If you take a table of numbers where the sum of squares in each row and column is 1, and you want to maximize the sum of fourth powers, the answer will only be permutations of coordinates. The fourth power sharply penalizes division: 0.5 gives only 0.0625, while 1 remains 1. It's advantageous to give the entire weight of a row and column to one cell, filling it with +1 or -1, and the rest with zeros. That is a signed permutation matrix — such a rotation mixes nothing, merely swapping axes, sometimes flipping them.
Mathematicians showed that any departure from whole ones and zeros can be 'nudged' toward greater contrast, increasing the contribution of the largest element. The principle is reminiscent of pulling slopes toward a mountain peak: the peak becomes higher. Surprisingly, similar rules govern the behavior of elementary particles in the standard model, where symmetries select extreme states. And in spectroscopy, narrow spectral lines are similar sharp peaks, arising when excitation is concentrated in a single transition.
as written by the authors arXiv:2607.12431
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.
If you replace the fourth power with the second, the sum turns out the same for all rotations and reflections — that's the property of length preservation. But take a power higher than two, and 'peaky' configurations immediately stand out.