Auditing Random Tensor Networks: A Quantitative Anatomy of Bond-Dimension Geometry

The random-tensor-network route to holographic geometry reads a spacetime out of a graph whose edges carry bond dimensions: entanglement entropies are minimal cuts at leading order, the cut weights are the geometric areas, and the boundary theory inherits subregion duality and error correction. This paper is an audit of that route, in the spirit of the companion audits of entanglement equilibrium and of mutual-information metric reconstruction: every assumption beyond established mathematics is isolated as one of six explicit hypotheses T1 to T6 and each is proved, checked numerically, or recorded as input. The audit yields: (i) on trees the random-tensor entropy is exactly Page, the quantum correction to the geometric area law is bond-dimension independent and the geometric limit does not remove it, verified by Monte Carlo to 10^-4; (ii) the continuum limit of the min-cut law is a property of the region-graph alignment, not of the geometry, showing that the quasi-uniform families that carried previous audits do not support the random-tensor continuum limit at all; (iii) with uniform bond dimensions the boundary entropy scaling is geometry-blind, meaning the celebrated holographic profiles must be put into the bond-dimension function by hand and geometry is strictly an input of the model; (iv) single-boundary-site mutual information is a finite-size effect and vanishes with system size, showing that the random-tensor route's geometry lives in region-level cut homology in sharp contrast to single-site mutual information routes. The ledger records the status of every hypothesis.