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

A new geometric framework models how concepts survive, distort, or die when they travel between minds with different internal value systems, according to a paper posted on arXiv. The work represents each agent as a personalized vector space and treats evaluative concepts as structured vectors whose transmission depends on linear interpretation maps [1][2]. The framework, authored by Chainarong Amornbunchornvej, formalizes evaluative concepts as "abstract beings" — structured vectors that propagate across agents. An abstract being survives communication only if it avoids the null spaces of the interpretation maps that mediate transmission, yielding a structural criterion for intelligibility, miscommunication, and what the paper terms "concept death" [2]. The model shows that conceptual distortion, motivational drift, and the limits of mutual understanding arise from purely algebraic constraints rather than failures of rationality or shared information [2]. This approach draws on the broader mathematical tradition of using algebraic structures to represent relationships that are not directly observable — a strategy also seen in structural equation modeling, where latent variables and causal connections are estimated from observed data patterns [3]. A central result, the No-Null-Space Leadership Condition, characterizes leadership as a property of representational reachability rather than persuasion or authority [2]. The condition defines when one agent's conceptual space can fully represent another's, sidestepping traditional notions of influence. The paper argues that meaning preservation is grounded in structural compatibility, unifying insights from conceptual spaces, social epistemology, and AI value alignment [2]. The manuscript was submitted on 10 December 2025 and revised on 13 June 2026, growing from 63 KB to 91 KB in its current version [1]. It appears under the arXiv category for Artificial Intelligence (cs.AI) and is accessible through arXivLabs, a framework that allows community collaborators to develop experimental tools on the repository's abstract pages [1][9]. arXivLabs was formalized in 2020 to enable third-party contributions while enforcing guidelines on openness and user data privacy [10]. By grounding meaning preservation in the geometry of vector spaces and linear maps, the framework offers a formal language for analyzing how concepts mutate or disappear as they traverse diverse cognitive geometries — a question with implications for both human communication and artificial systems [2].