Auditing Mutual Information: A Quantitative Anatomy of Metric Reconstruction from Entanglement

The mutual-information route to emergent geometry reads a Riemannian metric off the entanglement of a network state: lengths from a logarithmic functional of pairwise mutual information, distances from shortest paths. This paper is an audit of that route, in the spirit of the companion audit of entanglement equilibrium: every assumption beyond established mathematics is isolated as one of six explicit hypotheses M1 to M6, and each is proved, checked numerically, or refuted on the same exactly solvable pair (two-qubit) sector, supplemented by two synthetic models (a power-law chain, and a quasi-uniform versus designed point-set comparison on S2). The audit yields: M1, the exponential decay law I ~ I* exp(-d/xi), is not a technicality but the load-bearing assumption: it is equivalent to additivity of the length functional, and for power-law mutual information the reconstruction on the complete edge set produces a logarithmic remetrization d_k(u,v) = a + b ln d_h(u,v), verified by exact all-pairs shortest paths - a qualitative failure that no conformal rescaling repairs. Metric convergence (M2), measured by exact Dijkstra reconstruction on S2, holds with rate sup(d_k - d_h) <= 7.3 x 10^-3 at N = 3600 for quasi-random (golden-spiral) sets, while a designed banded equal-area set is disconnected at N <= 900 (isolated polar bands) and inflates the error by a factor of approximately 12 - local isotropy, not density uniformity, controls reconstruction. The area law with finite G (M4) carries a systematic deficit epsilon in [-0.16, -0.02] on quasi-random sets, consistent with the quasi-uniform bound xi + delta/xi. A pre-registered control protocol attributing the deficit to density fluctuations is executed and refuted (the registered correlation has the wrong sign, r = -0.37), locating the leading systematic in the boundary discretization of the regions, exactly the alternative pre-registered in the companion paper. Curvature extraction (M5) is one-sidedly biased (d_k >= d_h always) and only marginally resolvable at accessible sizes. The audit locates the quantitative obstructions precisely: in the decay law (M1), in the edge convention, and in the boundary layer. The ledger records the status of every hypothesis.