Abstract
<title>Abstract</title> <p> A confining, non-Abelian gauge connection Uij ∈SU(N) defined on the links of a dynamical graph has been proposed as the microscopic origin of classical spacetime geometry. The metric is reconstructed from the coset (soldering-form) sector of an enlarged connection, and black-hole and boundary-cut entropy are obtained as the logarithm of a combinatorial microstate count, matched—via the Freedman–Headrick theorem—to the bit-thread (max-flow/min-cut) formulation of holographic entanglement entropy. That matching is exact as a statement about counting, but it is not, by itself, a statement about the entanglement entropy of any specific quantum state: nothing in the combinatorial construction distinguishes a classical ensemble of flux occupation numbers from a genuine quantum wavefunction, so its connection to the entanglement/tensor-network program of emergent geometry remains an analogy rather than a derivation. This paper closes that gap. We first restate the flux-gauge connection ensemble in full—its partition function, its block-link renormalization-group coarse-graining with closed-form Migdal–Kadanoff coefficients, its emergent metric and Newton’s-constant identification, and its combinatorial boundary-cut entropy—so that the paper is self-contained. We then show, by an exact character-expansion/Peter–Weyl derivation standard in lattice gauge theory but not previously applied to this construction, that the SU(N) link ensemble is a tensor network: each flux-tube segment (link) carries a bond dimension equal to the dimension of the SU(N) representation it transports, and each flux-tube junction (node) carries a gauge-invariant intertwiner tensor, fixed by representation theory rather than assumed. Only a single, narrowly scoped statistical assumption—on the distribution of intertwiners within the multiplicity space that gauge invariance already guarantees—is needed to complete this exact, deterministic tensor network into a genuine random tensor network in the technical sense of Hayden–Qi–Kim–Preskill (HQKP). We further identify the renormalization-group layers as the isometric maps of a multi-scale entanglement renormalization ansatz (MERA), giving the emergent radial holographic direction a literal tensor-network origin rather than a scaling-dimension analogy. We then investigate, analytically and numerically, how the entanglement entropy of such a network reproduces the Ryu–Takayanagi/min-cut law. An exactly solvable limit built from classical (GHZ-type) perfect tensors reproduces the combinatorial law exactly for single-cut (boundary-anchored) regions but is shown to be degenerate for interior, two-sided regions, exposing a genuine gap in the combinatorial entropy argument: classical flux-occupation counting does not, by itself, certify multi-region (entanglement-wedge-level) quantum entanglement structure. We then construct a genuinely quantum (non-classical, Gaussian-random) tensor-network realization of the link ensemble and confirm numerically, on chains of up to sixteen sites, (i) a clean area-law plateau in which the entanglement entropy of an interior region becomes independent of its size once the region is large enough, in place of continued volume-law growth, and (ii) a linear scaling of the plateau entropy with the logarithm of the bond dimension, S(A) = κNcut(A) ln D, fit with R2 = 0.997 and an O(1) efficiency coefficient κ≈0.64 for this finite, non-isometric network. We interpret κ<1 as a genuine, quantifiable finite-network correction relative to the idealized D →∞isometric-projector limit in which HQKP prove κ →1, and we list the construction of that idealized limit, directly on the SU(N) link ensemble, as the key open numerical task. We close by proposing an explicit correspondence between the tensor-network minimal surface of this paper and the Cartan soldering-form metric of the gauge-connection construction, framed as a falsifiable consistency conjecture rather than an established result. <bold>PACS numbers:</bold> 04.60.Pp, 11.15.Ha, 03.67.Bg, 03.65.Ud </p>