Collatz Conjecture: Binary Structure Analysis and Trajectory Behavior
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This paper advances the Collatz conjecture by analyzing binary representations of natural numbers through fractional parts. We introduce a direct non-recursive relation for intermediate mantissas $\sigma_j$ in binary decompositions and prove their equidistribution using Weyl's theorem. The self-correcting dynamics of $\sigma_j$ ensure a balance between 1s and 0s, leading to an asymptotic density of 1/2 for 1s in binary expansions of $3^n$. This yields a probabilistic estimate: in approximately half of all cases, the binary expansions have many leading zeros, ensuring rapid descent. Theorems estimate zero density in powers of three and demonstrate sequence decrease for large $n$. Numerical verifications and updated figures support the findings, providing strong evidence for convergence in large cases.