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: n = (2ᵏm − 1) / 3, where 2ᵏm ≡ 1 (mod 3). The graph is not generated, explored, or traversed dynamically; all vertices and admissible edges exist simultaneously as part of a fixed, a priori structure, independent of any traversal, iteration, or convergence assumption. 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 independent of numerical size, ordering, or iteration. Even integers are incorporated canonically via their unique 2-adic factorization, forming vertical pillars above their odd parts without introducing additional branching. Structural completeness. The block-dependence principle establishes that 1 is the unique block-free integer. Depth is defined intrinsically by block membership: each member of Block(m) has depth equal to the depth of m plus 1, and an integer outside every block has depth 0. Since depth is non-negative and strictly decreases from child to parent, every upward chain terminates at a block-free integer of depth 0. That terminus satisfies Eₖ(x) = x; solving yields x(2ᵏ − 3) = 1, forcing x = 1 and k = 2. Therefore 𝒞(1) is the only block-closed component and exhausts all positive integers. Since the admissible edge set is the exact inverse of the 3n+1 map, the Collatz conjecture follows as a structural corollary: every positive integer lies in the single rooted component anchored at 1, and every forward trajectory under the Collatz map reaches 1. Residue structure and addressing. Groups of 3, 9, 27, … consecutive triples within any block exhaust all residue classes modulo 9, 27, 81, … respectively, closing exactly at successive powers of 3. This property holds within every block. As 3ᵗ grows, integers congruent at coarser resolution become distinguishable—each successive power of 3 reveals finer identity distinctions, with all identities present simultaneously and progressively revealed rather than generated. In the limit t → ∞, every element of the block receives a unique infinite address determined by its position within the hierarchy of groups of 3, 9, 27, …, independently of numerical magnitude. The development of this addressing structure and its applications in positional algebra are reserved for future work. The primary contribution 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 present in the family of arithmetic identities governing ℕ⁺. The CTG departs from generative approaches that build the inverse tree layer by layer, entangling structure with convergence. It resolves the circular definition inherent in any such construction by declaring all admissible edges to exist a priori and replacing traversal with block-closure and the depth argument as the organizing principles. This version extends the prior work and context section with a kinship criterion — a structural organization of ℕ⁺ must impose a network of parent, block, and ancestry relations connecting every positive integer into a single coherent whole, rather than merely classifying or partitioning — and a comparative discussion of alternative relational structures: the divisibility poset imposes kinship but lacks unique parenthood; the Stern–Brocot and Calkin–Wilf trees have unique parenthood but their domain is ℚ⁺, with the integers appearing only as the right spine, recovering nothing but the standard linear ordering. The CTG is the only structure whose domain is exactly ℕ⁺, with unique parenthood, and with the inverse Collatz map as the organizing rule.