I had the intuition several years ago that bubbles might be an intrinsic property of the power law itself. One of my first attempts to model them was through a log-periodic framework—I knew it was connected to power laws, but I hadn’t explored that connection deeply enough at the time.

@moneryordebt later revived this line of thinking and pushed the log-periodic approach further.
Recently, I finally had the time to sit down and work through the connection more carefully.
If the hypothesis is correct—that bubbles are log-periodic—then both the long-term trend and the oscillatory bubble behavior can be unified within a single framework: a power law with a complex exponent.
The math is in the comments, but the result is truly elegant.