# Computation Over Geometry: Meaning Identity Is Computed, Not Shipped in the Embeddings

arXiv: https://arxiv.org/abs/2609.28290 · HTML: https://mlsanity.com/paper/2609.28290/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 23 references that are in the map.
- First posted on arXiv 23 Sep 2026.

## Builds on (most cited first, 22 of 22)

- [Sentence-BERT: Sentence Embeddings using Siamese BERT-Networks](https://mlsanity.com/paper/1908.10084/) (2019)
- [Retrieval-Augmented Generation for Large Language Models: A Survey](https://mlsanity.com/paper/2312.10997/) (2023)
- [SimCSE: Simple Contrastive Learning of Sentence Embeddings](https://mlsanity.com/paper/2104.08821/) (2021)
- [Understanding intermediate layers using linear classifier probes](https://mlsanity.com/paper/1610.01644/) (2016)
- [Text Embeddings by Weakly-Supervised Contrastive Pre-training](https://mlsanity.com/paper/2212.03533/) (2022)
- [BEIR: A Heterogenous Benchmark for Zero-shot Evaluation of Information Retrieval Models](https://mlsanity.com/paper/2104.08663/) (2021)
- [ColBERT: Efficient and Effective Passage Search via Contextualized Late Interaction over BERT](https://mlsanity.com/paper/2004.12832/) (2020)
- [BERT Rediscovers the Classical NLP Pipeline](https://mlsanity.com/paper/1905.05950/) (2019)
- [SemEval-2017 Task 1: Semantic Textual Similarity - Multilingual and Cross-lingual Focused Evaluation](https://mlsanity.com/paper/1708.00055/) (2017)
- [How Contextual are Contextualized Word Representations? Comparing the Geometry of BERT, ELMo, and GPT-2 Embeddings](https://mlsanity.com/paper/1909.00512/) (2019)
- [The Linear Representation Hypothesis and the Geometry of Large Language Models](https://mlsanity.com/paper/2311.03658/) (2023)
- [Passage Re-ranking with BERT](https://mlsanity.com/paper/1901.04085/) (2019)
- [MTEB: Massive Text Embedding Benchmark](https://mlsanity.com/paper/2210.07316/) (2022)
- [C-Pack: Packed Resources For General Chinese Embeddings](https://mlsanity.com/paper/2309.07597/) (2023)
- [PAWS: Paraphrase Adversaries from Word Scrambling](https://mlsanity.com/paper/1904.01130/) (2019)
- [Analysis Methods in Neural Language Processing: A Survey](https://mlsanity.com/paper/1812.08951/) (2018)
- [PAWS-X: A Cross-lingual Adversarial Dataset for Paraphrase Identification](https://mlsanity.com/paper/1908.11828/) (2019)
- [Information-Theoretic Probing for Linguistic Structure](https://mlsanity.com/paper/2004.03061/) (2020)
- [PARAPHRASUS : A Comprehensive Benchmark for Evaluating Paraphrase Detection Models](https://mlsanity.com/paper/2409.12060/) (2024)
- [Differential syntactic and semantic encoding in LLMs](https://mlsanity.com/paper/2601.04765/) (2026)
- [Model Internal Sleuthing: Finding Lexical Identity and Inflectional Features in Modern Language Models](https://mlsanity.com/paper/2506.02132/) (2025)
- [On the Persistent Effects of Lexicality in Large Language Models](https://mlsanity.com/paper/2606.02750/) (2026)

## Abstract (arXiv)

Meaning identity (whether two sentences say the same thing after wording changes) is treated in retrieval and RAG as a geometric fact about independently encoded sentence vectors. We show that, for frozen off-the-shelf encoders and language models, it is not: identity is computed when both sentences share one forward pass, and is not a property of the embedding geometry those systems ship. On overlap-matched PAWS-X, purpose-built encoders (BGE, E5, GTE, MiniLM, E5-Mistral-7B) reach English confirm AUC only 0.55-0.65 (dense peak 0.70). Independently encoded last-token states of Llama 3, Mistral, and Qwen do no better; late fusion of the two vectors stays near chance. The same probe on a joint forward pass reaches 0.90-0.96 from 1.5B to 32B, collapses under partner shuffle, is mid-depth, saturates near 0.94 by 3B, and appears more weakly in GPT-2 XL (0.76). The gap holds beyond Llama-style models on other causal LMs, bidirectional encoders (DeBERTa, RoBERTa), and encoder-decoders (Flan-T5, T5, BART). Fixed or linear readers over frozen independent encodings never unlock identity; nonlinear pair readers recover part of it only on the full 49k-pair PAWS train split (0.68-0.87). Off-the-shelf rerankers split: BGE-reranker-large reaches 0.94, while MS-MARCO and Jina stay at 0.55-0.64. Independently trained families compute the same relation and a 1.5B joint reader can distill it from unlabelled teacher scores, while no linear function of the teachers own independent vectors can. Bi-encoders can be fine-tuned to fit PAWS (0.87-0.93), but transfer and STS-B suffer. Cosine compares wording neighbourhoods; identity is a cheap computed operator, not a property of either sentence vector.