Canonical Partition of the Set of Odd Integers via Linear Chains Generated by the Affine Operator 4x + 1

This note presents a structural decomposition of the set of positive odd integers.We show that the odd integers can be uniquely partitioned into a countable infinite family of disjoint subsets, called "Affine Chains," each generated by a unique primitive element or "Kernel" via the operator T(x)=4x+1.The work establishes existence and uniqueness conditions for this decomposition, provides a deterministic algorithm for extracting Kernels, and classifies Kernels according to their modular properties.These results introduce a non-standard coordinate system for odd integer arithmetic. Author: Alberto Lladó MolinsORCID: 0009-0008-6738-9766 This work is original and represents preliminary results intended as a basis for future developments.