While attempts to prove the Collatz conjecture have so far culminated in the work of Terence Tao (2019), which establishes "Almost all orbits of the Collatz map attain almost bounded values" [3], several attempts have been made to generalize the Collatz sequences 3n + 1, but many of them produced sequences that lack the essential structural properties of the original Collatz dynamics. Among these, the most promising known generalization is the oneproposed in 2022 by Naouel Boulkaboul [2], which takes the form 3n + 3^k and leads sequences to converge toward 3^k. In this work, we propose a new methodological generalization of the Collatz sequences based on a two-part transformation (1 + 2^k)n + S_k(n) if n mod 2^k ̸= 0, and n/2^kif n mod 2^k = 0, where S_k(n) is a correction function preserving the generalized singularitypreviously revealed in [1]. This revised formulation ensures that all rank-1 branch beginnings exhibit the generalized singularity in binary form.