Emergent Distance and Metricity of Mutual Information in 1D Quantum Chains

We develop and formalize a phase diagnostic based on the information-distance $d_E = K_0/\sqrt{I}$ (mutual information $I$) for 1D quantum chains. Calibrating with the Euclidean benchmark $I(r)\propto r^{-2}\mapsto d_E(r)\propto r$ 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 $I(r)\sim r^{-X}$ with $0<X\le 2$ 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 $\Delta(r,r)=d_E(2r)-2d_E(r)$ is asymptotically non-positive; in gapped regimes $I(r)$ decays exponentially and $\Delta(r,r)\gg 0$. The result is a practical, falsifiable large-scale diagnostic based solely on site-averaged two-site mutual information.