In my recent WMCF presentation, I described the possibility that the harmonic series may provide a kind of hidden numerical scaffold beneath several larger pattern claims I have explored over the years. This paper steps away from interpretation and focuses only on the mathematics itself. It studies where the harmonic partial sums first cross evenly spaced levels and shows that these crossing points form an orderly sequence with predictable exponential growth. The goal here is to present the underlying harmonic framework in a clear, rigorous, standalone form so interested viewers can examine the mathematical core directly.
Paper can be read on this site, or downloaded for later reading