NC1 is contained in L, and L is contained in NC2

Let $C_n$ have depth $d=O(\log n)$. Evaluate its output by a depth-first walk. Store the current root-to-gate child-choice word (at most $d$ bits), a constant amount of traversal state per current level encoded in that word, and $O(\log n)$ counters. Whenever a gate address is needed, recompute it from the output by replaying the stored choices, using logspace uniformity to obtain each gate record. This uses $O(\log n)$ space; bounded fan-in lets the walk combine the child values. Thus ${\mathrm{N}\mathrm{C}}^1\subseteq \mathrm{L}$.

Let $M$ be a logspace decider. On inputs of length $n$ it has polynomially many configurations, each indexed with $O(\log n)$ bits, and its start, accepting, and one-step adjacency predicates are logspace uniform. Boolean matrix squaring computes paths of length at most $2^i$ from those predicates. There are $O(\log n)$ squarings, and each Boolean matrix product has bounded-fan-in depth $O(\log n)$ and polynomial size, so start-to-accept reachability has depth $O({\log }^{2}n)$ and polynomial size. The gate indices and wiring are generated in logspace from $(n,i,u,v)$. Hence $\mathrm{L}\subseteq {\mathrm{N}\mathrm{C}}^2$.

The two simulations prove both stated containments without storing a logarithmic stack of logarithmic-size gate addresses.