I wish to understand why P is a subset of PSCPACE, that is why a polynomial-time langauge does have a polynomial-sized circuit. I read many proofs like this one here on page 2-3, but all the proofs use the same technique used in the Cook-Levin theorem to convert the computation of M on an n-bit input x to a polynomial sized circuit.
What I don't understand is that the resulting circuit is dependent on the input x, because what is being converted into a circuit is the computation of M on the specific input x. By definition of PSIZE, the same circuit must work for all the inputs in a fixed length, and thus is not dependent on one specific input.
So how is the process of creating a poly-sized circuit family for a poly-time deterministic Turing machine works exactly?