The Stochastic Renormalization Group β-Function for Growth-Mediated Coordination Systems: Full Symbolic Derivation, Integration, and Emergence of the IR Fixed Point

We present the full symbolic derivation of the stochastic renormalization-group (RG) β-function for growth-mediated coordination systems. Starting from a deterministic RG flow that admits a stable infrared (IR) fixed point at the Hausdorff dimension η∗ ≈ 2.52299 of the three-dimensional percolation backbone, we incorporate multiplicative noise whose amplitude grows with throughput. We derive the associated Fokker–Planck equation, linearize about the fixed point, and obtain the exact mean escape time from the basin of attraction via the backward Kolmogorov equation. The resulting escape-time scaling ⟨ℓesc⟩ ∼expλ∆η2c /(σ2(Φ)η∗2) is obtained in closed form. We further construct the explicit mapping between RG time ℓ and physical time t, including the intermittency correction arising from the multiplicative noise. The derivation establishes that finite-scale instability is a generic, inevitable consequence of stochastic RG flow near a deterministically stable IR fixed point, while the location of the instability remains non-universal and exponentially sensitive to noise amplitude and geometry. The framework provides the rigorous mathematical foundation for the stochastic RG treatment of coordination breakdown in turbulence,biological transport networks, and large-scale socio-economic systems.