A Structural Approach to the Collatz Conjecture
Year: 2026
Type: preprint
Abstract: This preprint develops a structural and set-theoretic approach to the Collatz conjecture, with emphasis on the dynamics of odd integers. The usual Collatz process is compressed into odd-to-odd steps, and the sets Am are defined according to the exact number of compressed steps required to reach the terminal state 1. Their union is denoted by U. Independently, the odd integers are partitioned through residue classes and finite-depth constraint families C, providing a complete arithmetic covering of the odd domain without omission. The paper carefully distinguishes between arithmetic coverage and membership in U, proves the parent-child inclusion relation for the partition constraints, and clarifies the logical role of the partition in relation to the Collatz dynamics. Several potential referee questions concerning admissibility, coverage, inclusion, and the role of sum identities are also addressed.
Author: Darabi, Saed
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212428387
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