Phase Transition of Information-Thermodynamic Networks via Stochastic Topology and Ensemble Renormalization Group — Endogenous Emergence of Fractal Structures Driven by Thermal Fluctuations and Co-evolution of Cognitive Manifolds —

This paper extends the "Information-Thermodynamic Tensor Network (ITTN)," which describes the dynamics of macroscopic organizational collapse under information explosion, to construct an "Annealed Disorder" model where topology and individual cognitive manifolds co-evolve within a shared thermal bath. Eliminating the rigid-body network approximation from previous studies, we reformulate the system as an "entropic elastic body" that processes information uncertainty and goal biases. The input information (AI output tensors) is defined as a scale-free multivariate Gaussian measure, and microscopic dissipation heat is reduced to algebraic matrix traces of bias and variance. Furthermore, we present a novel Hamiltonian that integrates Stochastic Gradient Langevin Dynamics (SGLD) on Grassmann manifolds with differentiable topology. Through the process of the Tensor Renormalization Group (TRG), we analytically derive that nonlinear scaling of dimensionality ($\rho \in (0,1)$) and the formation of "3-to-4-person cliques" endogenously emerge from the cancellation of internal heat based on the Information Bottleneck principle. Thus, we mathematically prove that the fractal hierarchical structures formed by organizations to avoid cascading failure are pure thermodynamic attractors. Code: https://gist.github.com/Melnus/60bd6081ea6f5bd16289145003470f49