Counterfactual Divergence Singularity: A Theoretical Model of High-Similarity Instability and Micro-Counterfactual Drift in Large Language Models

Large language models (LLMs) demonstrate remarkable fluency and contextual sensitivity, yet their behaviour under minimal semantic perturbations remains poorly understood. In particular, near-identical paraphrastic inputs—expected to yield stable and equivalent responses—often produce disproportionately divergent generative behaviour. This paper introduces the Counterfactual Divergence Singularity (CDS) , a theoretical and empirical framework that characterizes a previously underexplored instability regime in transformer-based language models. This work formalizes micro-counterfactual perturbations as infinitesimal semantic variations in embedding space and show that, as semantic similarity approaches unity, divergence metrics exhibit a hyperbolic blow-up. Using a controlled experimental pipeline based on FLAN-T5-generated paraphrases and Sentence-BERT semantic embeddings, this paper empirically demonstrate that divergence, curvature, and embedding drift collectively reveal a singular boundary where semantic proximity no longer guarantees behavioural stability. Despite minimal embedding displacement and near-maximal cosine similarity, reciprocal divergence measures increase sharply, exposing a structural nonlinearity in the input–output mapping of LLMs. The results suggest that LLM semantic manifolds are locally non-smooth and that standard embedding-based similarity metrics fail to capture instability near the identity boundary. This phenomenon provides a geometric and mathematical explanation for prompt sensitivity, counterfactual failure, and certain forms of hallucination observed in generative systems. By reframing these behaviours as consequences of latent-space singularities rather than isolated decoding artifacts, this work contributes a novel theoretical lens for evaluating robustness, interpretability, and reliability in large language models.