Abstract
<title>Abstract</title> <p>Spin networks, spin foams, and tensor networks are the three principal pre-geometric structures of background-independent quantum gravity, yet the dynamical input of the theory—the spin-foam vertex amplitude—is convention-ally fixed by simplicity constraints rather than derived from a physical principle. We propose an entropic selection principle: the physical spin-foam amplitude is a coarse-graining quantum channel on spin-network states along which a relative-entropy functional is exactly monotone, with dissipation vanishing precisely on the fixed-point sector. Three structural results are proved unconditionally: (i) monotonicity of the functional along any completely positive trace-preserving coarse-graining flow, by the data-processing inequality; (ii) an exact dissipation identity decomposing the total entropy drop into non-negative per-step productions; and (iii) vanishing dissipation of topological BF-type amplitudes, which act as conditional expectations onto their flat sector and are therefore the entropically reversible case. The selection principle recasts the question of which pre-geometric structure is fundamental in operational terms: spin foams are trajectories of an entropy flow on spin-network states, tensor networks are its representation, and fundamentality is stability of the flow’s attractors. As a fully controlled pilot test we realize the framework in a Z2 finite-group spin net, combining the Migdal–Kadanoff coarse-graining map with exact finite-volume enumeration of the Gibbs ensemble. The dissipation landscape σ(K) = D(pK ∥pR(K))/N is found to vanish, to machine precision, exactly at the three renormalization-group fixed points (K = 0, K∗≃ 0.6094, K → ∞) and to be strictly positive elsewhere, while the relative entropy to the attracting fixed point decreases monotonically along every flow tested, confirming the entropic classification of pre-geometric fixed points in this model. A proved/open status ledger delineates the unconditional theorems from the conjectural extension to SU(2)k spin nets and Lorentzian Engle–Pereira–Rovelli–Livine amplitudes.</p>