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

arXiv: https://arxiv.org/abs/2512.09831 · HTML: https://mlsanity.com/paper/2512.09831/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 44 references that are in the map.
- First posted on arXiv 10 Dec 2025.

## Builds on (most cited first, 8 of 8)

- [Sentence-BERT: Sentence Embeddings using Siamese BERT-Networks](https://mlsanity.com/paper/1908.10084/) (2019)
- [MiniLM: Deep Self-Attention Distillation for Task-Agnostic Compression of Pre-Trained Transformers](https://mlsanity.com/paper/2002.10957/) (2020)
- [Transformer Feed-Forward Layers Are Key-Value Memories](https://mlsanity.com/paper/2012.14913/) (2020)
- [Concept Bottleneck Models](https://mlsanity.com/paper/2007.04612/) (2020)
- [Word Translation Without Parallel Data](https://mlsanity.com/paper/1710.04087/) (2017)
- [Artificial Intelligence, Values and Alignment](https://mlsanity.com/paper/2001.09768/) (2020)
- [A mathematical theory of semantic development in deep neural networks](https://mlsanity.com/paper/1810.10531/) (2018)
- [Coordination Event Detection and Initiator Identification in Time Series Data](https://mlsanity.com/paper/1603.01570/) (2016)

## Built on by (most cited first, 2 of 2)

- [Collective Counterfactual Planning: Coordination, Consent, and Verification under Representational Constraints](https://mlsanity.com/paper/2608.17932/) (2026)
- [Functional Emotion Without Character: Large Language Models, Aristotelian Disposition, and the Limits of Behavioral Alignment](https://mlsanity.com/paper/2609.22362/) (2026)

## Abstract (arXiv)

This paper develops a geometric framework for modeling concepts, 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. Evaluative concepts are formalized as structured vectors, abstract beings, whose transmission is mediated by linear interpretation maps. An abstract being survives communication only if it avoids the null spaces of these maps, yielding a structural criterion for intelligibility, miscommunication, and concept death. Within this framework, I show how conceptual distortion, motivational drift, 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 conceptual dynamics across heterogeneous agents.