Abstract
<title>Abstract</title> <p>We propose and test a background-independent effective statistical model in which classi- cal spacetime geometry and Einstein gravity emerge from the coarse-grained connectivity of a confining flux-tube network. The microscopic substrate is a dynamical graph of integer-valued flux occupation variables on the links of a lattice, motivated by the confinement of non-Abelian gauge theories and the area-law behaviour of Wilson loops. We show that the network admits a percolation phase structure and that a block-link renormalization-group (RG) transformation flows the system to a nontrivial interacting fixed point at which a scale-invariant connectivity field ρ(x) emerges. Promoting the emergent metric to a conformal functional of ρand integrating out fast connectivity fluctuations 'a la Sakharov, we obtain an induced Einstein–Hilbert action whose Newton constant is fixed by the connectivity stiffness of the network. The two-point function of ρat the fixed point yields a massless spin-two propagating mode, and the diffeomor- phism group emerges as the continuum limit of graph-relabelling symmetry. Black-hole entropy is recovered as the logarithm of the number of flux links severed by a minimal connectivity cut, giving an area law whose coefficient we identify with the graph form of the Freedman– Headrick max-flow/min-cut construction. We report first-principles Monte Carlo tests of the link ensemble on lattices up to L= 96: a sharp percolation transition consistent with the known three-dimensional threshold pc ≃0.3116, and a boundary-cut entropy that obeys an area law to R2 ≃0.9996. We state every logical step as a Postulate, a Definition, or a Result, and we delib- erately restrict this paper’s claims to what is analytically controlled or numerically confirmed, leaving more speculative extensions of the framework (a candidate mechanism for Yang–Mills gauge symmetry, a string-theoretic embedding, and a cosmological inflation scenario) outside its scope.</p>