Paper 18: Scale-Dependent Turing Pattern Formation via CNRS-H Digit Strings in the Gierer–Meinhardt System

Classical Turing analysis of the Gierer–Meinhardt activator-inhibitor model answers whether pattern formation occurs at a single, chosen length scale; it does not, on its own, say up to which scale a pattern persists as the underlying transport parameters change continuously across a biological hierarchy (cell, tissue, organ). We call this the scale-extinction problem. We formulate it as a local hierarchy-jet problem, solve it exactly for a controlled analytic benchmark, and quantify how accurately the first boundary crossing can be extrapolated from a finite calibration window, by extending the CNRS-H multi-scale framework – which represents smooth scale-dependent functions as Taylor-coefficient (”digit”) sequences – to nonlinear reaction-diffusion systems. The two aims are one construction: applying CNRS-H’s structural properties (digit-shift differentiation and Cauchy convolution multiplication, derived self-containedly in Appendix A) to the Gierer–Meinhardt system across a continuous range of biological length scales both extends the framework to a nonlinear setting and yields a closed-form scale-dependent Turing condition. We define a scale-dependent Turing margin M(s) – the natural two-component specialisation of the general diffusion-driven-instability condition of Villar-Sepulveda and Champneys (2023) – and show, for the analytic diffusion profiles used in the illustrative example below, that successive orders of its CNRS-H jet, centred at a single reference scale s0, recover the scale sexit at which Turing instability ceases with rapidly decreasing error, using purely local derivative data and without knowledge of the diffusion coefficients away from s0. For one illustrative choice of scale-dependent diffusion profiles and a fully self-consistent Gierer–Meinhardt parameter set — an illustrative pair of unequal effective-transport attenuation profiles — Turing instability is active at and below the reference cellular scale and becomes extinct above a scale-profile-specific threshold (sexit ≈0.538 nats, ℓ≈17 µm for this profile; the numerical value is a property of the chosen illustration, not a predicted biological constant).