I want to say:
This technique is based on an observation we made about/on the theorem of …“
Question: In the above sentence which one should I use, “about” or “on” or neither ?
To clarify the theorem has already exist but we made a new observation about/on the theorem.
Either is correct, and you can find current examples of both.
In my opinion, “observations on” is slightly more old-fashioned and formal. Here is a Google ngrams search showing the declining use of “observation on” and the increasing use of “observation about”: https://books.google.com/ngrams/graph?content=observation+on%2Cobservation+about&year_start=1800&year_end=2000&corpus=15&smoothing=3&share=&direct_url=t1%3B%2Cobservation%20on%3B%2Cc0%3B.t1%3B%2Cobservation%20about%3B%2Cc0