Construction of the New Rule via Isomorphic Transformation in the Collatz Conjecture, and the Proof of Absolute Convergence via Bit-Budget Balance Analyzed from a Game of Life Perspective

While the rules governing the Collatz Conjecture are elementary, formulating a rigorous proof presents a formidable challenge. The primary source of this complexity is entirely attributable to the n÷2 phase. Under the traditional rules, executing 3n+1 on any given integer n forces an even output if n is odd, and an odd output if n is even. Since the traditional rule dictates dividing by 2 whenever n is even, any number entering the 3n+1 operation must strictly be an odd integer. Consequently, the output of 3n+1 under the traditional rule is systematically guaranteed to be an even integer. Because it is always even, executing successive 3n+1 operations consecutively is structurally impossible. In sharp contrast, the n÷2 operation can be executed continuously for an unpredictable number of consecutive steps. The system may revert to an odd integer after a single division, or it may trigger a long sequence of multiple consecutive divisions. If a near-infinite chain of 0s aligns within the lower bits under binary representation, the n÷2 operation would theoretically repeat near-infinitely.As long as this traditional branching mechanism remains active, any analytical approach or higher-dimensional methodology inevitably hits the impenetrable barrier of the continuous n÷2 divisions, blocking any further progress. Therefore, the core thesis of this paper is to investigate an innovative methodology to completely eliminate the n÷2 branching rule from the analytical system. In this paper, the traditional parity branching (n÷2) rule of the Collatz Conjecture is entirely eliminated via a unidirectional equivalent transformation. By deploying a deterministic velocity calculation (bounding inequality) that cross-references the maximum capacity threshold for carrier bit-growth at the highest active register against the absolute minimum propagation velocity of lower bit elimination (0-propagation/Eater), this paper rigorously substantiates and demonstrates the comprehensive structural mechanism through which every single integer achieves complete convergence into singular points, analyzed strictly from a Game of Life perspective.