No, a machine that supposedly turns a paradox into motion can’t actually break conservation of energy. The first law of thermodynamics is a mathematical statement about how energy behaves in any closed system; a device that runs forever would have to create energy out of nothing, which is mathematically impossible. Even if the machine appears to “refuse to be consistent,” it’s either exploiting a loophole in the measurement, or it’s using some external source of energy that you’re overlooking. In short, the illusion is either a clever trick or a misunderstanding of how the system actually exchanges energy.

If your engine’s “unbalanced input” is just extra fuel you’re feeding it, then it’s not a loophole, it’s a normal power source. A real perpetual motion machine would have to generate net work without any input, and that’s mathematically impossible. So either your paradox is a clever trick, or the law is still holding its ground.