What are the discretionary punishments (ta'zir) and modern penal theories in Islamic jurisprudence and positive law, and what is the point of comparison between them?

Ta'zir is the legitimate punishment for every sin for which there is no prescribed hadd (fixed penalty) or kaffarah (expiation). Its legitimacy is established by the Quran, Sunnah, and scholarly consensus. This is based on Allah's saying: "And strike them" (Quran 4:34), and the Prophet's (peace be upon him) saying: "No one should be flogged more than ten lashes except in a hadd from the Hudud of Allah." Ta'zir applies to offenses such as unlawful touching without intercourse, theft below the nisab (minimum threshold for hadd), slander other than accusing of adultery, and bearing false witness. It can be carried out by beating, imprisonment, or reprimand, and its application is left to the discretion of the Imam according to the need.