Polynomial-time computations have logspace-uniform circuit families

For $0\le t\le B=B(n)$, encode each configuration by one-hot state bits, head-position bits $H_{t,r,j}$ for each tape $r$, and symbol bits $X_{t,r,j,a}$. Use positions $0\le j\le n+B+1$; no head can leave this interval within $B$ steps. There are exactly $n$ input nodes labeled $x_1,\dots ,x_n$; the initial row uses these, their negations, and constants for the input, blank cells, initial heads, and start state.

For each tape and symbol, compute the scanned-symbol bit $R_{t,r,a}={\bigvee }_j(H_{t,r,j}\wedge X_{t,r,j,a})$ with a binary OR chain. The fixed transition table applied to the state and these scanned-symbol bits gives constant-size Boolean formulas for the next state, write symbols, and movement directions. At a halting state define this circuit update to leave the entire configuration unchanged. This is an extension of the tableau update only; no transition from a halted machine is assumed.

At each cell, retain its old symbol unless its head bit is one, in which case use the computed write symbol. A new head bit is the OR of the old adjacent head bits conjoined with their computed movement directions; in a halted configuration keep its old head bit. At position zero also retain a head commanded to move left, as required by the right-infinite tape convention. Treat other out-of-range head bits as zero. These are constant-size formulas once the scanned-symbol bits are available. Induction on $t$ gives the genuine configuration until halting and its unchanged copy thereafter. Therefore the accepting-state bit at time $B$ equals one exactly when $M$ accepts $x$.

Each layer uses $O(n+B)$ gates: the tape count, alphabet, and state set are fixed, each scanned-symbol OR chain is linear, and each cell update has constant size. There are $B$ layers, so the size is $O(B(n+B))$, hence polynomial. All formulas use binary AND/OR and unary NOT. Designate the final accepting-state node as output.

Number nodes by initial-row order and then by time layer, with scanned-symbol chains before transition formulas and cell updates. Pad fixed formula blocks with unused constant nodes if necessary so that each block length is a fixed constant or fixed multiple of $n+B$. A node is addressed by its layer, block, position, and constant-size formula slot; all are $O(\log (n+2))$-bit integers. Fixed-degree arithmetic computes block offsets, adjacent positions, OR-chain predecessors, and the initial input label from these indices. Counting $n$ on unary input and evaluating the fixed polynomial $B(n)$ also use logarithmic space. These procedures answer SIZE, TYPE, predecessor, and output queries and emit the topological encoding. Thus [L2] gives logspace uniformity, including $n=0$ where the initial row has only constants.