C02.1 · Interpretation

The cubic norm and the real lift are both legitimate algebraic objects, but they answer different questions. The cubic belongs to an element in a rank-three Jordan algebra; the real lift acts on the entire carrier. An eigenvalue of the latter is not automatically one of three physical masses, and “real matrix” is not a sufficient eigensolver contract.

For the displayed diagonal frame, three orthogonal idempotents are manifest. The cubic identity bounds the size of an orthogonal idempotent family: a linear combination with distinct coefficients would otherwise require a minimal polynomial of degree greater than the cubic. This explains the algebraic rank, without equating that rank with a physical family count.