Introduces a new framework using Base-4 representation to demonstrate maximum peak values in Collatz Conjecture sequences, indicating structural insights.
The Collatz Conjecture (3n+1 problem) states that the iterative application of a specific piecewise map to any positive integer will always reach the trivial cycle {4, 2, 1} in a finite number of steps. In this paper, we introduce a Base-4 (quaternary) representation framework to explicitly isolate and visualize the underlying parity bifurcations of the conjecture. By mapping arithmetic operations to geometric structural transformations—modeled as the expansion of "distorted rectangles" and their subsequent collapse into "harmony squares"—we establish a rigid algebraic bound governing local peak values. Furthermore, we provide a deterministic proof regarding the maximum peak values within any bounded finite interval dominated by pure quaternary sequences.
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河盛 et al. (2026) studied this question.
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