Ascending chain condition for four-dimensional log canonical thresholds

Prove the ascending chain condition for log canonical thresholds in dimension four, thereby ruling out infinite strictly increasing sequences of log canonical thresholds arising in the four-dimensional construction of curves in negative extremal rays.

Background

The proof constructs successive birational models by increasing suitable log canonical thresholds. Termination of that process would follow directly from the ascending chain condition in dimension four. The paper avoids relying on this unresolved statement by using the known ascending chain condition at codimension-three points, since the relevant thresholds can be localized to three-dimensional singularities. Thus the four-dimensional ACC remains an explicitly identified unresolved issue even though it is not needed for the conditional cone theorem established in the paper.