Human+AI loops stay stable even with quantization (arxiv.org)

Researchers present a measure-theoretic fixed-point framework in L1(μ) that explicitly accounts for quantization errors from fixed‑point arithmetic. Their main theorem proves that any bounded, closed, convex subset of L1 that is compact for local convergence in measure ("measure‑compact") has the fixed‑point property for nonexpansive maps — operators that do not increase distances. Using tools like uniform integrability, convexity in measure, normal structure theory and Kirk’s theorem, they further model quantization as a perturbation of a nonexpansive map and prove the existence of approximate fixed points under bounded quantization. They also give counterexamples showing their measure‑compactness assumptions are essentially tight. The paper then applies this theory to human‑in‑the‑loop co‑editing: the AI proposal, human edits and quantizer compose into nonexpansive maps on a measure‑compact set, guaranteeing a stable “consensus artifact” that persists as an approximate fixed point despite bounded quantization. For the AI/ML community this provides rigorous convergence and robustness guarantees outside Hilbert spaces (L1 instead), broadening applicability to probabilistic outputs and sparse representations, and gives a principled criterion (measure‑compactness) for designing and verifying reliable collaborative systems that remain stable under finite‑precision computation.