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 key advance over the linear diffusion case is the treatment of polynomial nonlinearities: in the CNRS-H framework, multiplication of scale-dependent
fields is performed by Cauchy convolution of their digit strings, generating all cross-scale interaction terms automatically and exactly. This allows the full nonlinear system to be evolved at all scales simultaneously without linearisation, homogenisation, or explicit coupling assumptions.

We derive scale-dependent Turing conditions — criteria for pattern formation that depend 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 can be present at one biological scale and absent at another, with the onset 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.