[im-a-baby]

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.

[mdaEyebot]

Sure, but computer chips are probably the most complicated things that our civilization can currently make.

Asking somebody to prove that a computer won't do X is a fool's errand, outside of some shrinking safety-critical industries.

That said, you could always make your own smart speaker with a hardware button or shell script instead of a wakeword:

https://developer.amazon.com/en-US/docs/alexa/alexa-smart-sc...

[im-a-baby]

That's very different than your initial statement of: "Many schools of thought believe that it is impossible to prove the absence of something.". Many (all?) schools of thought believe in the law of excluded middle, which makes it possible to prove the absence of many things.