C02.1 · Derivation

From the product to the cubic identity

The diagonal entries of \(X^\#\) are \[ bc-n(x),\qquad ca-n(y),\qquad ab-n(z). \] Its off-diagonal coordinates in the order \(z,y,x\) are \[ \bar y\,\bar x-cz,\qquad \bar x\,\bar z-by,\qquad \bar z\,\bar y-ax. \] These follow by multiplying the two matrices in \(X\circ X\); the remaining terms subtract the appropriate diagonal trace.

Substitute these expressions into \(X\circ X^\#\). Diagonal terms reduce to the displayed cubic \(N(X)\); off-diagonal terms cancel. Rather than certify this from a few numerical examples, the script gives all coordinates independent commuting symbols and expands every residual coefficient. All `generic.cubic_coordinate_polynomials_zero` coordinates vanish. The computed norm has degree \(3\) and \(89\) monomials in this basis (`generic.norm_degree`, `generic.norm_monomials`).

Consequently \[ X^{\circ3}-t(X)X^{\circ2}+S(X)X-N(X)I=0. \] This is an equation in the Jordan algebra. The code also checks \[ S(X)=\frac{t(X)^2-t(X\circ X)}2,\qquad dN_X(Y)=\operatorname{Tr}(X^\#\circ Y) \] as exact polynomial identities.

The Jordan identity itself is checked independently of numerical samples by constructing \(L_X\) and \(L_{X\circ X}\) for the same generic \(X\) and expanding every entry of their commutator. All \(729\) coordinate polynomials vanish (`Jordan_identity.generic_commutator_coordinate_polynomials_zero`). The additional deterministic integer samples are only supplementary controls; no universal conclusion is inferred from them.

This gives a complete finite polynomial specification for an independent implementation: another reader can construct the displayed product and calculate the same residuals without reading this script.

The trace form is not a positive Euclidean norm

Define \[ B(X,Y)=\operatorname{Tr}(X\circ Y). \] In the declared coordinates its matrix is \[ B=\operatorname{diag}(1,1,1;\ 2\eta;\ 2\eta;\ 2\eta), \quad \eta=\operatorname{diag}(1,1,1,1,-1,-1,-1,-1). \] Thus each octonionic block contributes both norm signs. The code obtains inertia \((15,12,0)\), ordered positive/negative/zero, matching the source's raw split signature.

Every basis multiplication operator satisfies \(L_A^T B=B L_A\); linearity in \(A\) makes those basis checks sufficient for the general trace-self-adjoint identity. It is **not** the assertion \(L_A^T=L_A\). A deterministic non-diagonal sample explicitly fails Euclidean symmetry.

An element cubic is not an operator cubic

Take the exact diagonal element \(D=\operatorname{diag}(1,2,3)\). Its element polynomial is \[ p_D(\lambda)=(\lambda-1)(\lambda-2)(\lambda-3), \] and \(p_D(D)=0\) in the Jordan sense. But on the off-diagonal block joining the first two diagonal entries, \(L_D\) acts by their arithmetic mean \(3/2\). Therefore \[ p_D(L_D)|_{J_{12}}=\frac38 I\ne0. \] This is directly executed as `diagonal123.operator_cubic_on_J12_scalar`; the full operator residual has \(16\) nonzero entries.

The code prints the actual multiplication spectrum. In this example the mean of the first and third diagonal entries coincides with the middle diagonal eigenvalue, producing a multiplicity merger. For a generic diagonal element the diagonal eigenvalues and pairwise means need not coincide. Appendix A p.139 already separates the anchor values from the pairwise means; that distinction must accompany the p.138 “Jordan spectrum” wording.

A split-Hermitian element with nonreal roots

Let \(W\) have \(z=\ell\) and all its other coordinates zero. Since \(\bar\ell=-\ell\), it is Hermitian under the stipulated octonion involution. Since \(\ell^2=1\), \[ W\circ W=-(E_1+E_2). \] Hence \[ W^{\circ2}+E_1^{\circ2}+E_2^{\circ2}=0 \] with nonzero terms. This is an explicit failure of formal reality, not merely an indefinite auxiliary metric.

The element characteristic polynomial is \[ p_W(\lambda)=\lambda(\lambda^2+1). \] Its nonreal roots are computed in `split_witness.element_roots`. The full real lift has characteristic polynomial \[ \lambda^9(\lambda^2+1)(\lambda^2+\tfrac14)^8, \] printed in an equivalent factored rational form by `split_witness.L_characteristic`. Thus it is a real matrix with nonreal eigenvalues, despite being self-adjoint for \(B\).

There is no contradiction: self-adjointness for an indefinite bilinear form is not the positive-inner-product spectral theorem. Nor should these complex characteristic roots be described as a real primitive-idempotent decomposition. Solving \(A\circ V=\lambda V\) on the full real vector space and requiring \(V\) to be a primitive idempotent are different problems.