We give a short-description decomposition of bounded-width CNFs. For every fixed $\varepsilon>0$, every width-$b$ CNF on $m$ variables is an exact disjoint union of at most $2^{\varepsilon m}$ width-$b$ CNFs, each determined by $2^{O_\varepsilon(b)}\log(m+1)$ satisfying assignments.

For every fixed $d\ge3$, we prove a worst-case size lower bound of $2^{\Omega_d((n\log n)^{1/(d-1)})}$ against depth-at-most-$d$ unbounded-fan-in AND/OR circuits with input negations for three polynomial-time Boolean families: inverse trace over binary fields, the Goldreich--Tal trilinear function, and OpenAI's polynomial evaluator with its original parameters. All parameter bits count as inputs. For inverse trace, every such circuit of size at most $S$, where $n\le S\le2^{c_dn^{1/(d-1)}}$, has uniform sign correlation at most $2^{-\eta_dn\log\log S/(\log S)^{d-1}}$. Here $c_d,\eta_d>0$ depend only on $d$. This improves on the classical parity correlation bound. The lower bounds combine the decomposition with restriction arguments; the correlation bound additionally uses common partial decision trees.

We also construct exact nonuniform AND/OR circuits for inverse trace of size $2^{O(n^{2/3})}$ at depth eight and $2^{O(\sqrt n\log n)}$ at depth nine. For every fixed integer $h\ge2$, the construction gives depth-$(4h+1)$ circuits of size $2^{O_h(n^{1/h}\log n)}$. The upper bounds implement finite-field evaluation and interpolation using circuits that enumerate intermediate states.