The first Vietoris--Rips homotopy transition of round spheres

For every \(n\geq3\), we identify the homotopy type of the strict Vietoris--Rips complex of the unit round sphere on an interval immediately above the regular-simplex threshold \(r_n=\arccos(-1/(n+1))\). It is the join of \(S^n\) with \(\SO(n+1)/A_{n+2}\), where \(A_{n+2}\) acts by the rotational symmetries of the simplex. This proves the first-transition interval conjecture for higher-dimensional spheres. A recent critical-diameter theorem supplies a common interval without further stationary configurations. We compute a closed labelled configuration realization at the threshold and use compact definable sublevel collars to transfer its homotopy type to the strict complexes.