My guess is that P is true for an exponentially small fraction of circuits. You could plausibly prove this with combinatorics (given that e.g. the first layer randomly puts inputs into gates, which means you could try to reason about the class of circuits that are the same except that the inputs are randomly permuted before being run through the circuit). I haven’t gone through this math, though.
What’s the proportion of circuits where P(C) is true?
My guess is that P is true for an exponentially small fraction of circuits. You could plausibly prove this with combinatorics (given that e.g. the first layer randomly puts inputs into gates, which means you could try to reason about the class of circuits that are the same except that the inputs are randomly permuted before being run through the circuit). I haven’t gone through this math, though.