Exceptional force, uncountably many solutions in the KPZ fixed point

We give a complete characterization of all eternal solutions $b(x,t)$ of the KPZ fixed point satisfying the asymptotic slope condition $\lim_{|x| \to \infty} \frac{b(x,0)}{x} = 2\xi$. For fixed $\xi$, there is exactly one eternal solution with probability one. However, in the second and third authors' work with Sepp\"al\"ainen, it was shown that there exists a random, countably infinite set of slopes, for which there exist at least two eternal solutions. These correspond to two non-coalescing families of infinite geodesics in the same direction for the directed landscape. We denote the two eternal solutions as $b^{\xi-}$ and $b^{\xi +}$. In the present paper, we show that, for the exceptional slopes, there are in fact uncountably many eternal solutions. To give the characterization, we show that these eternal solutions are in bijection with a certain set of bi-infinite competition interfaces. Each bi-infinite interface separates the plane into two connected components -- a left component and a right component. A general eternal solution with slope $\xi$ is equal to $b^{\xi-}$ on the left component and equal to $b^{\xi +}$ on the right component. For these bi-infinite interfaces in the exceptional directions, we uncover new geometric phenomena that is not present for directed landscape geodesics. Additionally, we show that this set of eternal solutions appears as the Busemann limits $\mathcal{L}(\mathbf {v_n};\mathbf {p}) - \mathcal{L}(\mathbf{v_n};\mathbf {q})$ for sequences $\mathbf {v_n}$ going to $-\infty$ in direction $\xi$.