Glushkov's Spectral Gap Analysis Method and the Discovery of Scale Transition in Prime Number Distribution: From Local Fibonacci-like Coherence to Asymptotic GUE Chaos

This preprint fixes priority for two novel interconnected formulations introduced for the first time on March 22, 2026: 1. Glushkov's Spectral Gap Analysis Method — a spectroscopic approach to prime gaps that introduces the concept of "quality factor of numerical resonance" Q (добротность числового резонанса). This quantifies local coherence in small prime gaps (dominant gap ≈6 for small values, Fibonacci-like correlation patterns for effective height u < 20), explaining why resonant structures persist strongly at low scales. 2. Discovery of the Scale Transition in Arithmetic — a phase transition in the distribution of prime numbers from a "cold" locally ordered regime (coherent Fibonacci-like structures and high resonance quality) to an asymptotic regime of statistical chaos consistent with Gaussian Unitary Ensemble (GUE) statistics at large scales. The transition features a computable crossover point ("melting point of order"), analogous to the Curie temperature in condensed matter physics. These concepts provide a physical interpretation of the persistent local numerical support for the Riemann Hypothesis: verification on the first millions/trillions of zeros occurs in the coherent low-temperature phase, while potential deviations may appear only at ultra-large scales where asymptotic randomness dominates. No prior literature (Riemann, Odlyzko, Clay Mathematics Institute works, or existing publications) has formulated this exact linkage between spectral resonance quality, Fibonacci coherence at small scales, and a scale-dependent phase transition to GUE chaos. Applications are outlined in lattice stability (superconductivity without noise) and cryptography (encryption schemes exploiting the chaos zone at large numbers). Priority fixation: The named method and discovery are original contributions of Oleg Yurievich Glushkov, dated March 22, 2026.