# Semantic Grounding Index: Geometric Bounds on Context Engagement in RAG Systems

arXiv: https://arxiv.org/abs/2512.13771 · HTML: https://mlsanity.com/paper/2512.13771/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Generation and context integration, one of 15 directions in Retrieval-augmented generation.
- Builds on 23 of its 39 references that are in the map.
- First posted on arXiv 15 Dec 2025.

## Placed in

- Retrieval-augmented generation / [Generation and context integration](https://mlsanity.com/rag/generation-and-context-integration/)

## Builds on (most cited first, 23 of 23)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Retrieval-Augmented Generation for Knowledge-Intensive NLP Tasks](https://mlsanity.com/paper/2005.11401/) (2020)
- [Sentence-BERT: Sentence Embeddings using Siamese BERT-Networks](https://mlsanity.com/paper/1908.10084/) (2019)
- [TruthfulQA: Measuring How Models Mimic Human Falsehoods](https://mlsanity.com/paper/2109.07958/) (2021)
- [SimCSE: Simple Contrastive Learning of Sentence Embeddings](https://mlsanity.com/paper/2104.08821/) (2021)
- [REALM: Retrieval-Augmented Language Model Pre-Training](https://mlsanity.com/paper/2002.08909/) (2020)
- [Survey of Hallucination in Natural Language Generation](https://mlsanity.com/paper/2202.03629/) (2022)
- [A Survey on Hallucination in Large Language Models: Principles, Taxonomy, Challenges, and Open Questions](https://mlsanity.com/paper/2311.05232/) (2023)
- [MiniLM: Deep Self-Attention Distillation for Task-Agnostic Compression of Pre-Trained Transformers](https://mlsanity.com/paper/2002.10957/) (2020)
- [Understanding Contrastive Representation Learning through Alignment and Uniformity on the Hypersphere](https://mlsanity.com/paper/2005.10242/) (2020)
- [Text Embeddings by Weakly-Supervised Contrastive Pre-training](https://mlsanity.com/paper/2212.03533/) (2022)
- [Semantic Uncertainty: Linguistic Invariances for Uncertainty Estimation in Natural Language Generation](https://mlsanity.com/paper/2302.09664/) (2023)
- [Towards General Text Embeddings with Multi-stage Contrastive Learning](https://mlsanity.com/paper/2308.03281/) (2023)
- [Retrieval Augmentation Reduces Hallucination in Conversation](https://mlsanity.com/paper/2104.07567/) (2021)
- [The Internal State of an LLM Knows When It's Lying](https://mlsanity.com/paper/2304.13734/) (2023)
- [SummaC: Re-Visiting NLI-based Models for Inconsistency Detection in Summarization](https://mlsanity.com/paper/2111.09525/) (2021)
- [HaluEval: A Large-Scale Hallucination Evaluation Benchmark for Large Language Models](https://mlsanity.com/paper/2305.11747/) (2023)
- [Hurdles to Progress in Long-form Question Answering](https://mlsanity.com/paper/2103.06332/) (2021)
- [Spherical Text Embedding](https://mlsanity.com/paper/1911.01196/) (2019)
- [LettuceDetect: A Hallucination Detection Framework for RAG Applications](https://mlsanity.com/paper/2502.17125/) (2025)
- [FaithBench: A Diverse Hallucination Benchmark for Summarization by Modern LLMs](https://mlsanity.com/paper/2410.13210/) (2024)
- [Uncertainty Quantification in Large Language Models Through Convex Hull Analysis](https://mlsanity.com/paper/2406.19712/) (2024)

## Abstract (arXiv)

When retrieval-augmented generation (RAG) systems hallucinate, what geometric trace does this leave in embedding space? We introduce the Semantic Grounding Index (SGI), defined as the ratio of angular distances from the response to the question versus the context on the unit hypersphere mathbbSd-1.Our central finding is semantic laziness: hallucinated responses remain angularly proximate to questions rather than departing toward retrieved contexts. On HaluEval (n=5,000), we observe large effect sizes (Cohen's d ranging from 0.92 to 1.28) across five embedding models with mean cross-model correlation r=0.85. Crucially, we derive from the spherical triangle inequality that SGI's discriminative power should increase with question-context angular separation θ(q,c)-a theoretical prediction confirmed empirically: effect size rises monotonically from d=0.61 -low θ(q,c), to d=1.27 -high θ(q,c), with AUC improving from 0.72 to 0.83. Subgroup analysis reveals that SGI excels on long responses (d=2.05) and short questions (d=1.22), while remaining robust across context lengths. Calibration analysis yields ECE=0.10, indicating SGI scores can serve as probability estimates, not merely rankings. A critical negative result on TruthfulQA (AUC=0.478) establishes that angular geometry measures topical engagement rather than factual accuracy. SGI provides computationally efficient, theoretically grounded infrastructure for identifying responses that warrant verification in production RAG deployments.