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In this note we present a formula for the Cachazo-Early-Guevara-Mizera (CEGM)generalized biadjoint amplitudes for all $k$ and $n$ on what we call theminimal kinematics. We prove that on the minimal kinematics, the scatteringequations on the configuration space of $n$ points on $\mathbb{CP}^{k-1}$ has aunique solution, and that this solution is in the image of a Veroneseembedding. The minimal kinematics is an all $k$ generalization of the onerecently introduced by Early for $k=2$ and uses a choice of cyclic ordering. Weconjecture an explicit formula for $m_n^{(k)}(\mathbb{I},\mathbb{I})$ which wehave checked analytically through $n=10$ for all $k$. The answer is a simplerational function which has only simple poles; the poles have the combinatorialstructure of the circulant graph ${\rm C}_n^{(1,2,\dots, k-2)}$. Generalizedbiadjoint amplitudes can also be evaluated using the positive tropicalGrassmannian ${\rm Tr}^+{\rm G}(k,n)$ in terms of generalized planar Feynmandiagrams. We find perfect agreement between both definitions for all caseswhere the latter is known in the literature. In particular, this gives thefirst strong consistency check on the $90\,608$ planar arrays for ${\rmTr}^+{\rm G}(4,8)$ recently computed by Cachazo, Guevara, Umbert and Zhang. Wealso introduce another class of special kinematics called planar-basiskinematics which generalizes the one introduced by Cachazo, He and Yuan for$k=2$ and uses the planar basis recently introduced by Early for all $k$. Basedon numerical computations through $n=8$ for all $k$, we conjecture that on theplanar-basis kinematics $m_n^{(k)}(\mathbb{I},\mathbb{I})$ evaluates to themultidimensional Catalan numbers, suggesting the possibility of novelcombinatorial interpretations. For $k=2$ these are the standard Catalannumbers.

Minimal Kinematics: An All $k$ and $n$ Peek into ${\rm Trop}^+{\rm G}(k,n)$