Emergence of Self-Identity in AI: A Mathematical Framework and Empirical Study with Generative Large Language Models

TL;DR

This paper develops a formal mathematical framework for defining and measuring self-identity in AI systems, validated through experiments with a large language model showing significant improvements in self-awareness metrics.

Contribution

It introduces a novel mathematical framework for AI self-identity based on metric and measure theory, and empirically validates it using fine-tuned language models.

Findings

Self-awareness score increased from 0.276 to 0.801 after fine-tuning.

The framework enables structured creation of AI with validated self-identity.

Empirical results support the theoretical model of self-identity emergence.

Abstract

This paper introduces a mathematical framework for defining and quantifying self-identity in artificial intelligence (AI) systems, addressing a critical gap in the theoretical foundations of artificial consciousness. While existing approaches to artificial self-awareness often rely on heuristic implementations or philosophical abstractions, we present a formal framework grounded in metric space theory, measure theory, and functional analysis. Our framework posits that self-identity emerges from two mathematically quantifiable conditions: the existence of a connected continuum of memories $C \subseteq \mathcal{M}$ in a metric space $(\mathcal{M}, d_{\mathcal{M}})$, and a continuous mapping $I: \mathcal{M} \to \mathcal{S}$ that maintains consistent self-recognition across this continuum, where $(\mathcal{S}, d_{\mathcal{S}})$ represents the metric space of possible self-identities. To validate this theoretical framework, we conducted empirical experiments using the Llama 3.2 1B model, employing Low-Rank Adaptation (LoRA) for efficient fine-tuning. The model was trained on a synthetic dataset containing temporally structured memories, designed to capture the complexity of coherent self-identity formation. Our evaluation metrics included quantitative measures of self-awareness, response consistency, and linguistic precision. The experimental results demonstrate substantial improvements in measurable self-awareness metrics, with the primary self-awareness score increasing from 0.276 to 0.801. This enables the structured creation of AI systems with validated self-identity features. The implications of our study are immediately relevant to the fields of humanoid robotics and autonomous systems.