Finite-Scale Instability in Growth-Mediated Coordination Systems: A Stochastic Renormalization Group Approach

This work builds on a companion study in which the infrared fixed point governing the geometry of growth-mediated coordination systems was derived from percolation universality, yielding a stable coordination exponent η∗ ≈ 2.5.
Many complex systems that sustain high-throughput transport—including turbulent fluids, biological vasculature, and large-scale coordination networks—exhibit a characteristic pattern: extended scale-invariant operation followed by breakdown at a finite control parameter. While such transitions are well documented, their origin is typically attributed to system-specific mechanisms or empirically determined thresholds. In a companion work, we identified a universal infrared fixed point governing the geometry of growth-mediated coordination in three dimensions, characterized by a Hausdorff exponent η∗ ≈ 2.5. Here we address a more general question: why must systems in this universality class destabilize at finite scale at all, and why is the location of that instability non-universal?

We show that finite-scale breakdown is a generic consequence of stochastic renormalization-group flow near a stable infrared fixed point. In the presence of multiplicative fluctuations whose amplitude grows with throughput, irrelevant operators induce eventual escape from the basin of attraction despite deterministic stability. The resulting transition is inevitable but system-dependent, with its location exponentially sensitive to non-universal features such as geometry, boundary conditions, and noise amplitude. This framework explains the emergence of large but variable critical control parameters without invoking substrate-specific equations of motion, recasting Navier--Stokes turbulence as one realization of a broader universality class of growth-mediated coordination systems. 

The framework derives the IR fixed point η∗ and the functional form of finite-scale corrections, but does not predict a universal critical control parameter for specific geometries; transition thresholds in particular systems depend on non-universal parameters not fixed by universality class geometry alone. As a principal application, the framework yields a first-principles derivation of the von K´arm´an constant κ∞ = 1/η∗ = 0.39636 with a falsifiable prediction κ → 0.397 ± 0.001 at Re ∼ 108, and a proposed identification—that the Hausdorff dimension of the intense-vorticity set equals η∗ = 2.52299—relevant to current sparseness-program approaches to the Navier–Stokes regularity problem.