Prime Information Geometry: Scale-Invariant Multi-Fractal Structures and Shape-Conserving Properties in Logarithmically Normalized Prime Gap

Abstract This study posits, through large-scale numerical experiments, that scale-invariant self-similarity and a weak multi-fractal structure exist in the sequence of log-normalized prime gaps gn = (pn+1−pn)/log⁡pn g_n = (p_{n+1} - p_n)/\log p_n gn=(pn+1−pn)/logpn. Analysis of the generalized Hurst exponent h(q) and generalized fractal dimension Dq across a wide five-order-of-magnitude scale from $10^6$ to $10^{11}$ revealed that the h(q) curve forms a "weakly twisted sloping sheet" that translates almost entirely while preserving its shape in the (q, log⁡10N, h) (q, \log⁻¹⁰ N, h) (q, log⁻¹⁰N, h) space. This structure persists robustly even under logarithmic scale transformations. These results suggest that, beyond the known fact that the global distribution of prime gaps collapses into an exponential distribution, the hierarchical structure of local complexity is preserved scale-invariantly. This provides a new perspective suggesting a deep connection between the statistical properties of prime sequence and random matrix theory, multi-fractal analysis, and information geometry. This paper presents a preliminary study offering strong numerical evidence for the existence of a scale-invariant statistical geometry inherent in the prime gap. Acknowledgements Regarding the Use of ChatGPT-4 Acknowledgements: Large-scale numerical calculations, statistical analysis, multi-fractal analysis (calculation of h(q) h(q) h(q) and Dq D_q Dq), and the creation of all figures and tables in this study were performed using OpenAI's ChatGPT-4 (and its successor models) as an auxiliary tool. AI assistance enabled efficient processing of vast amounts of prime number data and systematic analysis across multiple scales. We express our gratitude here.