= Exponentially narrow gluing-map cusp
{title2=$|\mu|<C(\delta),\quad C(1-\eta)\sim\eta A^{1/\eta}/e$}
= Gluing-map cusp
{synonym}
For fixed $0<A<1$ and $\delta<1$, the two folds of the <signed gluing-map reduction> enclose three scalar fixed-point branches. Writing $\delta=1-\eta$ gives width $C=\eta(1-\eta)^{-1}[A(1-\eta)]^{1/\eta}$. Since $(1-\eta)^{1/\eta}\to e^{-1}$, the displayed essential asymptotic follows. The fold curves meet with exponentially small width; this is not the ordinary algebraic cusp normal form of a smooth cubic vector field. A negative fixed point lifts to two symmetry-related flow cycles, so the cusp describes distinct return-map branches rather than counting every physical cycle separately.
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