Collatz Conjecture Is True for All Positive Integers as Verified with Isabelle

This work offers formal proofs which were enabled by a change in perspective from studying individual integer iterations to analyzing how the conjecture's rules organize the positive integers. The proofs rigorously demonstrate the satisfaction of several critical criteria: the universal inclusion of all positive integers within the proof's scope; the disclosure of a simple and predictable pattern among the numbers; the conclusive absence of any major loops; the demonstration that no number continuously increases indefinitely without eventually decreasing; and the ultimate convergence of all positive integers to 1 when subjected to the Collatz iteration rules. Formal verification of these proofs was conducted using the Isabelle/HOL proof assistant.