Paper 18: Local Taylor-Jet Estimation of a Scale-Dependent Turing Instability Boundary in the Gierer–Meinhardt System

We study a local extrapolation problem for reaction–diffusion systems whose measured coefficients vary smoothly with physical logarithmic scale. Following the normalized Scale Space convention, the scale coordinate is u = ln(ℓ/ℓ0); it is a physical logarithmic length coordinate and is not identified with a biological organizational level. In experimental use, ℓ must be a preparation-level characteristic length defined by one fixed measurement protocol across the family. The present paper does not introduce reaction–diffusion transport through the full four-dimensional Scale Space manifold. Instead it studies a one-parameter family {Ru} of ordinary spatial reaction–diffusion systems and asks how far a diffusion-driven linear instability boundary can be inferred from a finite local Taylor jet measured near one reference scale, with calibration regularity stated separately from the bridge-interval conditions needed to interpret or certify an out-of-window extrapolation. For a two-component system with scale-dependent diffusion matrix D(u) = diag(Da(u),Dh(u)) and reaction Jacobian J(u), we give the pointwise continuous wavenumber Turing margin M(u) = [B(u)^2 / 4Da(u)Dh(u)] −detJ(u), B(u) = Dh(u)Jaa(u) + Da(u)Jhh(u), and show that, when the homogeneous state is stable without diffusion, B(u) > 0 and M(u) > 0 are the corresponding two-species diffusion-driven instability conditions. This generalizes the fixed-reaction-Jacobian result of the previous version. The local Taylor coefficients are organized using the CNRS-H notation; their arithmetic is the standard Taylor-coefficient algebra, and the new contribution is the use of those local jets to estimate a non-local instability crossing. The previously validated Gierer–Meinhardt benchmark is retained as the fixed J special case. With Da(u) = 0.010e^(−u/3) and Dh(u) = e^(−2u), the first continuous mode instability exit is uexit = 0.537908, corresponding to ℓ/ℓ0 = euexit ≃ 1.712. Exact pointwise jets converge rapidly, while fitted jets exhibit an order–noise tradeoff: higher retained order becomes harmful when derivative estimates are noise dominated. In addition, because the benchmark margin is analytic, an eighth order Taylor enclosure with explicit Lagrange remainders certifies the crossing in 0.5379 < uexit < 0.5380 using local expansion data about u0 = 0. We then separate this continuous-wavenumber boundary from the finite-domain criterion, where the admissible Laplacian spectrum, domain size and bound ary conditions matter. The finite-domain growth rate Λ(u) = maxn αsp[J(u) − κn(u)^2 D(u)] can switch dominant modal branches and can fail to be differentiable at a Turing–Turing crossing even when each branch is smooth. Finally, we distinguish loss of linear instability from disappearance of nonlinear patterned states: subcritical branches, localized states, mode interactions and boundary-condition dependent bifurcation structure may persist beyond a linear crossing. Thus Paper 18 is a local estimator for a precisely defined linear stability boundary, not a general pattern-extinction theorem.