Affine Spherical Mass and a Gaussian Kinematic Formula for Tropical Two-Fans

Abstract

Let $F$ be an effective balanced rational two-dimensional fan in a rank-four lattice. We introduce an affine spherical mass \[ M_*(F)=\inf_{covol_g(N)=1}M_g(F) \] and prove the two-sided comparison \[ \sqrt{\frac83}\,m(F)\leq M_*(F)\leq 3\sqrt{2\,q(F)}. \] Here $q(F)=deg(F\cdot F)$ is stable self-intersection and $m(F)$ is the least degree of $F$ under a primitive rank-two lattice quotient. For a surjection $A$, one has $deg(A_{\ast }F)=deg(F\cdot [\ker A_R])$; hence $m(F)$ is equivalently the least stable intersection degree with a complete weight-one rational two-plane. In particular, \[ q(F)\geq\frac4{27}m(F)^2. \] The proof combines an exact Gaussian--kinematic formula for mixed stable intersection with an affine normalization principle for stationary spherical links. In dimension four, the identification $G{r}^{+}(2,4)\simeq S^2\times S^2$ converts ambient isotropy into a sharp second-moment transversality estimate. We prove affine covariance for $M_{\ast }$, derive equality and stability conditions for the transversality estimate, and give examples where ambient isotropy coexists with zero mixed intersection, as well as a three-fan in six-space with zero self-intersection. Determining the optimal numerical constant remains open.