New lower bounds for kissing numbers in dimensions $25$--$29$ and $31$

The kissing number in dimension $d$ is the largest number of non-overlapping congruent spheres that can simultaneously touch a central sphere of the same size. We study dimensions $25$-$31$, where the best previous constructions are based on Leech lifting from the optimal kissing configuration in dimension $24$. Our method exploits the absence of contacts between the unlifted bulk and the block consisting of lifted and auxiliary vectors. Rotating this block while keeping the bulk fixed creates room for two antipodal points in dimensions $26$, $27$, and $28$, and one point in dimension $29$. Two further modifications yield improvements in dimensions $25$ and $31$: a nonorthogonal diagonal linear deformation of the lifted block admits two antipodal points in dimension $25$, while rotating only the additional coordinates of the lifted vectors admits four nonantipodal points in dimension $31$. Together, these constructions yield the new lower bounds ${\tau }_{25}\ge 197058$, ${\tau }_{26}\ge 198552$, ${\tau }_{27}\ge 200046$, ${\tau }_{28}\ge 204522$, ${\tau }_{29}\ge 209497$, and ${\tau }_{31}\ge 238354$.