It is not that three rings changes bend radius. As stated earlier, bend radius is established at either end. One ring is staright forward. Two rings will “equalize” and end up side by side. When you add three or more rings the middle ring(s) have to offset or disspace a bit and that changes the entry and/ exit angle just a touch, therefore altering load.
Think of a multi block system in a tree canopy where seperate blocks share the load with rope angles trending more obtuse than acute. That is the concept, just in a much smaller space.
Don’t go crazy. There are diminishing returns as you add rings and ultimately load must be transfered and the first law of thermodynamics appliies to all rigging in this universe. As for splicing prowness? I say game on!
The original reccommendation of three rings over one or two is quite simple. In the original “extream” breaking tests (as in lifting lagre pieces and dropping them till failure on purpose ((described to me years ago by David)) two or less rings parted the rope the riggning point. Three rings and the rope parted at the piece. Pick your poison.
Tony