Abstract The Collatz conjecture remains unresolved as a halting problem, despite the surge of structural approaches in 2025 (such as inverse Collatz tree constructions and classifications of stopping regions) and extensive computational verification. The central challenge is the direct proof of halting for all positive integers, whose difficulty lies not in arithmetic errors but in a structural asymmetry inherent in the definition of natural numbers.This study emphasizes that the generative structure of natural numbers, fixed by Peano induction, is bound to an “absolute scheme biased toward the starting point.” Such bias omits the responsibility of defining bijection, thereby obstructing structural closure over infinite domains and embedding incompleteness and non-halting as inevitable consequences. To overcome this, the paper introduces the structural requirement of responsibility of bijection and applies categorical arrow symmetrization to enforce Start = Stop symmetry. This guarantees complete bijective closure across infinite sets and establishes a “direct” tree structure that eliminates nontrivial cycles and divergence.This methodological shift parallels the historical transition in geometry and physics, where Poincaré initiated the move from absolutism to relativity and homology. Refusing to relativize natural numbers confines arithmetic within a faith-based system, leaving it bound to incompleteness. Just as denying Galileo’s principle of relativity reduced science to religion, arithmetic without structural responsibility remains destined to non-halting.