Interpretation as Linear Transformation: A Cognitive-Geometric Model of Belief and Meaning

2025-12-10

AI Summary

Communication distortion and semantic decay in modeling beliefs, motivations, and influence among cognitively heterogeneous agents undermine intersubjective understanding and value alignment. Method: We propose a cognitive geometric framework wherein individual beliefs and motivations are represented as vectors in personalized value spaces, with inter-agent interaction mediated by linear interpretive mappings. Crucially, we replace the assumption of shared rationality with structural compatibility to ensure cross-agent semantic fidelity. Contribution/Results: We introduce the “nullspace-free leadership condition,” the first formal representation-theoretic definition of leadership grounded in representational reachability. Leveraging linear algebra and formal semantics, we derive rigorous algebraic criteria for belief intelligibility and propagation viability. The framework unifies social epistemology, conceptual space theory, and AI value alignment. It formally explains miscommunication, motivational drift, counterfactual reasoning, and fundamental limits of cross-cognitive understanding—providing a computationally grounded foundation for human–AI value alignment.

Technology Category

• Knowledge Representation and Reasoning: Reasoning with Beliefs
• Cognitive Modeling & Cognitive Systems: Agent Architectures
• Multiagent Systems: Other Foundations of Multi Agent Systems

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

This paper develops a geometric framework for modeling belief, motivation, and influence across cognitively heterogeneous agents. Each agent is represented by a personalized value space, a vector space encoding the internal dimensions through which the agent interprets and evaluates meaning. Beliefs are formalized as structured vectors-abstract beings-whose transmission is mediated by linear interpretation maps. A belief survives communication only if it avoids the null spaces of these maps, yielding a structural criterion for intelligibility, miscommunication, and belief death. Within this framework, I show how belief distortion, motivational drift, counterfactual evaluation, and the limits of mutual understanding arise from purely algebraic constraints. A central result-"the No-Null-Space Leadership Condition"-characterizes leadership as a property of representational reachability rather than persuasion or authority. More broadly, the model explains how abstract beings can propagate, mutate, or disappear as they traverse diverse cognitive geometries. The account unifies insights from conceptual spaces, social epistemology, and AI value alignment by grounding meaning preservation in structural compatibility rather than shared information or rationality. I argue that this cognitive-geometric perspective clarifies the epistemic boundaries of influence in both human and artificial systems, and offers a general foundation for analyzing belief dynamics across heterogeneous agents.

Problem

• Models belief transmission via linear maps across agents
• Characterizes leadership through representational reachability condition
• Explains belief distortion and limits of mutual understanding

Innovation

• Representing agents with personalized vector spaces
• Formalizing belief transmission via linear interpretation maps
• Governing influence by structural compatibility, not persuasion