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

Fix $0<A<1$ in the signed gluing-map reduction and write $\delta =1-\eta$ with $0<\eta <1$. The saddle-node curves of a signed power gluing map occur at parameter values $\mu =\pm C$, whereThus $C$ is the half-width of the three-branch interval, and $2C$ is its total width.

To see the scale, use $(1-\eta {)}^{1/\eta }\to e^{-1}$ and $(1-\eta {)}^{-1}\to 1$. ThenSince $A^{1/\eta }=\exp (\log A/\eta )$ and $\log A<0$, the width decays faster than every positive power of $\eta$. A parameter scan with an algebraic step size can therefore miss the coexistence interval entirely.

This is an essential exponential scaling, rather than the power-law fold separation in the ordinary smooth cubic cusp. The three branches counted here are scalar return-map branches. Their physical lifts differ: a negative scalar fixed point represents two symmetry-related cycles, while a positive one represents one cycle visiting both lobes. The singular zero value is a separatrix limit with infinite return time.