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In this paper we prove that points in the space $X(k,n)$ of configurations of$n$ points in $\mathbb{CP}^{k-1}$ which are fixed under a certain cyclic actionare the solutions to the generalized scattering equations on planar kinematics(PK). In the first part, we give a constructive upper bound: we show that thesesolutions inject into certain aperiodic k-element subsets of $\{1,\ldots, n\}$,and consequently that their number is bounded above by the number of Lyndonwords with k one's and n-k zeros. The proof uses a somewhat surprisingconnection between the superpotential of the mirror of $G(n-k,n)$ and thegeneralized CHY potential on $X(k,n)$. We also check the recent conjecture thatgeneralized biadjoint amplitudes evaluate to $k$-dimensional Catalan numbers onPK for several examples including $k=3$ and $n\leq 40$ and $(k,n)=(6,13)$. Wethen reformulate the CEGM generalized biadjoint scalar amplitude directly as aLaplace transform-type integral over ${\rm Trop}^+ G(k,n)$ and we use it toevaluate the amplitude on PK with the purpose of exhibiting how GFD's gluetogether.  We initiate the study of two minimal lattice polytopal neighborhoods of theplanar kinematics point. One of these, the rank-graded root polytope$\mathcal{R}_{k,n}$, in the case $k=2$, is a projection of the standard type Aroot polytope. The other, denoted $\Pi_{k,n}$, in the case $k=2$, is adegeneration of the associahedron. We check up to and including$\mathcal{R}_{3,9}$ and $\mathcal{R}_{4,9}$ that the relative volume of$\mathcal{R}_{k,n}$ is the multi-dimensional Catalan number $C^{(k)}_{n-k}$,hinting towards the possibility of deeper geometric and combinatorialinterpretations of $m^{(k)}(\mathbb{I}_n,\mathbb{I}_n)$ near the PK point.

Planar Kinematics: Cyclic Fixed Points, Mirror Superpotential, k-Dimensional Catalan Numbers, and Root Polytopes