It's pretty trivial to construct negatives that can be proven. E.g. take a solid colored object and prove it's not a specific color. You can prove the negative statement by proving a positive statement about the true color of the object, e.g. by direct observation, measuring the wavelength of emitted light, etc.

Similarly, you could prove a speaker doesn't eavesdrop by proving it performs a finite set of operations, none of which are eavesdropping.