i have a new theory for aru sekai fans & by new i mean someone else might have caught on already but i havent seen it said SO. the rute furutewoa melody is in kyuuyaku, touhikou, & now kannagi (maybe elsewhere bc i havent been looking for this motif before i only realized it today really & i havent had time to go thru all the songs again yet) & we have kyuu (past), something in touhikou im too tired to remember currently but i remember calling something similar a while ago when someone asked if there was a timeline to this, and now kannagi that uses the older language so im willing to bet this motif is a way of saying the events in the song happened in the "past". whether thats actually Long Ago or just to say its not the current situation or just happened before the songs without it i cant say for sure but i think it makes sense given what we have now.

just like we have the nami no ne no motif that signals which characters still have their "self" if its in the song or lost it if its not.

i have to do some more digging of course but with what i know right now this makes the most sense to me