The Mutual–Information Jewel Metric

This document is the full manuscript draft (Version 0.9, dated May 31, 2025) corresponding to the Priority Notice abstract publicly registered on Zenodo. This manuscript's SHA-256 hash is 01c35bce761471261d09ddf522f9467fc1d28dbc556ce13e9587f1ed039f25b9, as stated in the Priority Notice. The manuscript introduces the "MI/Distance Jewel," an emergent distance metric $d_E = K_0/\sqrt{I}$ derived from quantum mutual information. It establishes the "Jewel Theorem": if mutual information scales as $I(r) \propto 1/r^X$ with $0<X\le2$, then $d_E$ is a valid metric. The special case $X=2$, yielding Euclidean geometry $d_E \propto r$, is discussed and numerically benchmarked using DMRG simulations on the XXZ spin chain. This draft (v0.9) is shared for the primary purpose of substantiating the timestamped priority claim for the concepts, derivations, and results contained herein.