Dynamics Reveals Structure: Challenging the Linear Propagation Assumption

Neural networks adapt through first-order parameter updates, yet it remains unclear whether such updates preserve logical coherence. We investigate the geometric limits of the Linear Propagation Assumption (LPA), the premise that local updates coherently propagate to logical consequences. To formalize this, we adopt relation algebra and study three core operations on relations: negation flips truth values, converse swaps argument order, and composition chains relations. For negation and converse, we prove that guaranteeing direction-agnostic first-order propagation necessitates a tensor factorization separating entity-pair context from relation content. However, for composition, we identify a fundamental obstruction. We show that composition reduces to conjunction, and prove that any conjunction well-defined on linear features must be bilinear. Since bilinearity is incompatible with negation, this forces the feature map to collapse. These results suggest that failures in knowledge editing, the reversal curse, and multi-hop reasoning may stem from common structural limitations inherent to the LPA.

This work challenges a common assumption behind updating knowledge in large language models that a small, targeted change to a model will automatically spread to logically related facts. For example, if an update teaches a model that a statement is true, we would hope that it also adjusts what the model says about the negation, the reversed relation, or related multi-step conclusions. We show that this kind of automatic logical propagation is not guaranteed by the usual local update mechanisms used in neural networks. In particular, we identify a strong geometric structure that would be needed for updates to handle negation and reversed relations consistently, and we show that extending the same idea to multi-step reasoning leads to a fundamental obstruction. These results suggest that several known failures of language models, including difficulty with negation, reversed facts, and multi-hop reasoning after editing, may share a common cause. More broadly, our work argues that to make model updates reliable, we should study not only what a model represents, but also how its representations change when the model is updated.