We study the circuit cost of selecting $k$ objects satisfying orthogonality, zero XOR, or modular zero SUM. Our main lower bounds in nonuniform $\mathsf{AC}^0$ use a common framework of depth-zero projections from colored subgraph isomorphism. The projections use dimension, row count, or bit width $O(k\log n)$ and preserve circuit size, depth, and gate orientation. OV inputs satisfy pairwise distinctness within each array. Our direct XOR bounds require odd $k$, while SUM admits both parities. Stronger lower bounds for suitable source patterns transfer through the same framework to all three targets. For every fixed depth $d$ and sufficiently large fixed $k$, we prove unconditional lower bounds $n^{Ω(k)}$ for OV and $(n/k)^{Ω(k)}$ for XOR and SUM, with an absolute constant in the exponent independent of $d$ and $k$; only the threshold on $n$ may depend on $(d,k)$. For $k\to\infty$ with $k=n^{o(1)}$, combining our projections with Beame's small-clique lower bound yields $n^{Ω_d(\min\{\sqrt{k},\log n\})}$ at every fixed depth $d$. We strengthen the exponent to $Ω(k)$ at depth two in both orientations and at depth three with a top disjunction. The depth-three result allows unrestricted bottom fan-in and literal polarity; its proof bounds the probability that a planted copy forms a minterm of an arbitrary fixed CNF over a random background.

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