The Fractured Interlingua: Geometric Bottlenecks in Cross-Lingual Knowledge Editing

Transformer-based autoregressive multilingual language models (LMs) are often assumed to store factual knowledge in an interlingua: a language-agnostic latent space. If this assumption holds, a parameter update applied in a source language during knowledge editing should inherently propagate to its semantic equivalents in target languages. However, empirical benchmarks reveal that cross-lingual propagation of edits consistently fails, a degradation that prior work treats largely as an algorithmic symptom rather than a structural bottleneck. In this work, we hypothesize that this failure stems from language anisotropy, and we perform a rigorous geometric dissection of the cross-lingual propagation gap. Using associative memory-based "locate-then-edit" methods, including ROME, MEMIT, and AlphaEdit, as analytical frameworks, we derive a closed-form decomposition of this gap into key-space propagation, parallel residual mismatch, and orthogonal residual mismatch. We empirically find that cross-lingual propagation is consistently limited by a large orthogonal residual component, indicating that source-language edits often fail to span the target-language update directions required for successful transfer. Furthermore, we show that post-hoc correction using only source-language residual statistics faces a large residual-prediction barrier. Finally, we demonstrate that structural realignment via a targeted contrastive alignment objective substantially improves key-space propagation and partially improves cross-lingual edit transfer, while leaving value-space mismatch as a remaining bottleneck.