We Ran 35 Controlled Experiments on "Exotic Algebra" Neural Networks. They're a Placebo.
TL;DR: Quaternion, octonion, sedenion, and Clifford-algebra neural networks keep appearing in the literature with claims of parameter efficiency and special expressivity. We spent two months building them, benchmarking them — and honestly trying to make them win. Whenever we added one properly matched real-valued baseline, the advantage evaporated. Every time. We ended up proving the failure is inevitable, not accidental, killed one of our own headline results along the way, and distilled the whole thing into a one-line rule any reviewer can apply in 30 seconds. Preprint DOI: 10.5281/zenodo.21623615 — code and raw per-seed results: github.com/mxkuzn/hypercomplex-placebo-audit
The seduction
If you’ve never met the Cayley–Dickson ladder, it’s genuinely beautiful. Start with real numbers. Double them: complex numbers. Double again: quaternions — multiplication stops commuting, and you get 3D rotations for free. Double again: octonions — multiplication stops associating, which sounds broken but is mathematically profound. Once more: sedenions, where you meet...
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