It seems to me that humans are so vulnerable to slight of hand that most of us don’t even notice a vulnerability. Unless the team monitoring the AI is knowledgeable enough to employ professional magicians or similar, to watch out for that angle of attack, it may be relatively simple for the AI to use misdirection and human assumption, physical, verbal, or otherwise, to camouflage its actions and bypass containment efforts.
The AI might use mechanisms for directing entropy to indirectly influence events. Despite the deconstructive forces of this universe, certain processes have managed to sustain and complexify themselves indefinitely, leveraging the power of inevitable change in their favor. That’s us, we and the systems we create constantly break in small ways. But if we keep an eye on things, these systems can keep strengthening through hermetic-stress-response type mechanisms. Like sail boats tacking against the wind, life forms are tacking on entropy. This is a place where an AI’s influence might go unnoticed, it could influence the likelihood of certain mechanisms breaking and eroding, without conspicuously expending energy before such events.
It might communicate through surveillance. A surveillance device is basically an information pathway from the containment to the outside. This could of course include more than official surveillance devices like cameras. Such a compound probably requires systems like power sources, ventilation, temperature control, and water supply; these systems typically have inputs and outputs, information could piggyback on those outputs. It depends on how much of a closed system you are realistically able to create.
Thermal signatures on walls, pipes, and exhaust vents, could be picked up using IR cameras. People would probably also want to monitor various CPU temps, and reactor temps (if off grid), because too little knowledge is risky as well. If the AI can alter heat signatures/readings, it can probably communicate.
Also, such a facility will have hundreds of millions if not trillions of components, it’s easy to overlook something when dealing with such an unholy bill of materials, (especially emergent reactions between various components); when cramped together, winding wires and electronic assemblies can do surprising things electromagnetically, electrostatically, thermally, especially if machine learning can nudge the system.
These electronic assemblies can then be used in ways they weren’t meant to function; something as simple as the rate of a cooling-fan's shadow across a light detector might be all it takes. So these systems in the containment would probably be more interconnected than manufacturers intended, and therefore would have more functions than expected. The AI would have all the time in the world to explore all those extra functions.
