Paper 18: Multi-Scale Turing Pattern Formation via CNRS-H Digit Strings: Cross-Scale Coupling in the Gierer-Meinhardt System

We extend the CNRS-H multi-scale framework to nonlinear reaction-diffusion systems, demonstrating the approach on the Gierer–Meinhardt activator-inhibitor model across three nested biological scales (cell, tissue, organ). The CNRS-H framework represents smooth scale-dependent functions as Taylor-coefficient sequences; the key structural properties—digit-shift differentiation and Cauchy convolution multiplication—are derived self-containedly in Appendix A. Applied to nonlinear reaction-diffusion systems, Cauchy convolution of scale-dependent fields generates all cross-scale interaction terms automatically and exactly, allowing the full nonlinear system to be evolved at all scales simultaneously without linearisation of the scale dependence. We derive scale-dependent Turing conditions—criteria for pattern formation depending on the scale position s and its derivative ∂a/∂s, which are invisible to classical single-scale analysis. A numerical demonstration shows that Turing instability is active at the cell and sub-cellular scale (s < sexit ≈ 0.52 nats, ℓ < 17 µm) and becomes extinct at larger scales, with the extinction scale controlled by the cross-scale gradient encoded in digit[1] of the activator string. The scale gradient of the activator—a quantity with no classical analogue—emerges as a natural dynamical variable governing pattern formation across biological hierarchies, and is in principle accessible from scale-resolved transcriptomic data.