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

The Canonical Triple-Graph (CTG) The set of positive integers is fixed, yet its familiar linear ordering conceals a latent hierarchical structure. This paper describes that structure through an admissible associator on the odd integers:n = (2ᵏm − 1) / 3, defined whenever 2ᵏm ≡ 1 (mod 3). Odd integers not divisible by 3 ("active") admit infinitely many admissible exponents; odd multiples of 3 ("inactive") admit none. These admissible relations define directed edges m → n between odd integers, giving every odd integer n ≠ 1 exactly one parent—the odd part of 3n + 1. The resulting graph is rooted and acyclic, with unique ancestry for every non-root vertex. Grouping three consecutive admissible associates of a fixed integer yields a canonical triple (n, 4n+1, 16n+5): a uniform affine pattern recurring at every level of the hierarchy, with exactly one of the three entries divisible by 3 at each step. Every block is uniquely represented by its least (first) canonical triple, establishing a deterministic recursive rule for tracing the ancestry of any block back to the root. The central result of this paper is the unconditional proof of structural completeness (Hypothesis H). We prove that every odd integer belongs to C(1), the component rooted at 1. The proof bypasses traditional dynamical framing and relies strictly on the rigid structural coordinates of the CTG:1. The impossibility of infinite inward rays is established via Fermat's Method of Infinite Descent. Moving inward systematically strips the accumulated structural parameters, which cannot be reduced indefinitely.2. With the chain proven finite, the structural depth is a rigorously defined non-negative integer. By the fundamental arithmetic property that a strictly decreasing finite counter must reach zero, every chain is forced to terminate at the unique root n = 1. (A separate topological argument excluding cycles is unnecessary, as a cycle would preclude the depth from reaching zero). Consequently, every ancestor chain is mathematically forced to terminate at the unique fixed point n = 1. Since the admissible edge set is exactly the inverse of the Collatz (3n+1) map, the Collatz conjecture is resolved as a direct structural corollary. Even integers attach canonically via their unique 2-adic factorization, forming vertical pillars above their odd parts and extending the organization to all positive integers. A further structural observation shows that groups of 3, 9, 27, … consecutive triples exhaust all residue classes modulo 9, 27, 81, …, yielding in the limit a unique infinite address for every integer in the structure—a positional coordinate system independent of numerical magnitude, with applications reserved for future work. Changes in this version— Complete proof of Hypothesis (H): Section 9 unconditionally proves structural completeness. The proof establishes the impossibility of infinite rays (via infinite descent) and forces termination at the root via the inherent arithmetic truth of finite depth, rendering separate cycle arguments unnecessary.— Dual affinity emphasized: Section 8 highlights the dual affine nature of the graph (uniform horizontal affinity and path-dependent vertical affinity) as the algebraic engine of the proof.