The Phi Constant of Knowledge: Cross-Domain Geometric Invariance in Semantic Embedding Spaces

We present evidence that genuine cross-domain knowledge integration produces a specific geometric signature in semantic embedding spaces that is invariant across 37 model variants spanning 18 architectures, 4 knowledge domains, 13 languages from 9 linguistic families, and 190 years of historical text (1800-1990). The ratio between consecutive spectral bands follows the golden ratio phi = 1.618 (p = 0.020, N = 37 models), and the natural classification boundary falls at 1/phi^2 = 0.382. A 603-pair corpus with 14 pair types, tested across 28 pre-registered experiments, provides the most comprehensive evaluation of this hypothesis to date. AUC = 0.972 (best model). All code and data available at: https://github.com/kabuto1523/phi-constant-of-knowledge