Ryan JHANG's Law of Semantic Gravitation: Four Universal Statistical Invariants in Self-Organizing Semantic Spaces

Semantic embedding spaces are the numerical substrate of modern language models, yet their global statistical organization has never been characterized as a law. We report Ryan JHANG’s Law of Semantic Gravitation: when semantic states self-organize under competing tendencies, they settle into a statistical equilibrium exhibiting four universal statistical invariants — statistical equilibrium, heavy-tailed structure, effective- dimension reduction, and multi-scale fractality. Strikingly, the effective intrinsic dimension stabilizes at d ≈ 8 regardless of the ambient embedding dimension (testedacross 20–50 dimensions and multiple frequency bands), while cluster densities vary freely — exactly as a law of invariants predicts. We further show, via scale-invariance, that these power laws share a common mathematical origin with Neural Scaling Laws, and provide a conditional-coupling analysis of their causal relationship. This work anchors the phenomenological law and its invariants; the generating process is reserved for a subsequent release.