Local Taylor-Jet estimation of a scale-dependent turing instability boundary in the Gierer–Meinhardt system

Abstract Classical Turing analysis locates diffusion-driven instability boundaries directly when coefficient functions are known. We ask a narrower question: how accurately can such a boundary be extrapolated from a finite Taylor jet estimated over a narrow calibration window, without evaluating the system elsewhere? Using the standard two-species criterion, we define a scale-dependent Turing margin and estimate its first zero from local Taylor data. The construction requires positive, sufficiently differentiable measured diffusion coefficients on the calibration interval and excludes structural or transport transitions; extension beyond that interval additionally requires physical validity across the extrapolation bridge. For an analytic Gierer–Meinhardt benchmark, exact local derivatives recover the boundary rapidly, reaching sub-percent accuracy at fifth order. Finite-window fitted jets instead show an order–noise tradeoff: higher retained order becomes harmful when derivative estimates are noise dominated, and reliable recovery at the benchmark extrapolation ratio requires stringent measurement precision. The Taylor-coefficient algebra is standard; the contribution is the quantified accuracy, failure behaviour and applicability limits of local boundary extrapolation from finite-window measurements.