High-Rate Generation and Structural Properties of Primes via Aggressive Tree Pruning in Primorial-Like Sequences

Abstract: This paper presents an empirical and statistical analysis of a specialized prime generation algorithm designed to explore heavily structured multiplicative integer subsets. Utilizing an optimized multi-radix odometer mechanism coupled with aggressive tree pruning and O(1) power caching, we evaluate candidate integers derived from primorial bounded roots.

By pushing the computational threshold up to n = 8000, we mapped the state-space traversal into an Information Retrieval framework, empirically proving a strict zero-duplication rate and highly enriched precision. The generated sequence demonstrates a persistent asymptotic enrichment in prime density, converging to a limit of approximately 2.5175 times the Prime Number Theorem expectation.

Furthermore, topological analysis of the prime gaps reveals an exceptionally high Hurst exponent (H ≈ 0.73), strongly rejecting the null hypothesis of a memoryless random walk and indicating significant fractal-like long-range dependence. Finally, we expose an extreme, deterministic Chebyshev bias within the sequence, demonstrating that the primorial bounds fundamentally break probabilistic residue symmetry and induce a strict algebraic polarization.

Code and Data Availability: To ensure full reproducibility, this research is distributed as a complete open-science package:

• Source Code: The "Probabilistic-Prime-Explorer" analytical pipeline and generator are archived at https://doi.org/10.5281/zenodo.19383835 (v1.0.0).

• Dataset: The 4.3 GB raw computational logs and statistical summaries are available at https://doi.org/10.5281/zenodo.19382555.

Keywords: Number Theory, Prime Numbers, Probabilistic Algorithms, Chebyshev Bias, Fractal Analysis, Computational Mathematics.