See if this answers your question. With siblings, you are labeling Regions of DNA for each Sibling, Half-Identical Regions (HIR) or Full Identical Regions (FIR). The following is only to answer your question.
The first step is to take all your known ½ cousins and beyond relatives who have been auDNA tested and record the HIR, segment information, in a spreadsheet. Add two Columns Paternal and Maternal.
For each HIR, enter into only one of the columns, either Grandfather or Grandmother, leaving the other column blank.
For each combination of siblings, comparing each known DNA cousin, those segments that DO NOT TRIANGULATE, since there is no shared segment (no overlap), you will need to add a line into one of the siblings and enter the alternative path. If Grandfather is one sibling, you will add grandmother into the other.
But there are things you could/should do to increase your confidence.
1. When you add a line, identify the line as predicted.
2. When comparing siblings, about 25% of the DNA do not share either paternal or maternal grandparent. For those Regions, there should not be the same paternal or maternal grandparent.
3. Basically, look for inconsistencies.
Triangulation is used when you don’t know how you are related. So now add those unknown DNA Cousins with segments.
When you have 4 siblings, for most matches, you should not have to rely on predicted segments.
Generally speaking, we begin with the 4 possible paths via grandparents we are attempting to narrow down the path to 2 out of the 4.
You are attempting to create a decision tree based on a solid foundation.
The next step would be to identify within each of these 4 paths, the next level. For each grandparent, what HIR is from the great-grandmother and what is from the great-grandfather.
There is no need to triangulate.
I would suggest that you try Lazarus.
For a single known ½ first cousin or beyond, enter that person into group b, and the 4 siblings into group A. experiment with the size of the segments to see if it really makes a difference. You will see the segments that are used from each sibling to create the kit, and other information that may be educational.
In your original post, you said that you could identify the match as paternal, How, without also knowing if the match is via the paternal grandfather or grandmother.
Are you relying on yDNA?