The Symmetry Budget: How Much Group Structure Should a Network Architecture Assume?

A network architecture is, in large part, a symmetry assumption: convolution ties weights across translations, attention ties them across permutations, and a fully connected layer ties nothing. When the data are only approximately symmetric, tying trades approximation error against estimation error, and the right amount of tying is a property of the dataset, not of the architecture alone. We study this trade-off in an exactly solvable model: linear maps on signals over the cyclic group $C_{d}$, with architectures indexed by the subgroup lattice of $C_{d}$, so that a single integer $k \mid d$ interpolates between an unconstrained layer ($k=1$,$d^2$ parameters) and a fully equivariant one ($k=d$, $d$ parameters). We derive the excess risk $\delta_k^2 + \sigma^2 p_k/n$ in closed form, show that for a generic symmetry defect the risk is affine in $1/k$, and conclude that intermediate architectures are then never strictly optimal: the lattice collapses to an all-or-nothing choice with phase boundary $n^* = \sigma^2 d(d-1)/\varepsilon^2$. Structured defects break this collapse and restore interior optimum. Measurements agree with the theory on $60/60$ cells of a $(\varepsilon,n)$ phase diagram, recover the predicted exponent as $-1.92$ against $-2$, and exhibit the predicted $k^\star: 12 \to 4 \to 1$ ladder under graded defects; a two-layer nonlinear control reproduces the crossover.