The normal coefficient of a standard tube structure

Let $\rho, b > 0$ and let $(\bar J, J_\infty, S_\infty)$ be a standard tube structure (IsStdTubeStructure ρ b Jbar Jinf Sinf). Let $N(s,t) =$ normalCoeff Jbar (s, t) be the real $2\times 2$ matrix of $a \mapsto \big[(\partial_w \bar J(z, 0)\,a)(2\pi i z, 0)\big]_2$ at $z = e^{2\pi(s+it)}$. Then:

1. every entry of $N$ is $C^\infty$ on $\mathbb{R}^2$, and $N(s, t+1) = N(s, t)$;
2. every entry of $S_\infty$ is $C^\infty$;
3. $N$ is exponentially asymptotic to $S_\infty$ at rate $b$ as $s \to +\infty$ (IsExpAsymptoticTo (normalCoeff Jbar) Sinf b);
4. $N$ is the pull-back of a smooth zero-order term on the plane (IsPlaneCoeff): $N(s,t)\,a = 4\pi\bar z\,A(z)\,a$ for a $C^\infty$ map $A : \mathbb{C} \to \mathrm{End}_{\mathbb{R}}(\mathbb{C})$.

Role. $N$ is the zero-order term of the linearization of the $\bar J$-Cauchy–Riemann operator in the normal direction along $\mathbb{C} \times \{0\}$, in cylinder coordinates. The four properties are the hypotheses on the coefficient of a Cauchy–Riemann operator on the cylinder with a removable end at $s = -\infty$ and an asymptotic operator $-J_0\partial_t - S_\infty$ at $s = +\infty$.