Can a collection of one-dimensional views tell us whether two probability distributions are close?

Imagine two clouds of points. Comparing them directly means asking how much work it takes to move one cloud into the other. A simpler approach projects both clouds onto lines, compares those one-dimensional views, and averages over directions. This is called sliced Wasserstein distance.

The question is whether small differences in those views guarantee a small difference between the full distributions.

This candidate proves that they do under a particular condition. The transport cost uses a fixed power greater than one that is not an integer, and the distributions have a bounded moment of that same order. No bounded support or stronger moment assumption is needed. Even tiny amounts of probability travelling arbitrarily far away are covered.

The distinction between integer and noninteger powers is essential. Earlier counterexamples show that the corresponding uniform guarantee fails at integer powers. This result addresses the remaining noninteger case.

The proof tracks moment mass using logarithmic distance and direction. Classical transform formulas then help turn agreement between projections into control of full transport. The revised argument gives a stronger power-law bound, using a classical positive smoothing kernel.

A further consequence handles finitely many weighted directions, but retains an error term for angular coverage. Finite projections alone cannot identify every distribution.

There are important limits. The bound is not claimed to be sharp, its constant has not been numerically calibrated, and this is an unrefereed written-proof candidate, not a formally verified theorem. The small diagnostic tests do not prove the general result.

This is the Evidence Press briefing for the noninteger sliced Wasserstein release, dated the eleventh of October, twenty twenty-six. The page links the paper, source and checks. This synthetic voice briefing is an explanation, not additional evidence.