The cost function at the center of this paper looks almost like something from a physics textbook: c(x,y) = √(a² − |x−y|²). It charges you for moving mass from point x to point y, and it charges infinitely steeply as the distance |x−y| approaches a. Past that distance, the formula stops making sense entirely. Mathematicians call it the relativistic heat cost, and the hard cap on distance is the point: nothing travels farther than a, the way nothing travels faster than light.
Xiao-Tian Wu, in a preprint posted to arXiv on 29 July 2026, uses that cost to answer a regularity question. Optimal transport asks how to rearrange one distribution of mass into another as cheaply as possible. The answer is encoded in a function whose second derivatives satisfy a Monge-Ampère type equation, and a long-running question about such equations is how smooth their solutions have to be. Not whether solutions exist, but how well behaved they are once they do.
Where smoothness runs out
Smoothness here is measured on a finer scale than "differentiable or not." A function of class C¹ has a continuous first derivative. A function of class C^{1,β} has more: its derivative doesn't just vary continuously, it varies at a controlled rate, with β between 0 and 1 setting how controlled. Larger β means a tamer derivative. Analysts want the largest β they can prove, because that number governs what else can be deduced.
Wu constructs an explicit one-parameter family of generalized solutions on a ball, each with radial structure, meaning the solution depends only on distance from the center. Radial symmetry is what makes the construction tractable: it collapses a partial differential equation in n variables down to a planar autonomous system, two ordinary differential equations that can be analyzed by hand. Within that family, Wu picks out one specific solution and pins down its smoothness exactly. It belongs to C^{1,1/(2n-1)}, and it belongs to no better Hölder class. For every β larger than 1/(2n-1), the solution fails to be C^{1,β}.
The mechanism is visible in the reduced system. Wu tracks a phase variable s = ṙ, the rate of change of the radial coordinate, and shows it vanishes to order 2n−1 in the base variable. That degeneracy is not a technical nuisance sitting off to the side. It is the whole story: the exponent 2n−1 in the vanishing order becomes the exponent 1/(2n−1) in the smoothness class. In three dimensions the solution is C^{1,1/5}. In ten dimensions, C^{1,1/19}. The higher the dimension, the rougher the worst case.
What makes this kind of result useful is its direction. Proving a solution is smooth tells you about that solution. Constructing a solution that stops being smooth at a specific point on the scale tells you about every possible theorem: no one can prove a general estimate stronger than the example allows, because the example is a counterexample to it. Wu's construction is explicit, which is worth noting. It is a formula, not an abstract existence argument, so the exponent can be read off rather than inferred.
A second equation, same wall
The paper then carries the construction somewhere else. In dimension two, Wu transfers it to the special Lagrangian curvature equation, a geometric equation indexed by a phase parameter Θ. For every Θ strictly between 0 and π/2, Wu produces a sequence of smooth graphical solutions, surfaces given as graphs of functions, that converge uniformly to a limiting function of class exactly C^{1,1/3}. Each member of the sequence is perfectly smooth. The limit is not.
That gap is the conclusion. Wu writes that the two-dimensional special Lagrangian curvature equation therefore admits no pure interior C^{1,β} estimate for any β greater than 1/3. An interior estimate is a bound on how rough a solution can be well inside its domain, controlled only by information about the solution itself, with no assumptions about behavior at the boundary. If such an estimate existed above 1/3, it would apply uniformly to every solution in Wu's sequence, and the limit would inherit the bound. It doesn't. So the estimate can't exist.
Why it matters
Regularity theory is where partial differential equations get their reliability. Once you know solutions to an equation are smooth to a given degree, numerical schemes come with error guarantees, geometric arguments become available, and further theorems can be built on top. When a sharp threshold gets identified, the field stops searching above it.
The relativistic cost is not an arbitrary choice of test case. Its finite-speed constraint is exactly the feature that makes standard regularity machinery awkward, since the cost degenerates at the edge of its allowed range. Wu's example shows what that degeneracy actually costs, and does so dimension by dimension.
A few honest limits. This is a preprint, not yet peer reviewed. The solutions are radial, a symmetric special case, so the construction does not describe what generic solutions look like. And the result is negative in form: it rules out a class of estimates rather than establishing a positive theory. What it settles is where the ceiling sits. Anyone proving interior regularity for these equations now knows the exponent 1/(2n−1) is not a limitation of technique.