Capacity Stability of Complex Monge-Ampère Equations with Moving Prescribed Singularities

For complex Monge-Amp\`ere equations with moving big cohomology classes and prescribed model singularities of positive Monge-Amp\`ere mass, we prove that, under total variation convergence of the right-hand side non-pluripolar positive Radon measures, convergence of the prescribed model potentials in Monge-Amp\`ere capacity is equivalent to convergence in capacity of the associated normalized solutions. We further prove that the ceiling operator coincides with the singularity envelope for potentials associated to a big $(1,1)$-class, regardless of their Monge-Amp\`ere mass, thereby resolving a conjecture of Darvas-Di Nezza-Lu. Consequently, the singularity envelope is idempotent without the positivity assumption on the mass.