Every polynomial-time language has polynomial-size circuits

Fix an input length $n$. Encode a configuration of $M$ by the state, head position, and the $O(p(n))$ tape cells that can be visited in $p(n)$ steps. Each bit of the next configuration depends on only a constant-size neighbourhood of the current configuration, so it is computed by a constant-size Boolean subcircuit. Pad halting configurations so that this update is defined through time $p(n)$.

Wire $p(n)$ copies of the update layer in sequence, initialize the first layer from the $n$ input bits, and read the accepting-state bit in the final layer. There are $O(p(n))$ encoded bits per layer and $p(n)$ layers, so the resulting circuit $C_n$ has $O(p(n{)}^2)$ gates. Replacing its fixed local basis by the page basis changes this bound only by a constant factor.

For every $x\in \{0,1{\}}^n$, $C_n(x)=1$ exactly when $M$ accepts $x$. Thus $(C_n)$ is a polynomial-size family recognizing $L$, as required by [L2]. The construction proves existence of each $C_n$ and does not assert a uniform generator.