From Null Modes to Gauge Generators: A Bordered Faddeev--Jackiw Theory of Constraint Chains

A null mode of the bordered Faddeev--Jackiw matrix is not, in general, a gauge generator. We determine exactly what the residual kernel encodes and identify the additional condition required for genuine gauge symmetry. Writing $f^{(m)}=\begin{pmatrix}f^{(0)}&B\\-B^{\mathsf T}&0\end{pmatrix}$ and $Γ=