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.

The mathematical development in this paper was produced in dialogue with Claude.ai (Anthropic) in Spring 2026, directed by the author. Use of AI assistance is acknowledged in accordance with standard scholarly practice.