Capacity convergence of model singularities is equivalent to capacity convergence of normalized Monge-Ampère solutions, and the ceiling operator equals the singularity envelope for any big class.

Summary. The manuscript studies complex Monge–Ampère equations with moving big cohomology classes and prescribed model singularities of positive Monge–Ampère mass. Under total-variation convergence of the right-hand side non-pluripolar positive Radon measures, it claims that convergence of the prescribed model potentials in Monge–Ampère capacity is equivalent to convergence in capacity of the associated normalized solutions. Separately, it claims that the ceiling operator coincides with the singularity envelope for potentials associated to a big (1,1)-class, without any positivity restriction on Monge–Ampère mass, thereby resolving a conjecture of Darvas–Di Nezza–Lu and implying that the singularity envelope is idempotent without that positivity assumption.

Significance. If established, the capacity-stability equivalence would be a useful continuity tool for complex Monge–Ampère equations with moving singularities and cohomology classes. The identification of the ceiling operator with the singularity envelope without a mass-positivity hypothesis would resolve a named conjecture of Darvas–Di Nezza–Lu and clarify the algebraic structure of singularity envelopes (in particular their idempotence). These are standard, high-value targets in pluripotential theory; the abstract presents them as pure existence/stability and operator-identity statements against external benchmarks, with no parameter fitting.