Problem 24.9.3E8E7FDD4A1BF

Problem 24.9.3.E8E7FDD4A1BF Let $X$ fail the bounded mass property, and let $\theta $ be a smooth big background on $X$. Find a condition on a compatible Hermitian envelope system over $\theta $ (definition 16.29.6) that forces recovery of its ideal type at every finite parameter, and that reduces to the criterion of theorem 13.21.7, a positive limiting lower volume of the initial slices, when bounded mass holds. Specify the invariant and the uniform estimates that replace the finite deficit of upper volume, and describe their behavior under modifications.