C05.1 · Definitions/conventions

Let \(\phi=(1+\sqrt5)/2>0\), so \(\phi^2-\phi-1=0\), and set \[ J=\operatorname{diag}(\phi,1,\phi^{-1}),\qquad J^\#=\operatorname{diag}(\phi^{-1},1,\phi). \] Use the Jordan trace form \(B(X,Y)=\operatorname{Tr}(X\circ Y)\), not an arbitrary Euclidean norm on all split coordinates. On the diagonal subalgebra it is the ordinary positive dot product.

The Gram matrix of the ordered pair is \[ G=\begin{pmatrix}B(J,J)&B(J,J^\#)\\B(J^\#,J)&B(J^\#,J^\#)\end{pmatrix}. \] The internal mismatch angle is defined on this positive pair-plane by \[ \cos^2\theta_{\rm mismatch} =\frac{B(J,J^\#)^2}{B(J,J)B(J^\#,J^\#)}. \] It has no physical angle label in this article.

For the elementary Peirce normalization check, take the real unit in the \(J_{12}\) off-diagonal block. For the second factor, stipulate the vector \((1,1,1)\) in a positive three-copy space. The latter is a specified normalization convention, **not a derivation of the number of physical generations**.