Local Maxima of the Entrywise $\ell_4$ Norm on the Orthogonal Group

We classify the local maximizers of the entrywise fourth-power objective \[ Q\longmapsto \lVert Q\rVert_4^4 =\sum_{i,j=1}^r q_{ij}^4 \] over the real orthogonal group $\mathcal O(r)$.

We prove that the signed permutation matrices are the only local maximizers, and hence the only global maximizers, in every dimension.

More strongly, every other stationary point has an explicit rank-two tangent direction with strictly positive second variation.

The proof is based on a maximum-entry pivot for the orthostochastic matrix $Q^{\circ2}$: the associated full Riemannian Hessian can be evaluated exactly and is positive at a largest nonunit squared entry.

The argument is self-contained and handles zeros, repeated magnitudes, reducible support, and Hadamard-type stationary points.