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Study verifies Collatz conjecture outcomes in positive integers, suggesting no infinite divergence occurs.
We give a rigorous full proof of the Collatz conjecture via modulo-8 residue classification and 2-adic valuation. We construct the bijective residue dynamics for reduced odd Collatz iterations, restricting upward growth to limited residue classes with bounded amplification. Using exponent accumulation inequality and mathematical induction, we exclude non-trivial periodic orbits and infinite divergent trajectories. Combined with large-scale numerical test, we verify all positive integers eventually reach 1 under the Collatz map.
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Hu et al. (2026) studied this question.
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