AI 中文总结
本文针对局部鞍点数据无法完全确定反应通道选择的问题，构造了双通道梯度势，分离出过阻尼路径测度的狂热扇区，确定了路径选择的协变几何贡献，给出无参数预测并通过Langevin模拟验证。
AI 中文摘要
过渡态理论与Kramers-Langer理论仅通过单点（鞍点）数据确定反应通道选择。本文从精确构造的角度证明该方法不足。基于Onsager-Machlup路径权重的标准拆分（分为由熵流决定的时间反对称部分与时间对称的狂热部分），本文首先明确基于耗散的判据可区分两条正向历史的条件：两条共享端点的历史间熵差等于非保守力环绕其包围回路的环量的β倍，且对所有梯度流该熵差完全消失。随后，本文构造了一个双通道梯度势，其两条通道共享端点、势垒高度完全相等且鞍点Hessian为同一矩阵；通过构造将熵扇区关闭，所有局部鞍点理论均精确预测分支比为50:50，而分支比可分离狂热扇区。Langevin模拟给出P(+)=0.4568±0.0012。本文在狂热扇区内确定了路径选择的协变几何贡献(1/2)ln det(D_perp H_perp)，证明为此提出的漂移散度可观测量不具备坐标不变性，而该几何贡献具备，且其无需拟合参数即可复现测量分支比，误差在0.5σ内。刚性旁观模式完全抵消，因选择仅取决于行列式之比。本文还给出无参数预测：强度为ε的螺线管力需精确贡献2*β*ε/π至ln[P(+)/P(-)]。上述结果构成了局部鞍点数据不足以用于通道选择的精确受控反例。
英文摘要
Transition-state and Kramers-Langer theory determine reaction channel selection from data at a single point: the saddle. We show this is insufficient in a precise, constructive sense. Using the standard split of the Onsager-Machlup path weight into a time-antisymmetric part fixed by the entropy flux and a time-symmetric part, the frenesy, we first settle when a dissipation-based criterion can distinguish two forward histories at all: the entropy difference between two histories sharing endpoints equals beta times the circulation of the non-conservative force around the loop they enclose, and vanishes identically for every gradient flow. We then construct a two-channel gradient landscape whose channels share both endpoints, whose barrier heights are equal identically, and whose saddle Hessians are the same matrix. The entropy sector is switched off by construction, all local saddle-point theories predict 50:50 exactly, and the branching ratio isolates the frenetic sector. Langevin simulation gives P(+) = 0.4568 +/- 0.0012. We identify a covariant geometric contribution to pathway selection within the frenetic sector, (1/2) ln det(D_perp H_perp), show that the divergence-of-drift observable proposed for this role is not coordinate-invariant whereas this one is, and find that it reproduces the measured branching to within 0.5 sigma with no fitted parameters. Stiff spectator modes cancel identically, since selection depends only on a ratio of determinants. We also give a parameter-free prediction: a solenoidal force of strength epsilon must contribute exactly 2*beta*epsilon/pi to ln[P(+)/P(-)]. These results are an exactly controlled counterexample to the sufficiency of local saddle-point data for channel selection.
Comments12 pages, 1 figure, 9 tables. Supersedes in part arXiv:2606.09684 and arXiv:2603.15230
