The fourth observational note in the square-quantity-readout series (pure algebra / non-physical). It connects the conjugate complex norm ZZ-bar = x² + y² to the square-quantity readout Xᵢ = xᵢ², generalizes to an equal-footing multi-dimensional positive-definite sum of squares (sumⱼ Zⱼ Z-barⱼ = sumₙ xₙ² = R²), and observes a zero-sum-of-squares form obtained by adding the square (i zₖ) ² of imaginary-valued coordinates to this conjugate-norm backbone: sumⱼ Zⱼ Z-barⱼ + sumₖ (i zₖ) ² = 0, hence sumₙ xₙ² = sumₖ zₖ². The key point is that the conjugate product X X-bar and the non-conjugate square X² agree for real coordinates but diverge only for imaginary ones (Xₕ X-barₕ = zₕ² vs Xₕ² = -zₕ²) ; the minus sign arises solely from this divergence via i² = -1, not from a hand-placed metric signature. The note explicitly does NOT connect to physics: no time coordinate, no metric signature, no light cone / Minkowski / special or general relativity; coordinate labels are not identified with physical dimensions. Japanese and English drafts.
Noriaki Kihara (Thu,) studied this question.