Mapping prime gap chaos into quantized, bounded "energy tracks" (0 < K ≤ 0.125) using a non-linear feature transform. [OC] Visualization

What This Visualization Shows

This data visualization displays "Mapping prime gap chaos into quantized, bounded "energy tracks" (0 < K ≤ 0.125) using a non-linear feature transform. [OC]" and provides a clear visual representation of the underlying data patterns and trends. The visualization focuses on To analyze prime numbers using standard statistical tools, we have to deal with severe non-stationarity—the values grow to infinity and the gaps stretch logarithmically.

To explore this, I’ve been looking at a custom change of basis. For any consecutive prime pair a < b, let the local gap be g = b - a, and the macroscopic midpoint scale be m = (a + b) / 2. We can map the sequence into a normalized metric, K:

**K = 1 / (2 \* g²) - 1 / (8 \* m²)**

**The Bounding Mechanism:** As m → ∞, the finite-size correction term 1 / (8 \* m²) approaches zero, meaning K ≈ 1 / (2 \* g²) at large scales. Because the minimum possible gap for all primes past 3 is g ≥ 2, this transformation is designed to constrain the infinite sequence of prime gaps into a strictly bounded, global ceiling:

**0 < K ≤ 0.125**

By bounding the space, we can treat the prime sequence as a discrete dynamic system jumping between quantized "tracks" (Twin Primes sit near K ≈ 0.125, Cousin Primes near K ≈ 0.03125, Sexy Primes near K ≈ 0.0138).

*(Note: The dense blue region at the bottom represents the asymptotic floor of the transformation. Because K scales inversely with the square of the gap (1 / g²), the infinite long-tail of larger prime gaps (g ≥ 8) is compressed into a tight, bounded zone approaching 0, allowing the low-order gaps to float clearly at the top).*, which allows us to understand complex relationships and insights within the data through visual storytelling.

Deep Dive into the Topic

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