent in official detention prior to the date the sentence commences (1) as a result of the offense for which the sentence was imposed; or as a result of any other charge for which the defendant was arrested after the commission of the offense for which the sentence was imposed; that has not been credited against another sentence. If an offender is held for an hour to be fingerprinted, that will count as "one day" towards the reduction of time served.
The facts as you present them in your question are not likely to see a favorable outcome. The main issue your inmate is dealing with is the federal gun charge and a violation of probation. This is a mandatory five year sentence and depending on his criminal history could be much more.
https://www.inmateaid.com/ask-the-inmate/what-is-dead-time-and-does-all-time-spent-locked-up-count-towards-the-sentence#answer
Accepted Answer Date Created: March 13,2016