A metric-compatible positivity-preserving compact scheme for degenerate time–space fractional reaction–diffusion with variable delay

We develop a metric-compatible positivity-preserving compact scheme for a degenerate time–space fractional reaction–diffusion equation with variable delay. A fourth-order Riesz discretization and the second-order L 2-1 σ formula produce a candidate, which is projected onto the nonnegative cone in the inverse-diffusion metric. The same metric converts the nonlinear diffusion into a monotonicity product and transfers the candidate error to the accepted state with unit constant. We prove unique candidate solvability and exact nonnegativity of accepted states. Under explicit smoothness, zero-extension compatibility and causal step restrictions, complementary-kernel estimates yield an 𝑂 ⁢ ( 𝜏 2 + ℎ 4 ) error bound, with parameter dependence stated and no positive lower bound on the solution. A nonuniform Alikhanov implementation supports graded startup refinement, while the general nonsmooth convergence question remains separate from the smooth theorem. Numerical tests include analytical solutions, oscillatory and discontinuous delay reconstruction, 27 mesh–parameter contact configurations, a 576-case activation scan, and a published direct implicit baseline on a shared model. They distinguish correction distance from solution-error amplification: metric projection is nonexpansive for a fixed candidate, but is not uniformly more accurate than clipping over separately advanced trajectories. Dual active-set and full-solver benchmarks also separate inversion-free projection and fast operator actions from end-to-end runtime. These results delimit the accuracy, admissibility and computational scope of the proposed metric-compatible construction.