When queried the PQ will tell us what is the next vertex to attempt to delete. The error for each vertex is a function of its neighborhood. Each vertex is capable of recommitting its own error value if its neighborhood is in any way modified. Vertex local error is computed by finding the average plane through the neighborhood of the vertex and then calculating the the distance of the vertex to that average plane. Border vertices are treated differently in that the square root of the area of he border triangle is used as error.
Global error comes from the distribution of local error.
When a vertex is deleted its error is added to the the error of its neighboring vertices. Through this, the error is actually a global error bounds. The PQ is queried and the vertex index it returns will be attempted to be deleted from the mesh. With recursive loop splitting a proper triangulation is not guaranteed, if the triangulation fails the vertex will not be deleted at that time.