C04.1 · Definitions/conventions

For positive \(a,b,c\) with \(abc=1\), set \[ J=\operatorname{diag}(a,b,c),\quad J^\#=J^{-1},\quad U=a^2+b^2+c^2,\quad V=a^{-2}+b^{-2}+c^{-2}. \] The trace pairing gives \(T(J,J^\#)=3\), so the pair has Gram determinant \(UV-9\). DET-7 is therefore the equation \(UV=16\), not a consequence of unit determinant.

Route A adds equality of the unordered spectra of \(J\) and \(J^\#\). Route B instead requires \(a,b,c\) to be positive algebraic integers in \(\mathbb Q(\sqrt5)\). Here “positive” means positive in the chosen real embedding, not positive at both real embeddings. The positive golden unit \(\phi=(1+\sqrt5)/2\) has a negative conjugate; replacing positivity by total positivity would change the admissible class.