Sharp Hölder exponent for adjacent-cube swaps

Determine whether the sharp Hölder exponent in the discrete Shnirelman-type inequality can be achieved using only swaps of adjacent cubes, or whether the sharp exponent is an intrinsically fluid-dynamical phenomenon.

Background

The paper discusses discrete approximations in which a permutation of an N×NN\times N square tiling is implemented through swaps of adjacent cubes. Existing work establishes a discrete inequality with Hölder exponent $2/7$, improving an earlier exponent of $1/64$. The authors explicitly leave unresolved whether the sharp exponent can arise within the adjacent-swap and sorting framework itself, or whether it depends on genuinely continuous fluid dynamics.

References

It remains an open question whether the sharp H\"older exponent can be obtained through swaps of adjacent cubes (bridging the theory of permutons , and sorting algorithms with incompressible fluid motion) or it is a purely fluid dynamics feature.

— On Shnirelman's inequality in 2D: Irreversible behavior and Euler-Lagrange variational formulation  (2609.31294 - Schiffer et al., 25 Sep 2026) in Section 2, Subsection “Inequalities for Discrete Fluid Configurations”