The Collatz conjecture (3n+1 problem) is a classic unsolved problem in number theory. Its iteration rule is: for any positive integer n, if n is odd, map n 3 n+1; if n is even, map n n / 2. The question is whether repeated iteration necessarily converges to 1 for all initial positive integers. Decades of computational verification have covered all positive integers up to 2^68, all of which converge, yet a complete rigorous proof remains elusive. This paper does not attempt a new proof within pure number theory. Instead, it repositions the problem through the PFUSRC topological morphogenesis framework. The core thesis is: the Collatz conjecture is not about proving whether a convergence boundary exists — substituting n=1 directly verifies the minimal closed loop 1 4 2 1. The 4 2 1 cycle is itself the natural, complete minimal topological closure boundary within the number-line projection layer. The true essence of the problem is to demonstrate that all numerical trajectories share a single topological origin. The 4 2 1 closure unit can accommodate formal large-number mappings of arbitrary scale; tracing backward from any sufficiently large number converges to this minimal topological closure. The core argument is not that numerical sequences converge to the value 1, but that all numerical projection nodes can topologically trace back to the same ground-state anchor. This paper further distinguishes two concepts of infinity: mathematical infinity is merely a formal virtual extension at the scale level, while real higher-dimensional topological structures possess a closure boundary defined by the 55 curvature anchor points. The paper does not use “truncation” to negate the legitimacy of formal mathematical infinity; rather, it delineates the effective topological scope of this study — the 55-anchor closure bounds the “positive integer interval mappable to physical topological structures, ” and does not address the full domain of purely formal natural numbers. The number 1 in Collatz iteration is not an arithmetic endpoint, but the unique minimal closure anchor on the topological boundary. Using 26 and 27 as examples — adjacent on the number line but belonging to the middle and outer layers of the 55-anchor system — the dramatic difference in iteration path length (10 steps vs. 111 steps) and peak values is a direct manifestation of topological projection distance mapped onto the one-dimensional number axis. This paper further presents an alternative iteration rule topologically equivalent to standard 3n+1, demonstrating that the “mystery” of the Collatz conjecture does not reside in the numbers 3 and 2 themselves, but in the underlying “expansion-contraction” topological structure. This equivalence provides evidence that reduces the Collatz conjecture from a number-theoretic puzzle to a topological necessity. This paper provides a neutral paradigmatic comparison with Tao’s 2022 result that “almost all Collatz orbits attain almost bounded values, ” clarifying the hierarchical distinction between the two approaches: Tao approximates the proposition within pure arithmetic using analytic number theory and probabilistic estimates, while this paper transforms the entire conjecture into a topological convergence problem within a higher-dimensional geometric framework.
Zhenmin Wang (Sun,) studied this question.