I am applying for Fall 2018 for Ph.D. Programs in Physics. I came across a paper in my late sophomore year which I found extremely interesting so I wrote a mail to the author expressing my thoughts about his paper which included a slightly new interpretation of one key constant appearing in his paper. He was very glad to read my mail and responded quite enthusiastically--further suggesting that I should send my note to a Journal. I didn't send it to any Journal because I thought it was too unimportant to be separately published.
Then, coincidentally, I met with the guy when he visited my University to deliver a guest-talk, in my early final year. Since I knew he was coming, I had been playing around with the calculations and arguments related to his paper and I found a newer way to derive what he had derived in his paper using fewer axioms. So, I met with the guy and told him about my results. This time also he was very enthusiastic and glad to meet me. He again strongly suggested communicating my results to a Journal. We then had many discussions over email regarding my new calculations and his suggestions. We recently found an even more simpler proof of the same results and I am in the process of submitting a paper on the same to a Journal. I have also been in contact with the guy regarding several versions of the draft for this paper.
Thus, I thought it would be nice if he writes me a reference letter indicating what he liked about my thinking and arguments, and how important he thinks are my results. Moreover, it would show that I like to explore papers out of my interest and work on them without any academic obligations. But when I wrote to him about it, he told me that he would write me a reference letter if I want him to but he would not be able to write anything quantitatively (to quote, "for example, you were in top 10% of my class or something like that") and thus, he doubts whether the LoR would be really helpful. I think that even a qualitative LoR will do good, thus, I am thinking of asking him to write an LoR anyway.
So, before I do that, I would like to know whether such an LoR would do any good or not? And worse, can it somehow backfire? Also, I would like the answer to consider both the scenario -- this being my fourth LoR and this being one of my three LoRs.