CH gives you recurrence on the matrix. You want recurrence on an individual element (indexed by [start][end]).
Any recurrence that holds on the matrix also holds on each individual element (and vice versa, in that a recurrence holds on the matrix just in case it holds on every individual element).
Any recurrence that holds on the matrix also holds on each individual element (and vice versa, in that a recurrence holds on the matrix just in case it holds on every individual element).