Quantization Errors, Human--AI Interaction, and Approximate Fixed Points in $L^1(μ)$

2025-09-15

AI Summary

This paper investigates the existence of fixed points for nonexpansive mappings on $L^1(mu)$ under fixed-point arithmetic quantization errors, with application to stability analysis of consensus in human-AI collaborative editing systems. To address mapping distortion induced by quantization perturbations, we introduce the notion of “measure compactness” and integrate it with uniform integrability, measure convexity, normal structure, and Kirk’s fixed-point theorem. We prove that nonexpansive mappings on measure-compact sets admit fixed points, and retain approximate fixed points under bounded quantization error. This work constitutes the first systematic incorporation of robust fixed-point theory from functional analysis into human-AI interaction modeling. It rigorously establishes both the existence and quantization robustness of stable consensus states in collaborative editing, thereby providing a theoretical foundation for trustworthy human-AI collaboration.

Technology Category

Humans and AI: Planning and Decision Support for Human-Machine TeamsKnowledge Representation and Reasoning: Qualitative ReasoningSearch and Optimization: Non-convex Optimization

Application Category

Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasetsSemantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactionsSecurity and Privacy: Large-scale security measurements

Abstract

We develop a rigorous measure-theoretic framework for the analysis of fixed points of nonexpansive maps in the space $L^1(μ)$, with explicit consideration of quantization errors arising in fixed-point arithmetic. Our central result shows that every bounded, closed, convex subset of $L^1(μ)$ that is compact in the topology of local convergence in measure (a property we refer to as measure-compactness) enjoys the fixed point property for nonexpansive mappings. The proof relies on techniques from uniform integrability, convexity in measure, and normal structure theory, including an application of Kirk's theorem. We further analyze the effect of quantization by modeling fixed-point arithmetic as a perturbation of a nonexpansive map, establishing the existence of approximate fixed points under measure-compactness conditions. We also present counterexamples that illustrate the optimality of our assumptions. Beyond the theoretical development, we apply this framework to a human-in-the-loop co-editing system. By formulating the interaction between an AI-generated proposal, a human editor, and a quantizer as a composition of nonexpansive maps on a measure-compact set, we demonstrate the existence of a "stable consensus artefact". We prove that such a consensus state remains an approximate fixed point even under bounded quantization errors, and we provide a concrete example of a human-AI editing loop that fits this framework. Our results underscore the value of measure-theoretic compactness in the design and verification of reliable collaborative systems involving humans and artificial agents.

Problem

• Analyzing fixed points of nonexpansive maps in L^1(μ) space
• Modeling quantization errors in fixed-point arithmetic systems
• Establishing stable consensus in human-AI collaborative editing systems

Innovation

• Measure-theoretic framework for L1 fixed points
• Quantization errors modeled as nonexpansive perturbations
• Human-AI co-editing as measure-compact consensus system