Where Energy Is Spent Sets the Depth of Kinetic Proofreading

Kinetic proofreading buys accuracy with energy. Where the energy is spent, we show, caps how much accuracy it can buy. Drive layer $j$ of a $K$-checkpoint cascade and the error never falls below $\eeq^{K-j+1}$, where $\eeq$ is the error the same network reaches at equilibrium. The exponent is one plus the first Betti number of the downstream subgraph, which counts its independent cycles. Driving the first layer can recruit all $K$ checkpoints, whereas driving the final layer restricts the best possible scaling to $\eeq^2$, independently of network depth. We also derive the exact dependence on driving strength and show that the error either decreases monotonically or has a unique global minimum, beyond which stronger driving worsens accuracy. Drive placement therefore imposes a topological limit on nonequilibrium error correction.