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

Introduction: The set of positive integers admits a latent hierarchical structure beyond the familiar linear ordering. We introduce the Canonical Triple-Graph (CTG), a fixed directed graph on the positive integers defined by an admissible associator on odd integers: n = (2^k m - 1)/3, where 2^k m ≡ 1 (mod 3). This relation is interpreted algebraically as defining edges that exist a priori within a predetermined structure, not as steps of an iterative or dynamical process.

Method: We analyze the structural properties of the admissible associator and its induced adjacencies. Every odd integer not divisible by 3 admits infinitely many admissible exponents forming blocks of associates. These blocks decompose uniquely into canonical triples of the affine form (n, 4n+1, 16n+5), which expose uniform self-similarity throughout the structure. Even integers integrate canonically via 2-adic factorization, forming deterministic vertical pillars above their odd parts.

Results: The CTG forms a directed graph (forest) on the positive integers. The distinguished root 1 generates Block(1), an infinite set of odd integers each serving as the root of its own infinite self-similar tree with identical local structure governed by canonical triples. We prove structural completeness via a block-closure principle: every block-closed component must contain an element of Block(1), ensuring all odd integers belong to trees rooted in Block(1). The structure is acyclic with unique parenthood within the CTG.

Conclusions: The CTG provides a purely structural framework comprising infinitely many self-similar trees, all rooted in Block(1), organizing positive integers independent of numerical magnitude or dynamical interpretation. This reframes classical iteration questions as structural position and component membership within an a priori fixed combinatorial forest.