On the Connection Between the Conjugate Complex Norm and Square-Quantity Readout

The fourth observational note in the square-quantity-readout series (pure algebra / non-physical). It connects the conjugate complex norm ZZ-bar = x^2 + y^2 to the square-quantity readout X_i = x_i^2, generalizes to an equal-footing multi-dimensional positive-definite sum of squares (sum_j Z_j Z-bar_j = sum_n x_n^2 = R^2), and observes a zero-sum-of-squares form obtained by adding the square (i z_k)^2 of imaginary-valued coordinates to this conjugate-norm backbone: sum_j Z_j Z-bar_j + sum_k (i z_k)^2 = 0, hence sum_n x_n^2 = sum_k z_k^2. The key point is that the conjugate product X X-bar and the non-conjugate square X^2 agree for real coordinates but diverge only for imaginary ones (X_h X-bar_h = z_h^2 vs X_h^2 = -z_h^2); the minus sign arises solely from this divergence via i^2 = -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.