Exponentially narrow gluing-map cusp ($|\mu |<C(\delta ),C(1-\eta )\sim \eta A^{1/\eta }/e$)

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.