A Structural Descent Proof of Completeness of the Canonical Triple-Graph

This note presents an ontological argument for the structural completeness of the Canonical Triple-Graph (CTG), complementing the foundational CTG paper (Zenodo 10.5281/zenodo.18184678). The CTG framework provides a fixed, a priori directed structure on the positive integers, determined by admissible associators E_k(m) = (2^k · m - 1)/3 and their induced block organization. The structural completeness question asks whether every active odd integer (those not divisible by 3) belongs to the rooted component C(1) anchored at the distinguished integer 1. While the foundational CTG paper proved that any block-closed component must contain 1 (uniqueness of the anchored component), the question whether C(1) exhausts all active integers was left open as a structural conjecture. This note takes a different approach by examining the existential status of integers within the CTG framework. We combine three elements: 1. A priori existence: All positive integers exist as mathematical objects (Platonist foundation)2. Structural dependency: For every integer n ≠ 1, if n is active, then n belongs to some Block(m), establishing an existential dependency n ← m3. Grounding principle: Existential dependency chains cannot regress infinitely without grounding (philosophical principle borrowed from metaphysical grounding theory) The main result (Theorem 3.1) proves that these three elements together imply structural completeness: every active odd integer (except the generator 1) belongs to C(1), the component induced by 1. The proof proceeds by showing that any ancestry chain n ← m₁ ← m₂ ← ... must terminate at 1, since: (a) infinite chains violate the grounding principle, (b) circular chains are impossible by the structure of E_k, and (c) termination at any active integer other than 1 contradicts the proven uniqueness of 1 as the only block-free integer. This argument is explicitly hybrid, combining rigorous mathematical results about CTG structure with a philosophical principle about existential grounding. It differs fundamentally from dynamical or iterative approaches to related problems (such as the Collatz conjecture), instead examining what it means for an integer to exist within a structured mathematical framework. The note includes:- Formalization of existential dependency within CTG- Complete proof of ontological completeness (Theorem 3.1)- Discussion of the argument's philosophical foundations- Comparison to classical ontological arguments (Aquinas, Leibniz)- Analysis of the relationship to the Collatz conjecture- Implications for mathematical ontology and structured Platonism The document is structured as a rigorous mathematical note suitable for venues at the intersection of mathematics and philosophy of mathematics. It acknowledges that the grounding principle is philosophical rather than purely mathematical, making the work explicitly interdisciplinary. This represents a follow-up to the foundational CTG paper, articulating an argument that was developed but not included in the original publication due to strategic considerations regarding the reception of CTG as an independent framework.