HardyZInnerProductSpace

In the real-valued, square-integrable function space $L^2(\mathbb{R})$, the inner product between two random variables $X$ and $Y$ can be defined via the covariance function $C(h) = J_0(h)$:

$$ < X, Y > = E[(X - E[X]) * (Y - E[Y])] $$

This inner product essentially represents the covariance between $X$ and $Y$. In a stationary process, this covariance depends only on the distance $h$, which is the distance between the points in time at which $X$ and $Y$ are observed.

The choice of the Bessel function of the first kind of order zero, $J_0(h)$, as the covariance function comes from the ansatz that this function accurately represents the structure of the Hardy $Z$ function. This is not based on any empirical adjustment but arises from the intrinsic structural properties of the Hardy $Z$ function.

Then, we define a variance structure function $\gamma(h) = 1 - J_0(h)$, which is related to the covariance function through the equation $\gamma(h) = C(0) - C(h)$. With this, the covariance function $C(h)$ can be written as $C(h) = 1 - \gamma(h)$.

Consequently, the inner product between two random variables $X = Z(t)$ and $Y = Z(s)$ can be expressed as:

$$ < X, Y > = 1 - \gamma(h) $$

where $h = |t - s|$, and ${Z(t)}$ is a zero-mean, stationary Gaussian process. This definition of the inner product incorporates the variance structure function of the process, as it was derived from the properties of the Hardy $Z$ function.

What the inner-product measures

In this context, the inner product <Z(t), Z(s)> = 1 - γ(|t - s|) essentially measures the "dissimilarity" between values of the Hardy Z function at different points t and s.

This dissimilarity is framed in terms of the variance structure function γ(h) = 1 - J_0(h), which we've linked to the Bessel function of the first kind of order zero J_0(h). In a similar vein, if we were to apply this inner product to another function, it could be interpreted as measuring "dissimilarity" in a similar way to the Hardy Z function.

For a stationary process, this means that if the function's values at two points are highly correlated (i.e., they tend to rise and fall together), the inner product would be closer to 1. Conversely, if the function's values at two points are less correlated or negatively correlated, the inner product would be less than 1 or even negative.

So, this interpretation aligns with the idea that the inner product is measuring "dissimilarity" in terms of the variance structure function γ(h), which itself is a measure of "dissimilarity" or variability.

Dissimilarity is more aptly understood as novelty

The term "novelty" could be used to describe the concept of dissimilarity in this context. In a sense, the greater the dissimilarity between two points in the process, the more "novel" or different one is from the other. This aligns with the common usage of the term novelty to denote something new or unusual. Therefore, the inner product can be interpreted as a measure of novelty within the process, particularly when applied to the Hardy Z function.

Terence McKenna's Wonderfully Wacky TimeWaveZero aka "Novelty Theory" Idea

Terence McKenna, a famous ethnobotanist, mystic, psychonaut, lecturer, and author, spoke quite a bit about the concept of novelty. He proposed a theory, known as "TimeWave Zero" or "Novelty Theory", which suggests that time is a series of distinct stages which involve the ebb and flow of novelty - novelty, in this context, being defined as increase over time in the universe's interconnectedness, or organized complexity.

According to McKenna's theory, as we approach the temporal end point, or "eschaton", novelty increases exponentially. He associated this with the concept of a "singularity of infinite complexity", where everything that will occur has already happened in some sense. However, it should be noted that this theory has been widely critiqued within the scientific community due to its lack of rigorous mathematical basis and its reliance on subjective interpretations.

Regardless of its scientific merits or lack thereof, McKenna's Novelty Theory is an interesting philosophical idea that challenges our typical ideas of time and the universe.