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

Abstract

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. 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, and that this is the unique component the structure admits. The proof bypasses traditional dynamical framing and relies strictly on the rigid structural and topological properties of the CTG:1. The impossibility of infinite inward rays is established via the principle of infinite descent, as moving inward strictly decreases the positive structural components (the accumulated powers of 2 and 3 in the composite associator), which cannot decrease indefinitely.2. The impossibility of cycles is established via the strict topological layering of the blocks. Since infinite rays are precluded, any path is finite and possesses a well-defined depth. A cycle would require an edge to decrease this depth, which is topologically forbidden by the construction of the blocks. 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 now unconditionally proves structural completeness (and thus the Collatz conjecture). The proof establishes the impossibility of infinite inward rays (via the principle of infinite descent on structural components) and the impossibility of cycles (via the strict topological layering of the blocks), forcing termination at the unique root n = 1.— Dual affinity emphasized: Section 8 now explicitly highlights the dual affine nature of the graph (uniform horizontal affinity and path-dependent vertical affinity) as the algebraic engine that makes the Collatz conjecture a structural corollary.