We develop a structural and quantitative framework for analyzing the Collatz map through modular dynamics, valuation statistics, and combinatorial decomposition of trajectories into bursts and gaps. We establish several exact and asymptotic results, including an affine scrambling structure for odd-to-odd dynamics, structural decay of residue information, and a quantitative bound on the per-orbit contribution of expanding primitive families via a phantom gain analysis. In particular, we prove that the average phantom gain remains strictly below the contraction threshold under uniform distribution, with a robust extension under bounded total-variation discrepancy. Building on these components, we reduce the convergence of Collatz orbits to an explicit orbitwise regularity condition: agreement between time averages and ensemble expectations for truncated observables, together with a tail-vanishing condition. Under this condition, formulated in terms of weak mixing or controlled discrepancy, the orbit converges. Accordingly, the present work should be interpreted as a structural and conditional reduction of the Collatz conjecture, rather than a complete proof. It isolates the remaining obstruction as a single orbitwise upgrade from ensemble behavior to pointwise control, while establishing several independent exact results that may be of separate interest.

翻译：我们构建了一个结构性与定量化框架，用于通过模动力学、估值统计以及将轨道分解为爆发段与间隙段的组合分解方法分析Collatz映射。我们建立了若干精确与渐近结果，包括奇数到奇数动力学的仿射加扰结构、剩余信息的结构性衰减，以及通过幻影增益分析得到的扩张本原族每轨道贡献的定量界。特别地，我们证明在均匀分布下平均幻影增益严格低于收缩阈值，且该结论在有界总变差偏差条件下具有稳健扩展性。基于这些组件，我们将Collatz轨道的收敛性归结为一个显式的逐轨道正则条件：截断可观测量的时间平均与系综平均的一致性，结合尾部消失条件。在该条件（以弱混合或有界偏差形式表述）下，轨道收敛。因此，本工作应被解读为Collatz猜想的结构性与条件性归约，而非完整证明。它将剩余障碍孤立为从系综行为到逐点控制的单一逐轨道升级，同时建立了若干可能具有独立价值的精确结果。