The symmetric V-cycle can diverge under the multigrid axioms for cell-centred discretisations

The axiomatic convergence theory for multigrid methods applied to cell-centred finite-difference and finite-volume discretisations rests on two hypotheses: an imbalanced Galerkin condition (G3), which states that $R_{\ell-1}A_\ell P_{\ell-1}=2A_{\ell-1}$ with $R_{\ell-1}=\frac{1}{2}P_{\ell-1}^T$, and a weak approximation property $(A2)_\alpha$ of Bramble type. Under these hypotheses, together with Richardson smoothing, the symmetric W-cycle and the variable V-cycle are known to be uniformly convergent, while the uniform convergence of the standard symmetric V-cycle has remained open. We answer this in the negative by two constructions. First, for every smoothing count $m$ we exhibit hierarchies of every depth satisfying (G3), Richardson admissibility with $C_R=1$, and $(A2)_\alpha$ for every $\alpha\in(0,1]$ with the sharp level-independent constant $C_{A2}^2=4m$, whose symmetric $V(m,m)$-cycle error operator has spectral radius $\theta_m(1+2\theta_m)>1$ already on three levels, where $\theta_m=(1-\frac{1}{4m})^{2m}$, and growing geometrically with the depth; the family shows that any smoothing-count threshold $m_0$ that could restore uniform V-cycle convergence must grow at least quadratically in $C_{A2}$. Second, we prove that the same failure occurs in a completely standard discretisation: the cell-centred finite-volume hierarchy for a one-dimensional diffusion equation with a mesh-aligned coefficient jump $1:\kappa$ and harmonic (Samarskii) interface averaging satisfies (G3) exactly and $(A2)_{1/2}$ with a level-independent constant $C_{A2}=O(\kappa)$, yet for every $\kappa\ge3$ its symmetric $V(1,1)$-cycle with any admissible Richardson parameter, including the optimal one, diverges geometrically in the number of levels. In both constructions the W-cycle remains uniformly contractive, so the hypotheses separate the two cycles. All claims are verified numerically.