We develop a quantitative framework for the Collatz conjecture through a human-LLM collaboration, combining exact arithmetic structure, cycle-level probabilistic laws, and a conditional convergence reduction. The central quantitative result is the Per-Orbit Gain Rate theorem, which proves R <= 0.0893 < epsilon = 2 - log_2 3 ~= 0.415, leaving a safety margin of at least 4.65x. A robustness corollary shows that exact equidistribution is unnecessary: it suffices that sum_K delta_K < 0.557. This promotes the Weak Mixing Hypothesis (WMH) to the primary open condition. On the arithmetic side, we refine modular crossing methods and prove that by depth 13 about 91 percent of odd residue classes are already forced to descend below their start. On the odd skeleton, we prove the exact run-length identity L(n) = v_2(n+1) - 1, derive an exact one-cycle crossing criterion, and compute the exact one-cycle crossing density P_1cyc = 0.713725498.... A major breakthrough is that the odd-skeleton valuation process satisfies an exact finite-block law: every prescribed valuation block occurs on a single odd residue class with the expected density. Hence the valuation process is exactly i.i.d. geometric in the natural-density ensemble, and the induced run-compensate cycle types are exactly i.i.d. This yields an exact cycle-level large-deviation theory and an unconditional almost-all crossing theorem in cycle language. We also prove substantial classwise deterministic crossing: about 41.9 percent of odd starts lie in one-cycle residue classes where every representative crosses below its start, and about 50.4 percent lie in two-cycle residue classes with the same universal crossing property. The framework does not yet prove Collatz. The remaining gap is now sharply isolated as a pointwise problem: proving that every deterministic orbit realizes enough of the exact negative cycle drift to cross below its start.

翻译：我们通过人类-LLM协作构建了一个针对考拉兹猜想的量化框架，该框架融合了精确算术结构、循环级概率定律以及条件收敛约化方法。核心量化成果是轨道增益率定理，其证明R ≤ 0.0893 < ε = 2 - log₂3 ≈ 0.415，保留了至少4.65倍的安全裕度。鲁棒性推论表明精确均匀分布并非必要条件：只需满足∑_K δ_K < 0.557即可。这将弱混合假设提升为核心开放条件。在算术层面，我们改进了模交叉方法，证明在深度13时约91%的奇剩余类已被强制降至起始值之下。在奇骨架结构上，我们证明了精确游程恒等式L(n) = v₂(n+1) - 1，推导出精确单循环交叉判据，并计算出精确单循环交叉密度P_1cyc = 0.713725498...。重要突破在于奇骨架赋值过程满足精确有限块定律：每个预设赋值块都以期望密度出现在唯一的奇剩余类上。因此在自然密度系综中赋值过程完全服从独立同分布的几何分布，且导出的游程补偿循环类型也完全独立同分布。这催生了精确的循环级大偏差理论以及循环表述下的无条件几乎处处交叉定理。我们还证明了显著的类确定性交叉现象：约41.9%的奇起始点属于单循环剩余类（其中所有代表元都交叉降至起始值之下），约50.4%属于具有相同普适交叉性质的双循环剩余类。该框架尚未完全证明考拉兹猜想。当前遗留的差距被明确隔离为点态问题：证明每个确定性轨道都能实现足够的精确负循环漂移以交叉降至起始值之下。