Geometry, Performance, and Typology Are Independent Axes: A Null Result and a Path Toward Diagnostic Topology

Multilingual pretrained models are widely assumed to encode language identity in ways that reflect meaningful linguistic or functional structure, but whether representational geometry itself can serve as a reliable proxy for downstream quality remains untested at scale. We evaluate this directly: across four models (XLM-R, mBERT, mT0-small, mT5-small), six typologically diverse languages, and two downstream tasks (QA, semantic relatedness), we find that geometric distance between language representations does not reliably predict task performance, pooled, rank-normalized Spearman correlations are statistically indistinguishable from zero, with bootstrap confidence intervals spanning both positive and negative values. This holds even as typological alignment varies inconsistently across models, indicating that geometry, performance, and typology behave as largely independent axes. Motivated by this null result, we shift the evaluation question from prediction to diagnosis: can representational topology reveal specific, actionable pathologies, such as tokenizer-driven fragmentation, in low-resource language representations? We report a small single-language proof of concept using persistent homology on Amharic subword embeddings, showing a counterintuitive pattern that motivates a larger, controlled follow-up across resource-stratified low-resource languages.