for instance: a real number records the amplitude of a wave. A complex number records the amplitude and its phase. A quaternion records the phase of a phase. An octonion adds one more layer. the phase of a phase of a phase. These aren't what the wave "is made of"; they're the accounting ledger for tracking how interference patterns reference their own history.
The octonion algebra is the deepest ledger that stays consistent, a fourth doubling would introduce zero divisors and the bookkeeping would contradict itself (Hurwitz's theorem). The automorphism group of the octonions is the Lie group G₂, and it labels exactly the interference patterns that form closed loops through all three layers of this self-referencing record.
Most loops just cycle. But some shed their distinguishing detail. they forget which of the three layers they came from and collapse into standing waves. That forgetting produces three generations of fermions. The standing-wave conditions fix nine charged-fermion masses and the full CKM mixing matrix.
I would appreciate your feedback on the math in the following preprint: https://doi.org/10.5281/zenodo.20052650