Concerning the fractal nature of the Timewave: at first glance and for all practical purposes, the Timewave looks like a graph y = f(t), no fractal or iterative aspect is evident.
The question is whether the term ‘fractal’ is loosely used as a reference to its “density” in R^2 (i.e., if it were a graph proper of a function per se) or merely in terms of “weak” statistical self-similarity, i.e., assuming for now the definition of ‘fractal’ as that which entails recursion and self-similarity (on some scale).
The Quantum TW: Going beyond the TW as something like a mere R^2 graph with non-differentiable spikes here and there, we can informally view it as a strongly iterative fractal that is everywhere self-similar. “Telescoping” into any area of the TW would yield its own self-similar TW, and so on ad infinitum, or until we set the “time limit” as a unit of Planck time whose “vibration” is the TW itself. In this sense, every moment of time entails cosmogony and eschaton as each pulsating, discrete Planck unit of time entails incursion of absolute Novelty.
An aside: Informally viewing the TW as a quantum wave-function, the singularity f(t) = 0 implies its wave-function collapse. Speaking in terms of traditional metaphysics, it would be the actualization of pure potential, in this case the actualization of all history hitherto.