When Activation Patching Admits Many Faithful Circuits

Activation patching can establish faithfulness without determining a unique minimum circuit. In every additive $n$-component model with minimum $r$-faithful cardinality $1\le k<n$, complete order-$r$ evidence admits at most $\left(\genfrac{}{}{0ex}{}{n-1}{k}\right)$ minimum circuits. A patched ReLU model with $m$ patching sites attains the bound whenever $0\le r<m$ and $n-1\le \left(\genfrac{}{}{0ex}{}{m}{r+1}\right)$, giving an exact extremal characterization. For this construction, raising the patch-order bound to $r+1$ makes the full mask uniquely faithful and gives every proper mask faithfulness error at least one. At $k=\lceil n/2\rceil$, the exact maximum $\left(\genfrac{}{}{0ex}{}{n-1}{\lceil n/2\rceil }\right)$ is $2^{n-\frac{1}{2}{\log }_{2}n+O(1)}$. For fixed $r$, exponential multiplicity persists with per-component fan-in and parameter count bounded independently of $n$ and with uniform response norms through order $r+1$. The extremal family and transition remain stable under component perturbations and bounded departures from additive masking. Exact noiseless deterministic identification of an unknown extremal circuit requires $n-2$ queries of order at most $r+1$ in the worst case for $1\le k\le n-2$.