The cut property is the certificate behind the MST. For each chosen tree edge, removing it creates a fundamental cut, and that edge must be the cheapest crossing edge.
highlighted = computed this step
Cut certificate
There is one cut certificate for each MST edge, so this instance has 5 certificates. Motivation: removing a tree edge creates the exact split that can justify putting it back.
certificates=5
C-E cut
For edge C-E, the recomputed tree edge weight is 5. Why: it is the minimum edge crossing that fundamental cut, so this edge has a local certificate.
w(C−E)=5
B-C cut
For edge B-C, the recomputed tree edge weight is 2. Interpretation: the same cut-property check works on a different split, which makes the certificate repeatable rather than decorative.
w(B−C)=2
Diagram note
Certificate note: the cut diagrams show side S and the opposite side by role, with crossing edges certified from the recompute. Pixel positions are rounded for layout; every number shown is exact.