Relations as Group Elements: Algebraic composition in place of statistical association, in eight experiments

Modern large language models compose relations statistically: they learn the distribution of what follows what, and their reasoning degrades the moment a chain extends past the depths seen in training. This paper presents eight pre-registered experiments showing that when relations are instead represented as elements of a finite group — composed by arithmetic rather than inference — relational chains extrapolate perfectly to 6.4× and 12.8× beyond training depth, while a transformer of three times the parameter count falls to near chance. The results isolate a structural gap in current LLM systems: a swap test shows a transformer scoring 84% on relational questions without ever determining which entity is which, and a graph network with the correct message-passing bias degrades at the same rate. A 33,000-parameter model with the right algebra outperforms a 103,000-parameter transformer by 0.503 at depth 4, and a solver built on the composition law identifies 305 hidden relation phrases with zero errors — going silent, rather than guessing, where the system is underdetermined. The consistent finding is that the algebraic properties of a relation set are worth more than the capacity of the model operating on them: composition supplied as arithmetic works and extrapolates; composition left to be discovered as a byproduct of a downstream task is not discovered. The paper introduces Small Psychological Models (SPMs) as the class of compact systems built on this principle — models that own a domain, know the boundary of their own competence, and show their work — with direct implications for auditable, hallucination-free reasoning in high-stakes applications where current LLM architectures cannot certify their own outputs.