Semantic–Spatial Correlation Under Independence: A Large-Scale Characterization of the Null Distribution

AbstractSemantic–Spatial Correlation (SSC) quantifies the alignment between two multivariate structuresby computing a rank correlation between semantic and spatial distance matrices. As adistance-matrix dependence measure analogous to Mantel tests and related coefficients, SSC providesa quantitative handle for investigating structure–structure relationships across diverse researchdomains. Despite increasing use in behavioral and cognitive research, the null distributionof SSC under structural independence has not been systematically characterized. The present studyaddresses this gap through large-scale simulation: we generated synthetic semantic embeddings(N = 20 items, D = 100 dimensions) paired with spatial coordinates across n = 1,000 trials percondition, systematically varying embedding dimension (D = 50–500), sample size (N = 10–80),distance metrics, and spatial layouts. Across most conditions, mean SSC remained within ±0.003of zero, with 90% confidence intervals typically contained within approximately [−0.01, +0.01],though small-N conditions showed wider intervals as expected from sampling theory. Two onesidedequivalence tests (TOST) confirmed statistical equivalence to zero within a quasi-orthogonaldomain (|SSC| ≤ 0.10). Validation using pretrained multilingual BERT embeddings replicatedthese findings. These results establish a calibrated null baseline for SSC: under independent generation,SSC distributions concentrate tightly around zero regardless of dimensionality, sample size,metric choice, or spatial geometry. This characterization provides researchers with empiricallygrounded reference values for interpreting observed SSC in applications ranging from memoryresearch to neural embedding evaluation.Keywords: semantic–spatial correlation, null distribution, distance-matrix dependence, equivalencetesting, multivariate methods