Low-order CR--RT equilibrated-flux certification for semilinear problems on anisotropic meshes

We develop a low-order Crouzeix--Raviart--Raviart--Thomas (CR--RT) equilibrated-flux certification workflow for finite element approximations of semilinear diffusion--reaction problems, with particular emphasis on anisotropic mesh settings. Given a computed conforming finite element state $\tilde u_h$, the certification process is reduced to three computable quantities required by a Newton--Kantorovich argument: a dual-norm residual bound, a stability constant for the Fr\'echet derivative, and a Lipschitz bound for the derivative in a neighborhood of $\tilde u_h$. These components yield an explicit radius $\rho>0$, ensuring that the exact solution exists locally and uniquely within the ball $B(\tilde u_h,\rho)\subset V$. The residual bound is obtained from an $H(\mathrm{div})$-conforming $\mathbb{RT}^0$ certificate flux reconstructed through a Marini-type CR--RT route. The purpose of this route is not to replace general higher-order or local mixed equilibrated reconstructions, but to provide an explicit low-order construction whose algebraic structure is transparent on anisotropic simplicial meshes. Within the certified neighborhood, we further enclose selected quantities of interest $\mathcal J(u)$; the baseline enclosure follows from the verified inclusion, while an adjoint-based correction sharpens the resulting intervals. The numerical experiments report the behavior of the computable certification quantities for monotone semilinear models, including anisotropic mesh tests. Unless interval or outward-rounded scalar post-processing is explicitly used, the reported computations should be understood as floating-point evaluations of the derived rigorous estimators.