Every polynomial-time language has polynomial-size circuits

Fix an input length $n$ and enlarge the time bound to a polynomial $q(n)\ge n+1$ that covers the one-tape simulation in [L4]. Encode each of the first $q(n)+1$ one-tape cells by its tape symbol and a marker carrying the simulated state exactly at the head cell. An update to one cell depends only on the old codes of that cell and its immediate neighbours, since a one-tape head moves at most one cell; the boundary cell uses the fixed left-end rule. Each new cell code therefore has a fixed-size Boolean circuit. Give accepting and rejecting states absorbing local updates so the configuration remains defined through time $q(n)$.

Wire $q(n)$ copies of the update layer in sequence, initialize the first layer from the $n$ input bits, and OR the accepting-state markers in the final layer. There are $O(q(n))$ encoded bits per layer and $q(n)$ layers, so the resulting circuit $C_n$ has $O(q(n{)}^2)$ gates; initialization and the final OR also use $O(q(n))$ 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.