Channel selection at identically vanishing dissipation difference: isolating the frenetic sector of the overdamped path measure

Abstract

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.