Interference-like density striations in a superposition of shifted prime-counting step functions — is there a physical interpretation?

I'm looking at the family of step functions $\Phi_p(x) = \pi(2x - p)$, where $\pi$ is the prime-counting function and $p$ ranges over the primes. When I superimpose these functions for many values of $p$ (see attached figure, $n$ up to $5 \times 10^7$), the result is a dense region with internal striations, bounded above and below by the envelopes $\pi(x)$ and $\pi(2x)$.

The overall saturation toward the upper envelope as $n$ grows follows directly from the Prime Number Theorem — that part isn't surprising. What caught my attention is the striated internal structure, which looks a bit like an interference pattern between the shifted discontinuities of each $\Phi_p$.

Is there a standard physical framework — quantum chaos, GUE statistics, spectral analysis of superposed step functions, anything along those lines — that would predict or explain this kind of striation when you stack step functions shifted by a quasi-random sequence like the primes? Or is this just a visual artifact with no real spectral content behind it?

Happy to share more of the construction if it's useful.

Edit: Thanks to the comments below, here are precise definitions, along with a smaller-scale figure generated specifically to clarify this.

Where the points come from. Let $p_n$ denote the $n$-th prime, and let $\chi_{\mathbb{P}}(m)$ be the prime indicator ($1$ if $m$ is prime, $0$ otherwise). For each prime $p_n$ and each odd $m \le p_n$ with $m$ also prime, define

$$X(n,m) = \frac{p_n + m}{2}, \qquad Y(n,m) = \frac{p_n - m}{2}.$$

Since $p_n$ and $m$ are both prime, each point $X(n,m)$ marks the midpoint of a Goldbach partition $p_n + m = 2X$. The set $S(x)$ is the union of all such points, restricted to a domain bounded by $x$.

To make this concrete, I generated the smaller-scale figure below (up to $x=90$), plotting $S(x)$ as $(X(n,m), n)$:

Bounding. All points of $S(x)$ are trapped between $\pi(x)$ (below) and $\pi(2x)$ (above), since $n = \pi(p_n)$.

Connection to $\Phi_p$. The points of $S(x)$ coincide exactly with the jumps of the step-function family $\Phi_p(x) = \pi(2x - p)$, restricted to $p \le x$ (which guarantees $\Phi_p(x) \ge \pi(x)$). Wherever $\pi(x)$ and its shifted reflection $\pi(2x-p)$ jump simultaneously, a point of $S(x)$ appears.

The large-scale figure (original post). There, individual points aren't plotted; instead, for each $x$, the values of $\Phi_p(x)$ are aggregated over all primes $p \le x$, and color intensity reflects how many primes produce a value near that height. $n$ there is simply the variable $x$.

On "interference pattern." I mean the density inside $[\pi(x), \pi(2x)]$ is not uniform — there are streaks rather than a smooth gradient. These likely reflect the irregular spacing of primes (gaps between consecutive primes aren't constant), which could make several $\Phi_p$ cluster in value at some $x$ and spread out at others. This is a heuristic reading, not something I've proven — hence the question below.

Question. Is there a known framework (spectral statistics, GUE, classical results on the distribution of $\pi(2x-p)$ as $p$ varies) explaining this non-uniformity, beyond the PNT's asymptotic trend?