Semantic rulers: Continuous measurement of abstract concepts in LLM embedding spaces

Across scientific, economic, and design disciplines, evaluating abstract qualitative concepts—such as environmental sustainability or aesthetic density—traditionally relies on subjective human assessment. This inherent subjectivity limits the integration of these complex traits into automated quantitative workflows. To address this gap, this paper proposes a methodology that uses the latent embedding spaces of large language models and multimodal foundation models to construct continuous semantic scoring instruments. We introduce a counterfactual prompt engineering framework to extract concept vectors from text and image representations while controlling for domain-specific variation, thereby creating calibrated semantic scales. Across the examined domains, the results provide empirical evidence that the target concepts exhibit approximately linear geometric structure. We further evaluate the framework against an externally derived, geospatially grounded ordinal reference (NUS Global Streetscapes with Global Human Settlement Layer typologies), observing a substantial association between the semantic scores and physical settlement density without relying solely on AI self-consistency. The primary novelty of this work lies in extending the use of concept vectors from internal model interpretability toward external semantic scoring. To illustrate a bridge between natural language processing and applied mathematics, we integrate these continuous semantic distances into explicit bounded utility functions. The framework therefore provides a transparent mechanism for separating model-derived semantic scores from normative preference mappings, yielding continuous variables that may serve as components in broader decision support systems.