Candidate anharmonic reactive width from the classical normal form

Let $K_{\mathrm{CNF}}(I,J_2,\dots,J_n)$ be a truncated classical normal form near an index-1 saddle, where $I$ is the reaction integral and $J_k$ are the bath actions. On the normally hyperbolic invariant manifold (NHIM), set $I=0$. For each bath mode, define

$$J_k^{\max}(E)=\sup\left\{J_k\ge0:\exists (J_2,\dots,J_n)\text{ satisfying }K_{\mathrm{CNF}}(0,J_2,\dots,J_n)=E\right\}.$$

For energies sufficiently close to the saddle, candidate anharmonic reactive width. The transverse reactive width scale should be

$$c_{\mathrm{cand}}(E)=2\pi\min_{k\ge2}J_k^{\max}(E).$$

This is intended as a candidate width for a bounded full-dimensional neighborhood of the bottleneck, rather than as a proven formula for the exact Gromov width of an arbitrary reactive domain. The claim is suggested by the local normal-form geometry and remains unproved in the stated generality.