Abstract
We advance the study of the Collatz conjecture through an analysis of the binary representations of positive integers via fractional parts. We introduce a direct, non-recursive relation for the intermediate mantissas\( ~\sigma_j \) and prove the exact bound \( \sigma_j \le 1/(\ln 2\,(2^{m+1}-1)) \) after a run of \( m \)consecutive ones (Theorem 4), establishing that every such run must terminate. Using Weyl's equidistribution theorem applied to \( \{n \log_2 3\} \) (averaged over \( n \), not applied to any individual binary expansion), we establish that \( h(n) \le \frac{1}{2}L_n + o(L_n) \) holds for almost all \( n \), i.e.\ outside a set of natural density zero (Theorem 5); we explicitly identify this as the boundary of what the density method achieves. A complete mod-\( 8 \) transition table analysis proves \( Q \ge 2M - 2 \) for every Collatz window (tight constant \( C_0 = 2 \), Lemma 6), giving an exponential compression factor\( (3/4)^M \) deterministically. We identify two remaining open sub-problems: (1) extending the density bound from almost all \( n \) to every \( n \); and (2) ruling out non-trivial cycles, which is required to complete the descent argument. These open points are stated precisely rather than papered over. Numerical experiments confirm all theoretical bounds.