Controlling Logical Collapse in LLMs via Algebraic Ontology Projection over F2

TL;DR

This paper introduces Algebraic Ontology Projection (AOP) and Semantic Crystallisation (SC) to analyze and control the logical structure of large language models through algebraic methods, improving interpretability and consistency.

Contribution

It presents a novel algebraic framework for probing LLMs' internal ontological relations and identifies conditions to prevent logical collapse during inference.

Findings

AOP achieves up to 93.33% zero-shot inclusion accuracy.

SC predicts zero-shot accuracy without held-out data.

Model prompts act as algebraic boundary conditions.

Abstract

Do large language models internally encode ontological relations in a formally verifiable algebraic structure? We introduce Algebraic Ontology Projection (AOP), which projects LLM hidden states into the Galois Field F2 under Liskov Substitution Principle constraints, using only 42 relational pairs as algebraic keys. AOP achieves up to 93.33% zero-shot inclusion accuracy on unseen concept pairs (Gemma-2 Instruct with optimized prompt), with consistent 86.67% accuracy observed across multiple model families -- with no model tuning, but through prompt alone. This algebraic structure is strongly layer-dependent. We introduce Semantic Crystallisation (SC), a metric that quantifies F2 constraint satisfaction relative to a random baseline and predicts zero-shot accuracy without held-out data. System prompts act as algebraic boundary conditions: only their combination with instruction tuning prevents Late-layer Collapse -- a systematic degradation of logical consistency in the final layers, observed in 7 of 10 conditions. These findings reframe forward computation as an iterative process of algebraic organisation, and open a path toward LLMs whose logical structure is not merely approximated, but formally accessible.