The Canonical Triple-Graph: A Structural Organization of the Positive Integers

This work introduces the Canonical Triple-Graph (CTG), a fixed, a priori directed graph on the positive integers defined by a purely algebraic admissible relation on odd integers. The graph is not generated, explored, or traversed dynamically; all vertices and admissible edges exist simultaneously as part of a static global structure.

The admissible relation partitions odd integers into disjoint infinite blocks, each of which decomposes uniquely into canonical triples of affine form (n, 4n+1, 16n+5). These triples introduce no new vertices or edges and serve solely as canonical organizational units that expose a rigid and uniform local structure. Every odd integer except 1 has a unique parent, yielding a rooted, acyclic hierarchy that is independent of numerical size, ordering, or iteration.

Even integers are incorporated canonically via their unique 2-adic factorization, forming vertical pillars above their odd components. This extension preserves the admissible hierarchy and introduces no additional branching.

The primary contribution of the work is not the proposal of new arithmetic relations but the identification and systematic exploitation of canonical structural units and invariant local rules that reveal a global, self-similar organization already implicit in familiar identities. The framework replaces dynamical iteration by a static combinatorial organization and provides a deterministic structural coordinatization of the positive integers.

This paper is written in a contemporary academic climate characterized by widespread sensitivity toward Collatz-related research, by the increasing presence of AI-assisted tools, and by an observable pattern of immediate desk rejections, sometimes issued without engaging with the actual content of the work. The present paper does not address the Collatz conjecture. Any use of AI was limited strictly to assistance with English language formulation; all mathematical content, structure, and arguments are entirely the author’s own.