What is the reliable authority for judging the authenticity of Hadith, and what methodology did early scholars follow in that regard?

Judging the authenticity (sahih) or weakness (da'if) of a hadith relies on consulting the statements of specialized hadith scholars or their books on authentic and weak hadiths. Among the authoritative books whose authors intended to compile authentic hadiths are: Sahih al-Bukhari and Muslim, then Sahih Ibn Khuzaymah, Ibn Hibban, and Mustadrak al-Hakim. As for the books that focused on compiling weak and fabricated hadiths, they include: "Al-Mawdu'at" by Ibn al-Jawzi, "Al-La'ali al-Masnu'ah" by al-Suyuti, and "Silsilat al-Ahadith ad-Da'ifah wal-Mawdu'ah" by Al-Albani.