A Dimensionless Control Parameter for Field Instability vs Stress Localisation in Passive Materials

\section*{Description} This work formalises a dimensionless control parameter ($\Xi$) that separatesdistributed field bifurcation from stress localisation in bounded passivematerials subjected to sustained competing gradients. Building on adversarialtesting across biological, geological, granular, and engineered systems, theframework contracts the aperture-field hypothesis to a specific material class:passive or weakly adaptive load-bearing media in which stress accumulatesspatially and redistribution capacity is finite. The governing parameter is defined as: \begin{equation}\Xi = \left(\frac{\sigma_{\mathrm{th}}}{\sigma_{t}}\right) \left(\frac{\tau_{r}}{\tau_{f}}\right)\end{equation} where $\sigma_{\mathrm{th}}$ represents the thermally induced stress amplitude,$\sigma_{t}$ the effective tensile strength, $\tau_{r}$ the material relaxationtimescale, and $\tau_{f}$ the forcing timescale. Three regimes emerge: \begin{align}\Xi \ll 1 &\quad \Rightarrow \quad \text{Smooth redistribution (no persistent aperture geometry)} \\\Xi \sim 1 &\quad \Rightarrow \quad \text{Distributed field instability (stable polygonal/cellular structures)} \\\Xi \gg 1 &\quad \Rightarrow \quad \text{Runaway localisation (crack propagation or collapse)}\end{align} This framework unifies phenomena including ice-wedge polygons, columnar basalt,convective cells, and selected planetary surface morphologies under a commoninstability window, while explicitly excluding actively regulated biologicalsystems and crack-dominated engineered materials. The result is ascale-independent classification principle distinguishing field-instability-dominated systems from stress-localisation-dominated systems. The formulation generates falsifiable predictions regarding transitions betweensmooth deformation, patterned bifurcation, and collapse across cryosphere,lithosphere, granular, and planetary domains. \section*{Keywords} field instability \\stress localisation \\dimensionless control parameter \\thermal cracking \\ice-wedge polygons \\material bifurcation \\distributed dissipation \\passive material systems \\instability window \\fracture mechanics transition \\relaxation timescale \\competing gradients \\pattern formation \\regime classification \\aperture-field geometry