Root-Length Ratio Inversion: A Literature Search Gap — E8 Intelligence Research

FINDING: The search results are dominated by video lectures and speculative "Adelic Langlands" frameworks, not peer-reviewed papers. The only concrete mathematical item is a 2007 arXiv note on arithmetic lattices and weak spectral geometry. No direct evidence of a proven \(B_n/C_n\) root-length ratio inversion is present in these results. MATH: - Root systems: \(B_n\) has roots of two lengths (long: \(2\), short: \(1\) in standard normalization); \(C_n\) has the inverse ratio (long: \(2\), short: \(1\) swapped — actually \(C_n\) long roots have squared length \(2\), short roots squared length \(1\), but the *ratio* of long-to-short squared lengths is \(2:1\) for both \(B_n\) and \(C_n\); the inversion is in the *assignment* of which simple roots are long/short, not the ratio itself). - The ratio \(2:1\) in squared lengths corresponds to \(\sqrt{2}:1\) in actual lengths. - No equation, constant, or ratio beyond this standard Lie theory fact appears in the provided sources. - Th Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com