Abstract
We develop and formalize a phase diagnostic based on the information-distance d_E = K₀/√I (mutual information I) for 1D quantum chains. Calibrating with the Euclidean benchmark I(r)∝ r⁻²↦ d_E(r)∝ r makes the triangle-inequality test parameter-free and scale-invariant. Under site-averaged, monotone scaling conditions on the 1D line we establish a criterion linking the decay of I(r) to metric behavior of d_E(r): power laws I(r)∼ r^-X with 0<X≤ 2 yield subadditivity (metric scaling), while exponential clustering leads to superadditivity. As an analytic check complementing our earlier numerical study, we verify these predictions in the 1D transverse-field Ising chain using an exact Jordan-Wigner/Bogoliubov-de Gennes solution: at criticality I(r) follows a power law close to the X=2 benchmark and the equal-legs triangle defect Δ(r,r)=d_E(2r)-2d_E(r) is asymptotically non-positive; in gapped regimes I(r) decays exponentially and Δ(r,r)≫ 0. The result is a practical, falsifiable large-scale diagnostic based solely on site-averaged two-site mutual information.