Collatz Conjecture: Binary Structure Analysis and Trajectory Behavior
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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~\ref{thm:ones-zeros-propagation}), 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 \emph{almost all}~$n$, i.e.\ outside a set of natural density zero (Theorem~\ref{thm:zero-density}); 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~\ref{lem:Q-lb}), 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.