Fixed parameters and field-dependent sections

The action $T_{\xi }$ is first proved for a field-independent parameter. At each configuration, the section evaluates that action at $\xi =-w_{\mathrm{R}}(h)$. This proves an orbit identity.

Differentiating the section with respect to fields is another operation and introduces derivatives of $w_{\mathrm{R}}$. Those terms cannot be discarded in a dynamical symplectic reduction. The scalar symplectic corollary fixes $k$ and does not perform that reduction.