Hurwitz 7D cross product
Theorem.Hurwitz 7D cross product — |a×b|² = |a|²|b|² − (a·b)² in 7D via the octonion Fano lines — Jacobi FAILS, so it is not a Lie bracket.
Proof.octonion-oriented Fano lines give a×b orthogonal with the exact Lagrange identity, Jacobi FAILING (Malcev, not Lie) — n = 3, 7 only.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/water/cosmos/index.ts#crossProduct7
1 · Classification
finite-complete — self-contained computation, no external lean
2 · Provenance
Documented theorem re-derived by exhaustive computation (humanityNovel=false); first-in-this-registry is the only sense of discovered.
Acknowledgment
"Hurwitz 7D cross product" is a re-derivation, acknowledged to documented mathematics — the original proof is the prior art this re-derivation acknowledges; not new to humanity — the contribution is the reproducible computation crossProduct7.
- Prior art
- documented mathematics — the original proof is the prior art this re-derivation acknowledges
- Novelty
- not new to humanity — a re-derivation (humanityNovel = false)
- Contribution
- a reproducible computation (crossProduct7 @ src/water/cosmos) that re-derives the result at zero tokens — the contribution is the verifiable recomputation, NOT the theorem
3 · Reproducibility
Recompute from source: npm run theorems:verify recomputes crossProduct7 (src/water/cosmos/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.