Inspired by the Qur'anic parable of Al-Baqarah (2: 261) — a single seed producing seven ears, each bearing a hundred grains — and the tillering physiology of wheat, this paper presents a structural framework for the Collatz conjecture based on the Collatzogin Tree. The tree is a directed graph constructed from the forward Collatz function, built level by level, where each level partitions all positive integers according to their residue modulo \ (2^k-1\). This construction guarantees coverage: every positive integer appears in the tree. From the tree, we extract odd-only sequences and show that they obey the recurrence \ (aₓ+₁=4aₜ+1\) with total stopping times forming an arithmetic progression with common difference \ (2\). The branching distribution follows the Fibonacci sequence \ ( (1, 2, 3, 5, 8, 13, 21, 34) \). We prove that the tree admits no cycle other than \ (421\) and that no divergent trajectory exists. The convergence of all seeds (first terms of odd-only sequences) is verified computationally up to the bound of the constructed tree. This work integrates three domains: scriptural metaphor, plant physiology, and number theory.
Ogin Sugianto (Tue,) studied this question.
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