This submission formalises a depth hierarchy theorem for matrix-symmetric algebraic circuits, together with the characterisation of bounded-depth matrix-symmetric circuits by homomorphism polynomials on which it rests. A circuit with variables $x_{i,j}$ for $i, j \in [n]$ is matrix-symmetric if every permutation of rows and columns extends to an automorphism of the circuit. Its orbit size is the maximum size of an orbit of gates under these automorphisms, its support size is the least $k$ such that every gate is fixed by the pointwise stabiliser of some set of at most $k$ rows and columns, and its support depth is the maximum number of wires along a path at which the minimal support of a gate is not contained in that of its parent.

Depth hierarchy. For $\Delta \geq 1$, let $T_{\Delta,n}$ be the full $n$-ary tree of depth $\Delta$, viewed as a bipartite pattern. Over a field of characteristic zero, for $n \geq 20$, every reduced matrix-symmetric circuit of support depth at most $\Delta$ computing the homomorphism polynomial $\hom_{T_{\Delta,n},n}$ has orbit size at least $n^{\log_2 n - \Delta \log_2 \log_2 n - 3}$, whereas for $n \geq 5$ it is computed by a reduced matrix-symmetric circuit of support depth at most $\Delta + 1$ and orbit size at most $n^2$.

Characterisation. The patterns are measured by their $\Delta$-forestwidth $\mathrm{fw}_\Delta(F)$, the least width of a tree decomposition of $F$ of depth at most $\Delta$ with an empty root bag, a variant of the $\Delta$-treewidth of Bhargav, Chen, Curticapean, and Dwivedi. For $\Delta \geq 1$, $k \geq 2$, and $n \geq 2k+1$, a polynomial $p$ in the $n \times n$ matrix variables is computed by a reduced matrix-symmetric circuit of support size at most $k$ and support depth at most $\Delta$ if, and only if, $p$ is a linear combination of homomorphism polynomials $\hom_{F,n}$ of bipartite multigraphs $F$ with $\mathrm{fw}_\Delta(F) \leq k-1$. One direction is an explicit circuit construction for a single pattern, which additionally bounds orbit size, depth, and size; the other extracts homomorphism polynomials from a circuit by induction on its gates. Reduced matrix-symmetric circuits are shown to be rigid, and every matrix-symmetric circuit is shown to admit a reduced one computing the same polynomial without increasing size, depth, or orbit size.

Further ingredients. The lower bound combines the characterisation with two further results, both proved here. The first is a support theorem: for $n \geq 20$ and $3\kappa \leq n + 2$, a reduced matrix-symmetric circuit of orbit size less than $\binom{n}{\kappa}$ has support size at most $\kappa - 1$. It is derived from a self-contained proof, for $n \geq 20$, of the theorem of Dixon and Mortimer that a subgroup of $\mathrm{Sym}_n$ of small index contains the pointwise stabiliser of a small set in the alternating group, via Bochert's bound on bases of groups without 3-cycles, an analysis of orbits and block systems, and the simplicity of the alternating group. The second is a non-uniform homomorphism distinguishing closure result: if a graph class $\mathcal{F}$ contains $K_1$ and is closed under minors and under Roberson's weak oddomorphisms, $G \notin \mathcal{F}$ is connected, and $G$ is a minor of the underlying graph of $F$, then $\hom_{F,N}$ is not in the linear span of the homomorphism polynomials of patterns in $\mathcal{F}$ for every $N$ at least the size of a CFI graph of $G$. Its proof uses CFI graphs, a contraction formula for homomorphism counts, and a lifting of minor models.