It's too obvious that the Cahill family is in the Madrigal branch! Just have to add up all the strongholds they went to!
Book 2: One False Note[]
This was a Janus stronghold. Jonah Wizard said that Amy & Dan were intruders, he knew they're not itn the Janus branch.
Book 4: Beyond The Grave[]
This is an Ekat stronghold. Bae Oh came up to Amy & Dan, and used surcerity measures on them, then Nellie came and Saladin saved them by licking hummus. Go Saladin!!! They figured out that they weren't Ekaterinas.
Book 5: The Black Circle[]
It was an Lucian stronghold, but Irina Spasky wouldn't let them get the clue, and the NRR, she was able to prove that Amy and Dan weren't Lucians.
Book 7: The Viper's Nest[]
This is a possable Tomas stronghold. Mr. What-cha-ma-call-et also called surcrity mesures on them, so they aren't Tomas.
So all of this information proves that there is one more branch that Amy & Dan could possably belong to - and they do - the MADRIGALS!!!