When Verification Is Late: Delay Floors and Placement Flips in Corrected Multi-Agent Systems

<title>Abstract</title> <p>A cheap swarm of unreliable agents can be steered toward truth by a few strong correctors, but each acts only after it verifies, and verification takes time. We model this as a pinned, delayed grounded-Laplacian consensus and ask how much correction to buy, and where, once it is delayed. Three results follow. Delay imposes a floor on accuracy that no budget beats; it persists at finite grounding, and we give its constant exactly: the optimal strength-delay product w*tau is the Dottie number, the root of cos(xi) = xi. The floor is reached at a finite budget, so the budget-accuracy frontier has a ceiling that no delay-free placement theory has. Delay also flips the placement rule: past a closed-form budget, spreading correction over many correctors beats concentrating it. We calibrate the model to a corrected Qwen3.6-35B-A3B swarm. Code and data: https://github.com/YehudaItkin/pinning-delay-frontier.</p>